DP Maths AA HL - Functions - Trigonometric Functions

DP Maths AA HL - Functions - Trigonometric Functions - Full notes

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DP Maths AA HL | Trigonometric Functions

Trigonometric Functions

This topic contains 2 notes:

3.4 Radians and Unit Circle
3.7 Trigonometric Functions

DP MATHEMATICS AA · TOPIC 3: GEOMETRY AND TRIGONOMETRY

3.4

Radians, Arcs and the Unit Circle

Full notes for HL · Syllabus SL 3.4, 3.5

Contents

1	Radian measure
2	Arc length
3	Area of a sector and a segment
4	The unit circle
5	Exact values
6	Signs of trigonometric ratios in the four quadrants
7	Related angles and symmetry
8	Extending to all angles
9	Arc length and perimeter problems
10	Areas of compound regions
11	Angular speed and rotating objects
12	Exact values on the unit circle: a complete table
13	Solving equations using the unit circle
14	Ratios from a given ratio and quadrant
15	Radians in calculus and modelling
16	From the unit circle to the graphs
17	Checking answers with technology
18	Mixed review: worked problems
19	Additional exam-style practice
20	Exam-style questions with markscheme
21	Common misconceptions and links
22	Check your understanding
23	Summary and topic map
24	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
convert between degrees and radians
calculate arc length and sector area using radians
find the area of a segment
define sine and cosine using the unit circle, for any angle
recall exact values for 0, π/6, π/4, π/3, π/2 and their multiples
determine the signs of sin, cos and tan in each quadrant
use symmetry to find related angles
define tan θ as sin θ / cos θ

This booklet covers SL 3.4, 3.5, which is common to SL and HL. HL students extend this topic in booklet 3.9.

Guiding questions

Why do mathematicians measure angles in radians? How does the unit circle extend trigonometry beyond right-angled triangles?

How this topic is examined

Radians and the unit circle are foundations for the rest of the trigonometry and calculus in the course. Paper 1 tests exact values, arc and sector problems with π, and finding angles in a given interval; Paper 2 uses them in context.

Unless degrees are stated (with the ° symbol), angles are in radians. Set your calculator accordingly.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 Radian measure

Measuring angles by arc length.

Think about it

Why is a full turn 360°? Is there a way of measuring angles that comes from the circle itself rather than from a historical choice?

One radian is the angle at the centre of a circle subtended by an arc whose length equals the radius. Since the whole circumference is 2πr, a full turn is 2π radians. So:

To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. Common angles are usually written as fractions of π.

Radians are the natural unit in calculus: the result d/dx(sin x) = cos x is only true when x is in radians (5.6).

Degrees	0	30	45	60	90	120	135	150	180	270	360
Radians	0	π/6	π/4	π/3	π/2	2π/3	3π/4	5π/6	π	3π/2	2π

WORKED EXAMPLE 1 Conversions (basic)

Convert (a) 225° to radians; (b) 7π/12 to degrees; (c) 2.5 radians to degrees.

Solution

(a) 225 × π/180 = 5π/4A1

(b) 7 × 180/12 = 105°A1

Practice exercise 1

1. Convert 40° and 300° to radians.

2. Convert 3π/10 and 1.2 rad to degrees.

3. Which is larger: 2 rad or 120°?

2 Arc length

A fraction of the circumference.

A sector with angle θ (in radians) is the fraction θ/(2π) of the whole circle. So the arc length is (θ/2π) × 2πr:

Figure 1. A sector of radius r and angle θ.

WORKED EXAMPLE 2 Arc and perimeter (standard)

A sector has radius 8 cm and angle 1.2 rad. Find the arc length and the perimeter.

Solution

l = 8 × 1.2 = 9.6 cmA1

Perimeter = 8 + 8 + 9.6 = 25.6 cmA1

Common mistake

Using degrees in l = rθ. With degrees the formula is l = (θ/360) × 2πr.

Practice exercise 2

1. Find the arc length of a sector of radius 5 with angle 3π/5.

2. An arc of length 12 cm subtends 0.8 rad. Find the radius.

3. A sector has perimeter 30 cm and radius 9 cm. Find its angle.

3 Area of a sector and a segment

Sectors and the regions cut by chords.

A segment is the region between a chord and its arc: the sector minus the triangle formed by the two radii, whose area is ½r2 sin θ.

WORKED EXAMPLE 3 Sector area (basic)

Find the exact area of a sector of radius 6 cm and angle π/3.

Solution

½ × 36 × π/3M1

= 6π cm2A1

WORKED EXAMPLE 4 Segment area (exam level)

A chord AB subtends an angle of 2 rad at the centre of a circle of radius 10 cm. Find the area of the minor segment and the length of the chord.

Solution

Segment = ½(100)(2 − sin 2)M1

= 54.5 cm2A1

Chord AB2 = 100 + 100 − 200 cos 2, AB = 16.8 cmM1 A1

WORKED EXAMPLE 5 Finding the angle (exam level)

A sector has area 24 cm2 and arc length 8 cm. Find r and θ.

Solution

½r2θ = 24 and rθ = 8M1

Divide: ½r = 3, r = 6A1

θ = 8/6 = 4/3 radA1

Practice exercise 3

1. Find the area of a sector of radius 4 with angle 2.5 rad.

2. Find the exact area of the segment cut off by a chord subtending π/2 in a circle of radius 6.

3. A sector of angle 0.6 rad has area 30 cm2. Find its radius.

4 The unit circle

Trigonometry for any angle.

The unit circle has centre (0, 0) and radius 1. Measure the angle θ anticlockwise from the positive x-axis (clockwise for negative angles). The point where the terminal arm meets the circle has coordinates (cos θ, sin θ). This defines cosine and sine for every angle, not just acute ones.

For an acute angle, this agrees with SOH CAH TOA: the triangle has hypotenuse 1, so the adjacent side is cos θ and the opposite side is sin θ. tan θ = sin θ/cos θ, the gradient of the terminal arm, undefined when cos θ = 0.

Since the point lies on the circle x2 + y2 = 1, we get the Pythagorean identity cos2θ + sin2θ = 1 (3.6).

