DP Maths AI HL 3.7 Trigonometric Functions

DP Maths AI HL 3.7 Trigonometric Functions - Full notes

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DP Maths AI HL | 3.7 Trigonometric Functions

DP MATHEMATICS AI · TOPIC 3: GEOMETRY AND TRIGONOMETRY

3.7

Trigonometric Functions

Full notes for Applications and Interpretation HL

Contents

1	Periodic phenomena
2	Degrees and radians
3	Sine and cosine from the unit circle
4	The graph of y = sin x
5	The graph of y = cos x
6	The graph of y = tan x
7	Amplitude and principal axis
8	Period
9	Phase shift
10	The general sinusoidal model
11	Finding parameters from maximum and minimum values
12	Finding parameters from graphs and tables
13	Sinusoidal regression with technology
14	Solving trigonometric equations in context
15	Application: tides
16	Application: Ferris wheels and rotating objects
17	Application: temperature, daylight and sound
18	Mixed review: worked problems
19	Radians, arcs and sectors (HL)
20	Combining sinusoidal functions (HL)
21	Investigation: designing a Ferris wheel ride
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	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 practice exercise.

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
recognize periodic behaviour and its features
convert between degrees and radians
sketch y = sin x, y = cos x and y = tan x
identify amplitude, period, phase shift and principal axis
build sinusoidal models from maximum and minimum values, graphs and data
use sinusoidal regression
solve trigonometric equations in context using technology
model tides, rotating wheels, temperatures and daylight

HL booklet: this booklet contains all the SL content plus radian measure with arcs and sectors, and combinations of sinusoidal functions.

Guiding questions

How can trigonometric functions model things that repeat? How do the parameters of a sinusoidal model relate to the situation?

How this topic is examined

Sinusoidal models appear frequently on Paper 2: find the parameters of a model from a description or a graph, use it to predict, and find the times when a quantity is above or below a level. Paper 1 asks for amplitude, period and simple values.

State whether you are using degrees or radians, set your GDC accordingly, and give times in context (for example, 'between 2:15 a.m. and 6:40 a.m.').

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

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, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
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
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), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Periodic phenomena

Patterns that repeat.

Think about it

Plot the height of the tip of a clock's minute hand above the centre against time. What shape do you get? How long before the pattern repeats?

A periodic function repeats its values at regular intervals. The length of one complete cycle is the period. Tides, daylight hours, temperatures over a year, sound waves, heartbeats and rotating wheels are all periodic.

The sine and cosine functions are the basic periodic functions. Stretching and translating them produces a sinusoidal model that can fit almost any smooth periodic pattern.

2 Degrees and radians

Two units for angles.

A full turn is 360° or 2π radians. One radian is the angle that subtends an arc equal in length to the radius. To convert: degrees to radians, multiply by π/180; radians to degrees, multiply by 180/π. In modelling, time is often the input, and either unit can be used as long as the period is calculated consistently.

Degrees	30	45	60	90	180	270	360
Radians	π/6	π/4	π/3	π/2	π	3π/2	2π

GDC tip

Check the angle mode before every trigonometry question. A common source of lost marks is a GDC left in radian mode for a degree question, or vice versa.

Practice exercise 1

1. Convert 150° and 20° to radians.

2. Convert 2.5 rad to degrees.

3 Sine and cosine from the unit circle

Coordinates of a rotating point.

A point moving round a circle of radius 1 centred at the origin, at angle θ measured anticlockwise from the positive x-axis, has coordinates (cos θ, sin θ). As θ increases, the height sin θ rises and falls between −1 and 1, and repeats every 360°. This is why rotating objects produce sinusoidal graphs.

Figure 1. The unit circle: sin θ is the height of the point.

4 The graph of y = sin x

Start at zero, rise first.

y = sin x has period 360° (2π), amplitude 1, principal axis y = 0, maximum 1 at 90° and minimum −1 at 270°. It crosses the axis at 0°, 180°, 360°.

Figure 2. Two periods of y = sin x (x in radians).