Figure 2. The point on the unit circle at angle θ has coordinates (cos θ, sin θ).

5 Exact values

Learn the key angles.

θ	0	π/6	π/4	π/3	π/2
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	√3/3	1	√3	undefined

A memory aid: sin of 0, π/6, π/4, π/3, π/2 is √0/2, √1/2, √2/2, √3/2, √4/2; cosine runs the same values in reverse.

Practice exercise 4

1. Write down the exact value of sin(π/3) cos(π/6).

2. Evaluate tan2(π/6) + 1.

3. Evaluate 2 sin(π/4) cos(π/4).

6 Signs of trigonometric ratios in the four quadrants

All Students Take Calculus.

Since cos θ is the x-coordinate and sin θ the y-coordinate, their signs follow the quadrants. In the first quadrant all are positive; in the second only sine; in the third only tangent; in the fourth only cosine. A useful mnemonic, starting in the first quadrant and going anticlockwise, is All Students Take Calculus (A, S, T, C).

Quadrant	Angle range	Positive ratios
1st	0 < θ < π/2	all
2nd	π/2 < θ < π	sin
3rd	π < θ < 3π/2	tan
4th	3π/2 < θ < 2π	cos

7 Related angles and symmetry

Use the reference angle.

For any angle, find the reference angle α (the acute angle the terminal arm makes with the x-axis), take the value of the ratio for α, then attach the sign for the quadrant. Key symmetries:

WORKED EXAMPLE 6 Exact values beyond the first quadrant (standard)

Find the exact values of (a) sin(5π/6); (b) cos(4π/3); (c) tan(7π/4).

Solution

(a) 2nd quadrant, reference π/6: 1/2A1

(b) 3rd quadrant, reference π/3: −1/2A1

WORKED EXAMPLE 7 Using one ratio to find another (exam level)

Given sin θ = 3/5 and π/2 < θ < π, find cos θ and tan θ.

Solution

cos2θ = 1 − 9/25 = 16/25M1

2nd quadrant: cos θ = −4/5A1

tan θ = −3/4A1

Practice exercise 5

1. Find the exact value of cos(5π/4).

2. Find the exact value of sin(−π/3).

3. Given cos θ = −5/13 with π < θ < 3π/2, find sin θ and tan θ.

4. Find all θ in [0, 2π] with sin θ = √3/2.

8 Extending to all angles

Angles beyond 2π and negative angles.

Adding 2π brings the terminal arm back to the same position, so sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ: sine and cosine are periodic with period 2π. Tangent repeats every π. Negative angles are measured clockwise. This periodicity is the basis of the trigonometric graphs in 3.7.

WORKED EXAMPLE 8 Large angle (standard)

Find the exact value of cos(17π/6).

Solution

17π/6 − 2π = 5π/6M1

cos(5π/6) = −√3/2A1

Practice exercise 6

1. Find the exact value of sin(13π/4).

2. Find tan(−5π/3).

9 Arc length and perimeter problems

Combining arcs with straight edges.

Many problems combine arcs with straight lines: the perimeter of a sector, a segment, a running track, or a shape made from several sectors. Always identify which edges are arcs (use l = rθ with θ in radians) and which are straight (radii, chords, or sides of polygons). A chord can be found with the cosine rule or with 2r sin(θ/2).

WORKED EXAMPLE 9 Perimeter of a segment (standard)

A chord subtends 1.8 rad at the centre of a circle of radius 7 cm. Find the perimeter of the minor segment.

Solution

Arc = 7 × 1.8 = 12.6A1

Chord = 2 × 7 sin 0.9 = 10.97M1 A1

Perimeter = 23.6 cmA1

WORKED EXAMPLE 10 Running track (exam level)

A track has two straight sections of 85 m joined by semicircles of inner radius 36.5 m. Find the inner length of the track.

Solution

Two semicircles make a circle: 2π(36.5) = 229.3M1 A1

Total = 170 + 229.3 = 399 mA1

Practice exercise 7

1. Find the perimeter of a sector of radius 9 cm and angle 2π/3.

2. A pendulum of length 80 cm swings through 0.5 rad. Find the length of the arc traced by its tip.

3. Find the perimeter of a segment cut off by a chord subtending π/3 in a circle of radius 12.

10 Areas of compound regions

Sectors, triangles and segments together.

Compound regions are built by adding or subtracting sectors, triangles and segments. A useful strategy is to draw the region, label the radii and angles, and write the area as a sum before substituting numbers. Keep answers exact (in terms of π and surds) on Paper 1.

WORKED EXAMPLE 11 Overlapping circles (exam level)

Two circles of radius 6 cm have centres 6 cm apart. Find the area of their overlap.

Solution

Each circle's arc subtends 2π/3 at its centreM1

Each segment = ½(36)(2π/3 − sin(2π/3)) = 12π − 9√3A1

Overlap = 2(12π − 9√3) = 24π − 18√3 ≈ 44.2 cm2A1

WORKED EXAMPLE 12 Annular sector (standard)

Find the area between two concentric circles of radii 5 and 8 inside a sector of angle 1.2 rad.

Solution

½(64)(1.2) − ½(25)(1.2)M1

= 23.4A1

Practice exercise 8

1. A semicircle of radius 10 has a square inscribed with one side on the diameter. Find the area of the semicircle outside the square (side 4√5).

2. Find the area of a sector of radius 4 and angle 3π/4 minus the triangle with the same two radii.

3. A 'Reuleaux triangle' is formed from an equilateral triangle of side 2 by adding three segments. Find its area.

11 Angular speed and rotating objects

Radians per second.

An object rotating through θ radians in time t has angular speed ω = θ/t (rad/s). A point at distance r from the centre moves along an arc with linear speed v = rω, which follows directly from l = rθ. One revolution per second is 2π rad/s.

WORKED EXAMPLE 13 Wheel (standard)

A bicycle wheel of radius 0.35 m turns at 4 revolutions per second. Find its angular speed and the bicycle's speed.

Solution

ω = 8π rad/sA1

v = 0.35 × 8π = 8.80 m/sA1

WORKED EXAMPLE 14 Clock hand (standard)

A minute hand is 12 cm long. How far does its tip travel in 25 minutes?

Solution

θ = (25/60) × 2π = 5π/6M1

Arc = 12 × 5π/6 = 10π ≈ 31.4 cmA1

Practice exercise 9

1. The Earth rotates once in 24 hours. Find its angular speed in rad/s.

2. A point on the equator is 6370 km from the axis. Find its speed in km/h.

3. A fan blade of length 0.4 m turns at 1200 rpm. Find the speed of the tip.

12 Exact values on the unit circle: a complete table

All the special angles from 0 to 2π.

Combining the exact values for π/6, π/4 and π/3 with the quadrant signs gives exact values for sixteen special angles. Learning to generate them quickly (reference angle plus sign) is more reliable than memorizing a long table.

θ	2π/3	3π/4	5π/6	7π/6	5π/4	4π/3	5π/3	7π/4	11π/6
sin	√3/2	√2/2	1/2	−1/2	−√2/2	−√3/2	−√3/2	−√2/2	−1/2
cos	−1/2	−√2/2	−√3/2	−√3/2	−√2/2	−1/2	1/2	√2/2	√3/2

WORKED EXAMPLE 15 Evaluating an expression (standard)

Find the exact value of sin(4π/3)cos(5π/6) + tan(7π/4).

Solution

(−√3/2)(−√3/2) + (−1)M1

= 3/4 − 1 = −1/4A1

Practice exercise 10

1. Find the exact value of cos(2π/3) + sin(11π/6).

2. Find the exact value of tan(5π/6) × tan(π/3).

3. Evaluate sin2(3π/4) + cos2(5π/3).

13 Solving equations using the unit circle

All solutions in an interval.

The unit circle shows at once how many solutions an equation such as sin θ = k has in [0, 2π]: draw the horizontal line y = k and see where it meets the circle. For cos θ = k use the vertical line x = k; for tan θ = k use the line through the origin with gradient k.

WORKED EXAMPLE 16 Two solutions (standard)

Solve cos θ = −√3/2 for 0 ≤ θ ≤ 2π.

Solution

Reference angle π/6; cos negative in quadrants 2 and 3M1

θ = 5π/6, 7π/6A1

WORKED EXAMPLE 17 Extended interval (exam level)

Solve sin θ = ½ for −2π ≤ θ ≤ 2π.

Solution

Principal solutions π/6, 5π/6A1

Subtract 2π: −11π/6, −7π/6M1

−11π/6, −7π/6, π/6, 5π/6A1

Practice exercise 11

1. Solve tan θ = √3 for 0 ≤ θ ≤ 2π.

2. Solve sin θ = −1 for 0 ≤ θ ≤ 4π.

3. Solve 2cos θ + 1 = 0 for −π ≤ θ ≤ π.

14 Ratios from a given ratio and quadrant

Using a right-angled triangle.

Given one ratio and the quadrant, draw a right-angled triangle for the reference angle (using the ratio as side lengths), find the third side by Pythagoras, then attach signs from the quadrant. This is equivalent to using the identity cos2θ + sin2θ = 1.

WORKED EXAMPLE 18 Triangle method (standard)

tan θ = −8/15 and π/2 < θ < π. Find sin θ and cos θ.

Solution

Triangle 8, 15, 17M1

Quadrant 2: sin θ = 8/17, cos θ = −15/17A1 A1

Practice exercise 12

1. cos θ = 7/25 and 3π/2 < θ < 2π. Find sin θ and tan θ.

2. sin θ = −1/3 and π < θ < 3π/2. Find tan θ exactly.

15 Radians in calculus and modelling

Why radians are essential.

In calculus, the results d/dx(sin x) = cos x and lim sin x/x = 1 (as x → 0) only hold when x is in radians. For small angles in radians, sin x ≈ x and cos x ≈ 1 − x2/2. These approximations are widely used in physics (for example, the simple pendulum).

In models of periodic motion with time t, an expression such as sin(ωt) uses radians, so ω = 2π/period.

WORKED EXAMPLE 19 Small-angle approximation (standard)

Use sin x ≈ x to estimate sin(0.1) and find the percentage error.

Solution

Estimate 0.1; actual 0.099833A1

Error = 0.17%A1

Practice exercise 13

1. Estimate cos(0.2) using cos x ≈ 1 − x2/2.

2. Find ω for a wave with period 0.25 s.

16 From the unit circle to the graphs

Unwrapping the circle.

As θ increases from 0 to 2π, the height of the point (cos θ, sin θ) rises from 0 to 1, falls to −1, and returns to 0: plotting this height against θ gives the sine curve. The horizontal coordinate gives the cosine curve. Symmetries of the circle become symmetries of the graphs: sin(π − θ) = sin θ is a reflection of the sine curve in the line θ = π/2.

Reading values from the graphs is a useful check on exact-value calculations: for example, the graph of cosine is below the axis between π/2 and 3π/2, confirming that cos θ < 0 in quadrants 2 and 3.

Figure 3. Sine and cosine over one full turn.

WORKED EXAMPLE 20 Reading the graph (standard)

Use the graphs to state the values of θ in [0, 2π] for which sin θ > cos θ.

Solution

The curves cross at π/4 and 5π/4M1

π/4 < θ < 5π/4A1

Practice exercise 14

1. For which θ in [0, 2π] are sin θ and cos θ both negative?

2. How many solutions does sin θ = 0.3 have in [0, 4π]?

17 Checking answers with technology

Radian mode, tables and graphs.

Most errors in this topic come from the calculator being in the wrong mode, or from finding only one solution. On Paper 2, a graph of y = sin x with the line y = k shows every solution in an interval; on Paper 1, a quick sketch of the unit circle does the same job.

WORKED EXAMPLE 21 Graphical check (standard)

Solve 3 sin x = 1 for 0 ≤ x ≤ 2π and check graphically.