Practice exercise 2

1. Write down the values of x in [0°, 720°] where sin x = 1.

2. Write down sin 210° using the graph's symmetry.

5 The graph of y = cos x

Start at the maximum.

y = cos x has the same shape as sin x, translated 90° (π/2) to the left: cos x = sin(x + 90°). It starts at its maximum 1 at x = 0. Use cosine models when a situation starts at a maximum (or minimum, with a negative amplitude).

Figure 3. Cosine is sine shifted left by π/2.

6 The graph of y = tan x

Asymptotes, period 180°.

y = tan x = sin x/cos x has period 180° (π), no amplitude, and vertical asymptotes where cos x = 0 (x = 90°, 270°, ...). It is used less often in modelling, but appears in angle-of-elevation contexts.

Figure 4. y = tan x with asymptotes.

7 Amplitude and principal axis

Height of the wave and its centre line.

In y = a sin x + d, the amplitude |a| is half the distance between the maximum and minimum, and the principal axis y = d is the line midway between them:

WORKED EXAMPLE 1 Amplitude and axis (basic)

For y = 3 sin x + 5, find the amplitude, principal axis, maximum and minimum.

Solution

Amplitude 3, axis y = 5A1

Maximum 8, minimum 2A1

Practice exercise 3

1. A wave varies between −2 and 6. Find a and d.

2. State the range of y = −4 cos x + 1.

8 Period

Length of one cycle.

Increasing b squeezes more cycles into the same interval. To find b from a known period: b = 360°/period (degrees) or b = 2π/period (radians). If the input is time in hours and the period is 12.4 hours, b = 2π/12.4 = 0.507 (radians) or 360/12.4 = 29.0 (degrees).

Practice exercise 4

1. Find the period of y = sin 3x in degrees.

2. Find b for a period of 24 hours, using degrees.

3. Find b for a period of 0.02 s, using radians.

9 Phase shift

Shifting the start.

y = sin(b(x − c)) is shifted c units to the right. Always factor b out of the bracket before reading c: sin(2x − 60°) = sin(2(x − 30°)), a shift of 30°, not 60°.

Common mistake

Reading the phase shift of sin(2x − 60°) as 60°. Factor first: the shift is 30°.

10 The general sinusoidal model

Four parameters.

Parameter	Meaning
\|a\|	amplitude
b	number of cycles per 360° (or 2π); period = 360°/b
c	horizontal shift (phase shift)
d	principal axis y = d

Figure 5. Amplitude 2, period π, shift π/4, axis y = 1.

Practice exercise 5

1. State the amplitude, period (degrees), shift and axis of y = 5 cos(3(x + 20°)) − 2.

2. Write a sine model with amplitude 6, period 8 and axis y = 10, starting at the axis and increasing at x = 0 (radians).

11 Finding parameters from maximum and minimum values

A reliable procedure.

From a maximum (x1, ymax) and the next minimum (x2, ymin): a = (ymax − ymin)/2; d = (ymax + ymin)/2; period = 2(x2 − x1); b = 360°/period or 2π/period. With cosine, c = x1 (the position of a maximum).

WORKED EXAMPLE 2 From max and min (standard)

A quantity has a maximum of 18 at t = 2 and the next minimum of 6 at t = 8. Write a cosine model (degrees).

Solution

a = 6, d = 12A1

Period 12, b = 30A1

y = 6 cos(30(t − 2)) + 12A1

Practice exercise 6

1. Max 40 at t = 0, next min 10 at t = 5. Write a cosine model in degrees.

2. Min 2 at x = 1, next max 8 at x = 4. Write a model using −cos (radians).

12 Finding parameters from graphs and tables

Read carefully, then check.

From a graph, read the maximum and minimum values and their positions, and the distance between consecutive maxima. From a table, the largest and smallest values give a and d, and the time between them gives half a period. Check the model by substituting another point.