Solution

sin x = 1/3: x = 0.340 and π − 0.340 = 2.80M1 A1

The line y = 1/3 meets the sine curve twice in [0, 2π]R1

GDC tip

Set radian mode before any arc, sector or trigonometric-equation question. Check an exact answer by evaluating, for example, sin(5π/6) on the GDC and comparing with 0.5.

Practice exercise 15

1. Solve 4 cos x = −1 for 0 ≤ x ≤ 2π.

2. Solve tan x = 2.5 for 0 ≤ x ≤ 2π.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 22 Sector with maximum area (exam level)

A sector has perimeter 20 cm. Find the radius that maximizes its area and the maximum area.

Solution

2r + rθ = 20, so θ = (20 − 2r)/rM1

A = ½r2θ = r(10 − r) = 10r − r2A1

Maximum at r = 5: A = 25 cm2 (with θ = 2 rad)A1

WORKED EXAMPLE 23 Unit circle and identity (exam level)

Given sin θ = 0.6 with θ obtuse, find the exact value of sin(π − θ) + cos(π + θ).

Solution

cos θ = −0.8A1

sin(π − θ) = 0.6; cos(π + θ) = −cos θ = 0.8M1

Sum = 1.4A1

Practice exercise 16

1. A sector of area 12 cm2 has an arc length of 6 cm. Find its radius and angle.

2. Show that the area of a segment with angle θ is never greater than the area of the sector.

3. Find all θ in [0, 2π] with sin θ = cos θ.

19 Additional exam-style practice

Short questions in the style of Paper 1.

WORKED EXAMPLE 24 Paper 1 question (exam level)

The diagram shows a sector OAB of radius r and angle 0.8 rad. The area of the sector is 40 cm2. (a) Find r. (b) Find the perimeter of the sector.

Solution

(a) ½r2(0.8) = 40, r2 = 100: r = 10 cmM1 A1

(b) 20 + 8 = 28 cmA1

WORKED EXAMPLE 25 Paper 2 question (exam level)

A water wheel of radius 2.5 m turns at 3 revolutions per minute. (a) Find the angular speed in rad/s. (b) Find the speed of a point on the rim. (c) Find the area swept by one spoke in 4 seconds.

Solution

(a) 3 × 2π/60 = 0.314 rad/sA1

(b) 2.5 × 0.314 = 0.785 m/sA1

Practice exercise 17

1. Find the exact value of cos(7π/6) − sin(−π/3).

2. A sector of radius 8 has area 24. Find its perimeter.

20 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 12 minutes)

A sector OAB of a circle with centre O has radius 6 cm and angle AOB = θ radians. The perimeter of the sector is 20 cm.

(a) Find θ. [2]

(b) Find the area of the sector. [2]

Markscheme

Part	Answer	Marks
(a)	12 + 6θ = 20	M1
	θ = 4/3	A1
(b)	½(36)(4/3)	M1
	24 cm2	A1
(c)	24 − ½(36) sin(4/3)	M1
	24 − 17.49	A1
	6.51 cm2	A1

Examiner's comments. A calculator in degree mode gave wrong answers in (c).

Question 2 (Paper 1 style) (about 10 minutes)

Let cos θ = −2/3, where π < θ < 3π/2.

(a) Find the exact value of sin θ. [3]

(b) Find the exact value of tan θ. [2]

Markscheme

Part	Answer	Marks
(a)	sin2θ = 1 − 4/9	M1
	= 5/9	A1
	sin θ = −√5/3 (third quadrant)	A1
(b)	(−√5/3) ÷ (−2/3)	M1
	√5/2	A1
(c)	2/3	A1

Examiner's comments. The sign of sin θ must follow from the quadrant.

21 Common misconceptions and links

Students often think…	Actually…
l = rθ works in degrees	only in radians
sin is negative in the 2nd quadrant	only cos and tan are
tan(π/2) = 1	it is undefined
sin θ > 0 means θ is acute	it can be in the 2nd quadrant

How 3.4 connects to the rest of the course

Topic	Connection
3.1 Triangles	right-angle trigonometry
3.6 Identities	Pythagorean identity
3.7 Trig functions	periodic graphs
5.6 Calculus	derivatives of sin and cos in radians

22 Check your understanding

Section A: multiple choice

1. π/4 radians in degrees is

A. 45° B. 90° C. 30° D. 60°

2. The arc length with r = 5 and θ = 2 is

A. 5 B. 25 C. 10 D. 2.5

3. cos(2π/3) =

A. 1/2 B. −1/2 C. √3/2 D. −√3/2

4. In the 3rd quadrant, the positive ratio is

A. sin B. cos C. all D. tan

5. The sector area with r = 4 and θ = 0.5 is

A. 2 B. 4 C. 8 D. 16

Section B: structured

6. Find all θ in [0, 2π] with cos θ = −√2/2. [2]

7. A sector of radius r has angle 1.5 rad and area 27 cm2. Find r and the arc length. [3]

Answers

1. A 2. C 3. B 4. D 5. B

6. 3π/4, 5π/4.

7. ½r2(1.5) = 27: r = 6 cm; arc 9 cm.

23 Summary and topic map

Idea	Remember
Radians	π rad = 180°
Arc length	l = rθ
Sector	½r2θ; segment ½r2(θ − sin θ)
Unit circle	point (cos θ, sin θ)
Exact values	0, π/6, π/4, π/3, π/2
Quadrants	All Students Take Calculus
Periodicity	sin, cos period 2π; tan period π

Before the exam, make sure you…

•use radians in arc and sector formulas
•find the quadrant before the sign
•use reference angles
•set the calculator mode

Radians and the Unit Circle on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 4. Topic map for Radians and the Unit Circle.

24 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 2π/9; 5π/3

2. 54°; 68.8°

3. 2 rad = 114.6°, so 120° is larger

Practice exercise 2

1. 3π

2. 15 cm

3. 12/9 = 4/3 rad

Practice exercise 3

1. 20

2. 9π − 18

3. 10 cm

Practice exercise 4

1. 3/4

2. 4/3

3. 1

Practice exercise 5

1. −√2/2

2. −√3/2

3. sin θ = −12/13, tan θ = 12/5

4. π/3, 2π/3

Practice exercise 6

1. −√2/2

2. √3

Practice exercise 7

1. 18 + 6π = 36.8 cm

2. 40 cm

3. 4π + 12 = 24.6

Practice exercise 8

1. 50π − 80 = 77.1

2. 6π − 4√2 = 13.2

3. 2π − 2√3 ≈ 2.82

Practice exercise 9

1. 7.27 × 10−5 rad/s

2. 1668 km/h

3. 50.3 m/s

Practice exercise 10

1. −1

2. −1

3. 3/4

Practice exercise 11

1. π/3, 4π/3

2. 3π/2, 7π/2

3. ±2π/3

Practice exercise 12

1. −24/25; −24/7

2. 1/(2√2) = √2/4

Practice exercise 13

1. 0.98 (actual 0.98007)

2. 8π rad/s

Practice exercise 14

1. π < θ < 3π/2

2. 4

Practice exercise 15

1. 1.82, 4.46

2. 1.19, 4.33

Practice exercise 16

1. r = 4, θ = 1.5

2. segment = ½r2(θ − sin θ) ≤ ½r2θ because sin θ ≥ 0 for 0 ≤ θ ≤ π

3. π/4, 5π/4

Practice exercise 17

1. 0

2. θ = 0.75: 16 + 6 = 22

DP MATHEMATICS AA · TOPIC 3: GEOMETRY AND TRIGONOMETRY

3.7

Trigonometric Functions

Full notes for HL · Syllabus SL 3.7

Contents

1	The graphs of sin x and cos x
2	The graph of tan x
3	Amplitude and principal axis
4	Period
5	Horizontal translations (phase shift)
6	The general form a sin(b(x − c)) + d
7	Finding the equation from a graph
8	Modelling periodic phenomena
9	Sketching sinusoidal graphs step by step
10	Finding equations from graphs: more practice
11	Graph of y = tan x and its transformations
12	Solving trigonometric equations graphically
13	Modelling: tides and harbours
14	Modelling: rotating wheels
15	Modelling: temperature and daylight
16	Composite trigonometric functions
17	Mixed review: worked problems
18	Additional exam-style practice
19	Inverse trigonometric values on the GDC
20	Exam-style questions with markscheme
21	Common misconceptions and links
22	Check your understanding
23	Summary and topic map
24	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the Quick check.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
sketch y = sin x, y = cos x and y = tan x, showing key features
identify amplitude, period, principal axis and phase shift
sketch transformed trigonometric graphs
find the equation of a sinusoidal graph from its features
model periodic real-life situations with sinusoidal functions
solve problems with periodic models, including with technology

This booklet covers SL 3.7, which is common to SL and HL. HL students extend this topic in booklet 3.9.

Guiding questions

How can trigonometric functions model things that repeat, such as tides, daylight and sound? How do the parameters of a sine function change its graph?

How this topic is examined

Trigonometric functions appear on both papers. Paper 1 asks for sketches and exact features; Paper 2 has modelling questions (tides, Ferris wheels, temperatures) that require finding parameters and solving equations with the GDC.

Use radians unless degrees are stated. Label maxima, minima, intercepts and the principal axis on sketches.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working
Solve	find the values of the unknowns, using algebra or technology as allowed
Show that	obtain the given result; every step must be shown and the result must not be assumed
Hence / Hence or otherwise	use the previous result (or another valid method)
Sketch	a diagram showing the general shape and key features, labelled
Prove	a formal, rigorous argument from accepted facts
Justify / Explain	give valid reasons or evidence

How to use these notes

Each section teaches one part of the syllabus. You'll see these boxes throughout; each has one job:

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or proof of a key result, so you understand where formulas come from
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently on Paper 2
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given); examples are graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet
Quick check	short questions to check the key ideas

1 The graphs of sin x and cos x

Waves from the unit circle.

Think about it

As a point moves round the unit circle at a steady speed, how does its height change with time? How does its horizontal position change?

Plotting the y-coordinate of the unit-circle point against the angle gives y = sin x; plotting the x-coordinate gives y = cos x. Both are smooth waves with:

Period 2π (the graph repeats every 2π); amplitude 1 (the maximum distance from the centre line); range −1 ≤ y ≤ 1; principal axis y = 0. sin x has zeros at 0, π, 2π, ...; cos x has zeros at π/2, 3π/2, .... The cosine graph is the sine graph translated π/2 to the left: cos x = sin(x + π/2).

Figure 1. One period of y = sin x and y = cos x.

2 The graph of tan x

Asymptotes where cos x = 0.

Since tan x = sin x/cos x, tan x is undefined where cos x = 0, giving vertical asymptotes at x = π/2 + kπ. Its period is π, it has no amplitude (range ℝ), and its zeros are at x = kπ.

Figure 2. y = tan x has period π and asymptotes at x = π/2 + kπ.

Practice exercise 1

1. State the period and range of y = tan x.

2. Write down the zeros of y = cos x for 0 ≤ x ≤ 4π.

3. Write down the maximum value of y = sin x and the values of x in [0, 4π] where it occurs.

3 Amplitude and principal axis

Vertical stretch and shift.

For y = a sin x + d: the amplitude is |a| (a vertical stretch); the principal axis is y = d (a vertical translation). The maximum is d + |a| and the minimum d − |a|. Conversely, amplitude = (max − min)/2 and d = (max + min)/2. If a < 0, the graph is reflected in the principal axis.

Figure 3. Amplitude 3 and principal axis y = 2.

Practice exercise 2

1. State the amplitude, maximum and minimum of y = 4 cos x − 1.

2. A sinusoidal graph has maximum 13 and minimum 5. Find its amplitude and principal axis.

4 Period

Horizontal stretch.

For y = sin(bx) with b > 0, the graph completes b cycles in 2π, so the period is 2π/b (a horizontal stretch with scale factor 1/b). For tan(bx) the period is π/b. In degrees, the period of sin(bx) is 360°/b.

Figure 4. y = sin 3x has period 2π/3.