Month	1	3	5	7	9	11
Temp (°C)	4	9	17	22	17	9

WORKED EXAMPLE 3 From a table (exam level)

Model the temperatures with T = a cos(b(m − c)) + d, m in months, degrees mode.

Solution

Max 22 at m = 7, min about 4 at m = 1: a = 9, d = 13A1

Period 12, b = 30, c = 7A1

T = 9 cos(30(m − 7)) + 13; check m = 3: 9 cos(−120°) + 13 = 8.5 ≈ 9A1 R1

13 Sinusoidal regression with technology

Let the GDC fit the curve.

Most GDCs have SinReg, which fits y = a sin(bx + c) + d to data. Note the form: the GDC uses (bx + c), so the phase shift is −c/b. Compare the regression parameters with your estimates and comment on the fit.

GDC tip

Enter months in L1 and temperatures in L2, set radian mode, and run SinReg L1, L2. Plot the data with the curve to judge the fit.

Practice exercise 7

1. A regression gives y = 9.1 sin(0.524x − 2.09) + 13.0. Find the period and the phase shift.

14 Solving trigonometric equations in context

When is the quantity at a given level?

To find when a sinusoidal quantity equals a value, graph the model and the horizontal line and find the intersections within the required interval. For 'above' or 'below' questions, find the interval(s) between intersections. Give answers in context with units.

WORKED EXAMPLE 4 Times above a level (standard)

h(t) = 2.5 sin(30t) + 4 (degrees, t in hours). Find the times in [0, 12] when h > 5.

Solution

Intersections with h = 5: t = 0.786, 5.21M1 A1

0.786 < t < 5.21: about 4.43 hoursA1

Practice exercise 8

1. For h above, find the first time h = 2.

2. How long in each cycle is h below 3?

15 Application: tides

A classic model.

WORKED EXAMPLE 5 Harbour depth (exam level)

High tide is 9.4 m at 03:00, low tide 2.2 m at 09:12. Model the depth d (m) t hours after midnight, and find when a ship needing 4 m can enter after 03:00.

Solution

a = 3.6, d = 5.8, half-period 6.2 h, period 12.4 hA1

d(t) = 3.6 cos(2π(t − 3)/12.4) + 5.8A1

d = 4 when t = 7.13 and t = 11.27M1 A1

Safe until 07:08 and again from 11:16A1

Practice exercise 9

1. Using the model, find the depth at noon.

16 Application: Ferris wheels and rotating objects

Height of a seat.

A seat on a wheel of radius r, with centre at height H, rotating once every T minutes, has height h(t) = H − r cos(2πt/T) if it starts at the bottom. Amplitude r, axis H, period T.

WORKED EXAMPLE 6 Ferris wheel (standard)

A wheel of radius 15 m with centre 18 m above the ground turns once every 6 minutes, starting at the bottom. Find the height after 2 minutes and the time per revolution above 25 m.

Solution

h(t) = 18 − 15 cos(60t) (degrees)A1

h(2) = 18 − 15 cos 120° = 25.5 mA1

h > 25 for 1.96 < t < 4.04: 2.07 minutesM1 A1

Figure 6. Height of a seat on the wheel.

Practice exercise 10

1. A bicycle wheel of radius 0.35 m turns 3 times per second. Write the height of a point on the rim (starting at the top, axle 0.35 m above ground).

17 Application: temperature, daylight and sound

More periodic models.

Daily temperatures (period 24 h), yearly temperatures and daylight hours (period 365 days), and sound waves (period 1/frequency) are modelled with sinusoids. The frequency f of a wave (cycles per second, Hz) is 1/period, so b = 2πf in radians.

WORKED EXAMPLE 7 Daylight (exam level)

Daylight varies from 8.5 h (day 355) to 16.1 h (day 172). Find a model and the number of days with more than 14 hours of daylight.