Practice exercise 3

1. Find the period of y = cos(x/2).

2. Find b if y = sin(bx) has period π/2.

3. State the period of y = tan 4x.

5 Horizontal translations (phase shift)

Shift left or right.

y = sin(x − c) is a translation of y = sin x by c units to the right (2.11). When the coefficient b is present, factorize first: y = sin(2x − π/3) = sin(2(x − π/6)) is shifted π/6 to the right, not π/3.

Common mistake

Reading sin(2x − π/3) as a shift of π/3. Factor out the 2 first: the shift is π/6.

6 The general form a sin(b(x − c)) + d

Four parameters, four features.

Parameter	Effect
\|a\|	amplitude (vertical stretch)
2π/b	period (horizontal stretch with factor 1/b)
c	phase shift (horizontal translation)
d	principal axis y = d (vertical translation)

The same form with cosine is often more convenient when a graph starts at a maximum. To sketch: draw the principal axis; mark maximum and minimum levels; find one starting point from c; then mark quarter-periods, at which the graph is alternately at the axis, a maximum, the axis, and a minimum.

WORKED EXAMPLE 1 Features from the equation (standard)

For f(x) = 3 sin(2(x − π/4)) + 1, state the amplitude, period, phase shift, principal axis, maximum and minimum values.

Solution

Amplitude 3; period 2π/2 = πA1

Shift π/4 right; principal axis y = 1A1

Maximum 4, minimum −2A1

Figure 5. Principal axis y = 1 (dashed), amplitude 3, period π.

WORKED EXAMPLE 2 Finding the first maximum (exam level)

Find the coordinates of the first maximum of f(x) = 3 sin(2(x − π/4)) + 1 for x ≥ 0.

Solution

sin is maximal when 2(x − π/4) = π/2M1

x − π/4 = π/4, x = π/2A1

(π/2, 4)A1

Practice exercise 4

1. State the amplitude, period and principal axis of y = 2 cos(3x) − 4.

2. Find the phase shift of y = sin(3x + π).

3. Sketch y = −2 sin x + 1 for 0 ≤ x ≤ 2π, labelling the maximum and minimum points.

7 Finding the equation from a graph

Read the four features.

From a graph: d = (max + min)/2; |a| = (max − min)/2; b = 2π/period (measure the distance between two consecutive maxima); and c from the position of a convenient point. With cosine, c is the x-coordinate of a maximum; with sine, c is where the graph crosses the axis going up. Use a negative a if the graph starts by going down.

WORKED EXAMPLE 3 From features (exam level)

A sinusoidal graph has a maximum at (1, 8) and the next minimum at (4, 2). Write its equation in the form y = a cos(b(x − c)) + d.

Solution

d = 5, a = 3A1

Half-period = 3, so period 6 and b = 2π/6 = π/3M1 A1

Maximum at x = 1: c = 1, y = 3 cos(π(x − 1)/3) + 5A1

Practice exercise 5

1. A graph has maximum (0, 7) and minimum (π, −1). Find its equation as y = a cos bx + d.

2. A sine graph crosses its axis y = 2 going upward at x = 1, with period 8 and amplitude 5. Write its equation.

8 Modelling periodic phenomena

Tides, daylight, Ferris wheels.

Many repeating phenomena are modelled by sinusoidal functions: tide heights, hours of daylight over a year, temperature over a day, the height of a seat on a Ferris wheel. Identify the maximum and minimum values (giving a and d), the period (giving b), and the time of a maximum or crossing (giving c). Then use the model to answer questions, usually with a GDC.

WORKED EXAMPLE 4 Ferris wheel (exam level)

A Ferris wheel of radius 20 m has its centre 22 m above the ground and rotates once every 10 minutes. A seat starts at the lowest point at t = 0. (a) Write a model for the seat's height h(t). (b) Find the total time in one revolution for which the seat is above 30 m.

Solution

(a) h(t) = 22 − 20 cos(πt/5)M1 A1

(b) 22 − 20 cos(πt/5) = 30: cos(πt/5) = −0.4, t = 3.15 and 6.85M1 A1

Time above 30 m = 6.85 − 3.15 = 3.69 minutesA1

Figure 6. Height of a seat on the Ferris wheel over one revolution.

WORKED EXAMPLE 5 Tide model (standard)

The depth of water in a harbour is d(t) = 3 sin(0.5t) + 7 metres, t in hours after midnight. Find the maximum depth and the first time it occurs.

Solution

Maximum 10 mA1

0.5t = π/2, t = πM1

3.14 hours (about 3:08 a.m.)A1

Practice exercise 6

1. Daylight in a city varies from 9 h (21 Dec, day 355) to 15 h, with a period of 365 days. Write a model D(t) using cosine with t in days.

2. For d(t) = 3 sin(0.5t) + 7, find the times in [0, 12] when the depth is 8.5 m.

3. A temperature model is T = 6 sin(π(t − 9)/12) + 18. Find the maximum temperature and when it occurs (0 ≤ t < 24).

9 Sketching sinusoidal graphs step by step

Quarter-period points.

To sketch y = a sin(b(x − c)) + d: draw the principal axis y = d; mark the maximum d + |a| and minimum d − |a|; start at x = c on the axis (going up if a > 0); then every quarter-period the graph is at max, axis, min, axis. For cosine, start at a maximum.

Figure 7. Period 4, amplitude 2, shift 1, axis y = 3.

WORKED EXAMPLE 6 Key points (standard)

List the key points of y = 2 sin(π(x − 1)/2) + 3 for 1 ≤ x ≤ 5.

Solution

Period 4, quarter 1A1

(1, 3), (2, 5), (3, 3), (4, 1), (5, 3)A1

Practice exercise 7

1. Sketch y = −3 cos(2x) + 1 for 0 ≤ x ≤ π, listing key points.

2. Sketch y = sin(x + π/4) for 0 ≤ x ≤ 2π.

10 Finding equations from graphs: more practice

Read four features.

From a graph: d = (max + min)/2; |a| = (max − min)/2; period from two consecutive maxima (b = 2π/period); c from a maximum (cosine) or an upward axis crossing (sine). Check with one more point.