Solution

a = 3.8, d = 12.3, period 365, max at day 172A1

D = 3.8 cos(2π(t − 172)/365) + 12.3A1

D > 14 for t between 108 and 236: about 129 daysM1 A1

Practice exercise 11

1. A note has frequency 440 Hz. Find the period and b (radians).

2. A daily temperature varies from 14 °C at 05:00 to 28 °C at 17:00. Write a model.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 8 Mixed problem 1 (exam level)

f(x) = a sin(bx) + d has maximum 7, minimum −1 and period 90°. Find a, b, d and solve f(x) = 5 for 0° ≤ x ≤ 90°.

Solution

a = 4, d = 3, b = 4A1 A1

sin 4x = 0.5: 4x = 30°, 150°M1

x = 7.5°, 37.5°A1

Practice exercise 12

1. Write the equation of y = cos x after a vertical stretch by 3, a horizontal stretch by factor 1/2 and a translation 2 up.

2. Find the first two positive solutions of 3 sin(2x) + 1 = 2.5 (radians).

19 Radians, arcs and sectors (HL)

Arc length and area.

Radian measure makes arc and sector formulas simple. Angular speed ω (radians per unit time) gives linear speed v = rω for a point on a rotating object.

WORKED EXAMPLE 9 Sector (standard)

A sector of radius 12 cm has angle 1.4 rad. Find the arc length and area.

Solution

l = 16.8 cmA1

A = ½(144)(1.4) = 100.8 cm2A1

Practice exercise 13

1. A windmill blade of length 40 m turns at 1.5 rad/s. Find the speed of the tip.

2. Find the area of a sector with radius 5 and arc 8.

20 Combining sinusoidal functions (HL)

Sums and beats.

Adding two sinusoids with the same period gives another sinusoid with that period. Adding sinusoids with slightly different frequencies produces beats: the amplitude rises and falls slowly. Sums of sinusoids model sound, tides with several components and alternating currents.

Figure 7. Beats from two close frequencies.

Practice exercise 14

1. Use technology to find the maximum value of y = 3 sin x + 4 cos x.

2. Explain why sin 10x + sin 11x produces a slowly varying amplitude.

21 Investigation: designing a Ferris wheel ride

A Paper 3 style exploration.

A ride designer wants a wheel whose seats reach 60 m and board at 3 m, with each ride lasting exactly 3 revolutions in 24 minutes. 1. Find the radius and the height of the centre. 2. Write h(t) for a seat that boards at the bottom. 3. For how much of the ride is a seat higher than 50 m? 4. The designer wants passengers to spend at least 30% of the ride above 40 m. Does this design meet that requirement? 5. Suggest how the model could be adapted for a wheel that stops for 1 minute at the bottom on each revolution.

Answers. 1. r = 28.5 m, centre 31.5 m. 2. h = 31.5 − 28.5 cos(2πt/8). 3. h > 50 for 2.90 < t < 5.10 in each 8-minute revolution: 2.20 min per revolution, 6.60 of 24 min (27.5%). 4. h > 40 for 2.39 < t < 5.61 per revolution: 3.23/8 = 40%: yes. 5. Use a piecewise model with constant height during the stop and a time-shifted sinusoid afterwards.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

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

f(x) = 3 cos(2x) + 1, 0° ≤ x ≤ 360°.

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

(b) Write down the range of f. [2]

Markscheme

Part	Answer	Marks
(a)	3	A1
	180°	A1
(b)	−2 ≤ f(x) ≤ 4	A2
(c)	two periods, two intersections per period	M1
	4	A1

Examiner's comments. A sketch makes part (c) straightforward.

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

The depth of water at a pier is modelled by d(t) = 2.8 sin(0.5t − 1) + 6, t hours after 6 a.m. (radians).

(a) Find the maximum depth and the first time it occurs. [3]

(b) Find the period. [1]

Markscheme

Part	Answer	Marks
(a)	8.8 m	A1
	0.5t − 1 = π/2: t = 5.14	M1
	about 11:08 a.m.	A1
(b)	4π ≈ 12.6 h	A1
(c)	solve d(t) = 7.5	M1
	t = 3.13 and 7.15	A1
	4.02 hours	A1

Examiner's comments. Convert t to clock times where asked.