WORKED EXAMPLE 7 From two points (exam level)

A sinusoid has a minimum (2, −1) and the next maximum (8, 7). Write it as y = a cos(b(x − c)) + d.

Solution

d = 3, a = 4A1

Half-period 6: period 12, b = π/6A1

Maximum at x = 8: y = 4 cos(π(x − 8)/6) + 3A1

Practice exercise 8

1. A sine curve crosses its axis y = 0 going up at x = π/3 with period π and amplitude 2. Write its equation.

2. Write y = 4 cos(π(x − 8)/6) + 3 as a sine function.

11 Graph of y = tan x and its transformations

Asymptotes move with the graph.

y = tan(b(x − c)) + d has period π/b, vertical asymptotes where b(x − c) = π/2 + kπ, and passes through (c, d).

Figure 8. y = tan 2x has period π/2.

WORKED EXAMPLE 8 Asymptotes (standard)

Find the asymptotes of y = tan(x − π/4) for 0 ≤ x ≤ 2π.

Solution

x − π/4 = π/2 + kπM1

x = 3π/4, 7π/4A1

Practice exercise 9

1. State the period of y = 3 tan(x/2).

2. Solve tan 2x = 1 for 0 ≤ x ≤ π.

12 Solving trigonometric equations graphically

Intersections in an interval.

For equations like 3 sin(2x) + 1 = 2.5 or 2 cos x = x − 1, graph both sides over the required interval and read the intersections. Count solutions first using the period.

WORKED EXAMPLE 9 GDC (standard)

Solve 3 sin(2x) + 1 = 2.5 for 0 ≤ x ≤ 2π.

Solution

sin 2x = 0.5: four solutionsR1

x = π/12, 5π/12, 13π/12, 17π/12A2

Practice exercise 10

1. Solve 2 cos x = x − 1.

2. How many solutions has sin 4x = 0.3 in [0, 2π]?

13 Modelling: tides and harbours

Depth over time.

Tidal depth is modelled by d(t) = a cos(b(t − c)) + d, with period about 12.4 hours. High and low tide give a and d; the time of high tide gives c.

WORKED EXAMPLE 10 Harbour (exam level)

High tide 9.6 m at 04:00, next low tide 2.4 m at 10:12. A boat needs 5 m. Find when it can enter after 04:00.

Solution

a = 3.6, d = 6, period 12.4A1

d(t) = 3.6 cos(2π(t − 4)/12.4) + 6A1

d = 5 at t = 7.65 and 12.75M1 A1

Can enter until 07:39 and again from 12:45A1

Practice exercise 11

1. Using the model, find the depth at 08:00.

14 Modelling: rotating wheels

Height of a point.

A point on a wheel of radius r with centre at height H, turning once every T seconds, has height h(t) = H − r cos(2πt/T) if it starts at the bottom, or H + r sin(2πt/T) if it starts level with the centre and rising.

WORKED EXAMPLE 11 Ferris wheel (exam level)

A wheel of radius 25 m, centre 30 m high, turns every 8 minutes, starting at the bottom. Find the time per revolution above 45 m.

Solution

h = 30 − 25 cos(πt/4)A1

h = 45 when cos(πt/4) = −0.6: t = 2.82, 5.18M1 A1

2.36 minutesA1

Practice exercise 12

1. Find the height of the seat after 1 minute.

2. Find the speed of the seat (m/min).

15 Modelling: temperature and daylight

Annual cycles.

Average monthly temperature and daylight hours repeat every 12 months (365 days). Use the maximum and minimum to find a and d, and the timing of the maximum for c.

WORKED EXAMPLE 12 Daylight (exam level)

Daylight varies from 7.8 h (21 Dec, day 355) to 16.6 h (21 June, day 172). Find the number of days with more than 14 h.

Solution

D = 4.4 cos(2π(t − 172)/365) + 12.2A1 A1

D > 14: cos(...) > 0.409: |t − 172| < 66.9M1

about 134 daysA1

Practice exercise 13

1. Write a model for a temperature with maximum 32 °C in month 7 and minimum 12 °C in month 1.

16 Composite trigonometric functions

Sums and products.

Sums such as sin x + cos x and products such as x sin x appear in physics and engineering. Use technology to find their maxima, minima and zeros. sin x + cos x = √2 sin(x + π/4), a sinusoid with amplitude √2.

Figure 9. Sum and product of trigonometric functions.

WORKED EXAMPLE 13 Maximum (standard)

Find the maximum of y = sin x + cos x.

Solution

√2 sin(x + π/4)M1

Maximum √2 at x = π/4A1

Practice exercise 14

1. Find the maximum of 3 sin x + 4 cos x.

2. Find the first positive zero of x sin x − 1.

17 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 14 Mixed problem (exam level)

f(x) = 3 sin(2x) − 1, 0 ≤ x ≤ π. (a) Find the range. (b) Solve f(x) = 0.5. (c) Find the values of k for which f(x) = k has exactly two solutions.

Solution

(a) −4 ≤ f(x) ≤ 2A1

(b) sin 2x = 0.5: x = π/12, 5π/12A1

(c) Two solutions for −4 < k < 2 except k = −1 (three solutions, including the endpoints): −4 < k < −1 or −1 < k < 2R1 A1

Practice exercise 15

1. Find the period and amplitude of y = 5 − 2 sin(3πx).

2. Find the smallest positive x where y = 2 cos(x − π/6) has a maximum.

18 Additional exam-style practice

Paper 2 questions.

WORKED EXAMPLE 15 Paper 2 question (exam level)

The height of a buoy is h(t) = 1.2 sin(0.8t) + 0.3 cos(0.8t) metres. (a) Find the maximum height. (b) Find the period. (c) Find the first time h = 1.

Solution

(a) GDC: 1.24 mA1

(b) 2π/0.8 = 7.85 sA1

Practice exercise 16

1. For the buoy, find the total time in one period during which h > 0.5.

19 Inverse trigonometric values on the GDC

Principal values and all solutions.

The GDC returns one principal value from sin−1, cos−1 or tan−1. For a periodic model, add multiples of the period and use symmetry to find all solutions in the interval.

WORKED EXAMPLE 16 All solutions (standard)

Solve 4 sin(πt/12) = 3 for 0 ≤ t ≤ 24.

Solution

πt/12 = sin−1(0.75) = 0.848 or π − 0.848M1

t = 3.24 or 8.76A1

Practice exercise 17

1. Solve 5 cos(0.5t) = −2 for 0 ≤ t ≤ 4π.

20 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 2 style) (about 15 minutes)

The height of the tide at a port is modelled by h(t) = a cos(bt) + d metres, t hours after high tide. High tide is 9.2 m and low tide 1.6 m, occurring 6.2 hours later.

(a) Find a, b and d. [4]

(b) Find the height of the tide 2 hours after high tide. [2]

(c) A ship needs at least 4 m of water. Find the length of time in the first tidal cycle for which the ship cannot enter. [4]

Markscheme

Part	Answer	Marks
(a)	d = 5.4	A1
	a = 3.8	A1
	period 12.4, b = 2π/12.4	M1
	b = 0.507	A1
(b)	3.8 cos(2b) + 5.4	M1
	7.41 m	A1
(c)	h(t) < 4 between the solutions of h(t) = 4	M1
	t = 3.84 and 8.56	A1 A1
	4.71 hours	A1

Examiner's comments. Keep b unrounded throughout part (c).

Question 2 (Paper 1 style) (about 10 minutes)

Let f(x) = 2 sin(3x) + 1 for 0 ≤ x ≤ 2π/3.

(a) Write down the amplitude and period of f. [2]

(b) Find the coordinates of the maximum point. [2]

Markscheme

Part	Answer	Marks
(a)	2	A1
	2π/3	A1
(b)	3x = π/2	M1
	(π/6, 3)	A1
(c)	sin 3x = ½	M1
	3x = π/6, 5π/6	A1
	x = π/18, 5π/18	A1

Examiner's comments. Extending the interval for 3x to [0, 2π] is essential in (c).

21 Common misconceptions and links

Students often think…	Actually…
the period of sin 2x is 4π	it is π
sin(2x − π/3) shifts π/3	it shifts π/6
the amplitude is the maximum value	it is half the distance from max to min
tan x has amplitude 1	it has no amplitude

How 3.7 connects to the rest of the course

Topic	Connection
2.11 Transformations	stretches and translations
3.6 Equations	solving trig equations
3.4 Unit circle	definition of sin and cos
5.6 Calculus	derivatives of sinusoidal models

22 Check your understanding

Section A: multiple choice

1. The period of y = cos 4x is

A. π/2 B. 4π C. 2π D. π/4

2. The amplitude of y = −5 sin x + 2 is

A. −5 B. 2 C. 5 D. 7

3. The principal axis of y = 2 sin x − 3 is

A. y = 2 B. y = −3 C. y = 0 D. y = −1

4. tan x has asymptotes at

A. x = kπ B. x = 2kπ C. y = ±1 D. x = π/2 + kπ

5. A max of 10 and min of 2 give d =

A. 4 B. 6 C. 8 D. 5

Section B: structured

6. Sketch y = 3 cos 2x for 0 ≤ x ≤ π, labelling key points. [3]

7. Find the equation of a sine curve with amplitude 4, period 6π and principal axis y = 1, starting at the axis and increasing at x = 0. [3]

Answers

1. A 2. C 3. B 4. D 5. B

6. Max (0, 3), zeros at π/4 and 3π/4, min (π/2, −3), max (π, 3).

7. y = 4 sin(x/3) + 1.

23 Summary and topic map

Idea	Remember
sin, cos	period 2π, amplitude 1
tan	period π, asymptotes x = π/2 + kπ
Amplitude	\|a\| = (max − min)/2
Principal axis	y = d = (max + min)/2
Period	2π/b
Phase shift	factorize b(x − c)
Modelling	tides, wheels, daylight

Before the exam, make sure you…

•read max, min and period carefully
•factorize before reading the shift
•label key points on sketches
•keep parameters unrounded

Trigonometric Functions on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 10. Topic map for Trigonometric Functions.

24 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. π; ℝ

2. π/2, 3π/2, 5π/2, 7π/2

3. 1 at π/2 and 5π/2

Practice exercise 2

1. 4; 3; −5

2. 4; y = 9

Practice exercise 3

1. 4π

2. 4

3. π/4

Practice exercise 4

1. 2; 2π/3; y = −4

2. π/3 to the left

3. minimum (π/2, −1), maximum (3π/2, 3)

Practice exercise 5

1. y = 4 cos x + 3

2. y = 5 sin(π(x − 1)/4) + 2

Practice exercise 6

1. D(t) = −3 cos(2π(t − 355)/365) + 12

2. sin(0.5t) = 0.5: t = 1.05, 5.24 (next solutions are beyond 12)

3. 24 °C at t = 15

Practice exercise 7

1. (0, −2), (π/4, 1), (π/2, 4), (3π/4, 1), (π, −2)

2. shift π/4 left: zeros at 3π/4, 7π/4

Practice exercise 8

1. y = 2 sin(2(x − π/3))

2. y = 4 sin(π(x − 5)/6) + 3

Practice exercise 9

1. 2π

2. π/8, 5π/8

Practice exercise 10

1. x = 1.38 (one solution)

2. 8

Practice exercise 11

1. d(8) = 3.6 cos(2π(4)/12.4) + 6 = 4.41 m

Practice exercise 12

1. 30 − 25 cos(π/4) = 12.3 m

2. 2π(25)/8 = 19.6 m/min

Practice exercise 13

1. T = 10 cos(π(m − 7)/6) + 22

Practice exercise 14

1. 5

2. x = 1.11

Practice exercise 15

1. 2/3; 2

2. π/6

Practice exercise 16

1. h = 0.5 at t = 0.214 and 3.10: about 2.89 s

Practice exercise 17

1. 0.5t = 1.98 or 4.30: t = 3.96, 8.61

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