Question 3 (Paper 2 style) (about 10 minutes)

The average monthly temperature in a city has a maximum of 31 °C in July (m = 7) and a minimum of 11 °C in January (m = 1).

(a) Write a cosine model T(m), degrees mode. [3]

(b) Predict the temperature in October. [1]

Markscheme

Part	Answer	Marks
(a)	a = 10, d = 21	A1
	b = 30, c = 7	A1
	T = 10 cos(30(m − 7)) + 21	A1
(b)	T(10) = 21 °C	A1
(c)	cos(30(m − 7)) > 0.4: 4.78 < m < 9.22	M1
	May to September	A1

Examiner's comments. Interpret month values sensibly.

23 Common misconceptions and links

Students often think…	Actually…
the period of sin 3x is 3π	2π/3
the shift of sin(2x − 60°) is 60°	30°
amplitude = maximum	half of max − min
GDC mode does not matter	check degrees/radians

How 3.7 connects to the rest of the course

Topic	Connection
2.9 Transformations	stretches and shifts
3.2 Trigonometric ratios	unit circle
5.3 Derivatives	rates of change of sinusoids
2.2 Functions	modelling

24 Check your understanding

Section A: multiple choice

1. The period of y = sin 2x (degrees) is

A. 180° B. 720° C. 360° D. 90°

2. The amplitude of y = −5 cos x + 3 is

A. −5 B. 3 C. 5 D. 8

3. The axis of a wave from 4 to 16 is

A. y = 6 B. y = 10 C. y = 12 D. y = 4

4. tan x has period

A. 360° B. 90° C. 720° D. 180°

5. A wheel turning every 8 min has b (degrees) =

A. 8 B. 45 C. 360 D. 2π/8

Section B: structured

6. Write a sine model for a wave with max 12, min 2, period 20 and value 7 increasing at x = 0. [3]

7. Convert 3π/5 to degrees. [1]

Answers

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

6. y = 5 sin(18x) + 7 (degrees)

7. 108°

25 Summary and topic map

Idea	Remember
Periodic	repeats every period
sin, cos	period 360° (2π), amplitude 1
tan	period 180°, asymptotes
Model	a sin(b(x − c)) + d
Parameters	a = (max − min)/2; d = (max + min)/2; b = 360/period
Applications	tides, wheels, temperature, daylight, sound

Before the exam, make sure you…

•set the angle mode
•factor b before reading the shift
•check a model with another point
•give times in context

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 8. Topic map for Trigonometric Functions.

26 Answers to practice exercises

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

Practice exercise 1

1. 5π/6; π/9

2. 143°

Practice exercise 2

1. 90°, 450°

2. −0.5

Practice exercise 3

1. a = 4, d = 2

2. −3 ≤ y ≤ 5

Practice exercise 4

1. 120°

2. 15

3. 100π

Practice exercise 5

1. 5; 120°; 20° left; y = −2

2. y = 6 sin(πx/4) + 10

Practice exercise 6

1. y = 15 cos(36t) + 25

2. y = −3 cos(π(x − 1)/3) + 5

Practice exercise 7

1. 2π/0.524 = 12.0; shift 2.09/0.524 = 3.99 right

Practice exercise 8

1. t = 7.77 h

2. from 6.79 to 11.2: 4.43 hours

Practice exercise 9

1. d(12) = 3.6 cos(2π(9)/12.4) + 5.8 ≈ 5.25 m

Practice exercise 10

1. h = 0.35 + 0.35 cos(1080t) (degrees), or 0.35 + 0.35 cos(6πt)

Practice exercise 11

1. 0.00227 s; 880π

2. T = −7 cos(15(t − 5)) + 21 (degrees)

Practice exercise 12

1. y = 3 cos 2x + 2

2. x = 0.262, 1.31

Practice exercise 13

1. 60 m/s

2. ½ × 5 × 8 = 20

Practice exercise 14

1. 5

2. the waves move in and out of phase

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