DP Maths AA SL - Geometry and Trigonometry - Trigonometric Ratios and Identities

DP Maths AA SL - Geometry and Trigonometry - Trigonometric Ratios and Identities - Full notes

RevisionPrep

DP Maths AA SL | Trigonometric Ratios and Identities

Trigonometric Ratios and Identities

This topic contains 1 note:

3.6 Trig Identities and Equations

DP MATHEMATICS AA · TOPIC 3: GEOMETRY AND TRIGONOMETRY

3.6

Trigonometric Identities and Equations

Full notes for SL · Syllabus SL 3.6, 3.8

Contents

1	The Pythagorean identity
2	The tangent identity
3	Double angle identities
4	Proving trigonometric identities
5	Solving basic trigonometric equations
6	Equations with multiple angles
7	Quadratic trigonometric equations
8	Equations using identities
9	Solving graphically
10	Using the Pythagorean identity to simplify
11	Double angle identities in practice
12	Proofs of identities: strategies
13	Equations with multiple angles
14	Equations reducing to quadratics
15	Equations with tan
16	Graphical solutions and counting solutions
17	Trigonometric equations in context
18	Mixed review: worked problems
19	Additional exam-style practice
20	Exact values and identities together
21	Checking with technology
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 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
use the identity cos2θ + sin2θ = 1
use tan θ = sin θ / cos θ
use the double angle identities for sin 2θ and cos 2θ (three forms)
prove simple trigonometric identities
solve trigonometric equations in a finite interval, finding all solutions
solve equations involving multiple angles such as sin 2x
solve quadratic equations in sin, cos or tan
use identities to rewrite and solve equations, and solve graphically with technology

This booklet covers SL 3.6, 3.8, which is common to SL and HL.

Guiding questions

How can one trigonometric expression be rewritten in many equivalent forms? Why does a trigonometric equation usually have many solutions?

How this topic is examined

Identities and equations are central to Paper 1, often in 'show that' and 'hence solve' form: rewrite an equation using an identity to obtain a quadratic, then solve in a given interval. Paper 2 uses technology for equations that cannot be solved exactly.

Always find all solutions in the given interval, give exact values on Paper 1, and check the units (degrees or radians) of the interval.

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 Pythagorean identity

From the unit circle.

Think about it

A point on the unit circle has coordinates (cos θ, sin θ). What equation must those coordinates satisfy?

Because (cos θ, sin θ) lies on x2 + y2 = 1, for every θ:

Here cos2θ means (cos θ)2. Rearranged forms are just as useful: sin2θ = 1 − cos2θ and cos2θ = 1 − sin2θ.

WORKED EXAMPLE 1 Finding another ratio (standard)

Given cos θ = −12/13 and π/2 < θ < π, find sin θ.

Solution

sin2θ = 1 − 144/169 = 25/169M1

2nd quadrant, sin positive: sin θ = 5/13A1

Practice exercise 1

1. Given sin θ = 0.6 with θ obtuse, find cos θ.

2. Simplify 1 − cos22x.

3. Simplify (sin θ + cos θ)2 + (sin θ − cos θ)2.

2 The tangent identity

tan = sin/cos.

This is the gradient of the radius on the unit circle. It lets you convert equations such as sin x = √3 cos x into tan x = √3.

Practice exercise 2

1. Solve sin x = cos x for 0 ≤ x ≤ 2π.

2. Given tan θ = 2 with θ acute, find sin θ and cos θ exactly.

3 Double angle identities

Sine and cosine of twice an angle.

The three forms of cos 2θ come from replacing sin2θ or cos2θ using the Pythagorean identity. Choose the form that leaves only one trigonometric function in your equation. These identities are proved from the compound angle formulas (3.9, HL) or from De Moivre's theorem (1.12, HL); at SL they are given in the formula booklet.

WORKED EXAMPLE 2 Using a double angle (standard)

Given sin θ = 1/3 and 0 < θ < π/2, find the exact values of sin 2θ and cos 2θ.

Solution

cos θ = √(1 − 1/9) = 2√2/3A1

sin 2θ = 2(1/3)(2√2/3) = 4√2/9A1

cos 2θ = 1 − 2(1/9) = 7/9A1

Practice exercise 3

1. Given cos x = 3/5, find cos 2x.

2. Write sin 4x in terms of sin 2x and cos 2x.

3. Simplify 2 sin 3x cos 3x.

4. Write cos2x in terms of cos 2x.

4 Proving trigonometric identities

One side to the other.

Start from the more complicated side and use known identities, algebra (common denominators, factorizing, difference of squares) and the Pythagorean identity until you reach the other side. Write each step on a new line.

WORKED EXAMPLE 3 An identity (standard)

Prove that (1 − cos 2θ)/sin 2θ ≡ tan θ.

Solution

LHS = (1 − (1 − 2sin2θ))/(2 sin θ cos θ)M1

= 2sin2θ/(2 sin θ cos θ) = sin θ/cos θA1

= tan θ = RHSAG

WORKED EXAMPLE 4 Using Pythagoras (exam level)

Prove that sin4θ − cos4θ ≡ −cos 2θ.

Solution

LHS = (sin2θ − cos2θ)(sin2θ + cos2θ)M1

= (sin2θ − cos2θ)(1)A1

= −(cos2θ − sin2θ) = −cos 2θ = RHSAG

Practice exercise 4

1. Prove that (sin θ + cos θ)2 ≡ 1 + sin 2θ.

2. Prove that tan θ + 1/tan θ ≡ 2/sin 2θ.

3. Prove that cos4x − sin4x ≡ cos 2x.

5 Solving basic trigonometric equations

Find every solution in the interval.

To solve, for example, sin x = k in an interval: (1) find the principal value from the inverse function (or an exact value); (2) use symmetry to find the second solution in one period (sin x = sin(π − x); cos x = cos(−x), so the second cosine solution is 2π − x; tan repeats every π); (3) add or subtract multiples of the period to find all solutions in the interval. A sketch of the graph or the unit circle prevents missed solutions.

WORKED EXAMPLE 5 Exact solutions (basic)

Solve 2 cos x = −1 for 0 ≤ x ≤ 2π.

Solution

cos x = −½: reference angle π/3M1

cos negative in the 2nd and 3rd quadrantsR1

x = 2π/3, 4π/3A1

WORKED EXAMPLE 6 Degrees with a calculator (standard)

Solve 3 sin x = 2 for 0° ≤ x ≤ 360°.

Solution

sin x = 2/3: x = 41.8°A1

second solution 180° − 41.8° = 138.2°A1

Practice exercise 5

1. Solve tan x = −1 for 0 ≤ x ≤ 2π.

2. Solve sin x = −√3/2 for −π ≤ x ≤ π.

3. Solve 5 cos x = 2 for 0° ≤ x ≤ 360°.

6 Equations with multiple angles

Stretch the interval first.

For sin 2x = k with 0 ≤ x ≤ 2π, let u = 2x: then 0 ≤ u ≤ 4π, which covers two full periods. Find all u in the extended interval, then divide by 2. A multiple angle gives proportionally more solutions. For sin(x + π/6) = k, shift the interval in the same way.

WORKED EXAMPLE 7 Double angle equation (standard)

Solve sin 2x = √3/2 for 0 ≤ x ≤ 2π.

Solution

u = 2x, 0 ≤ u ≤ 4π: u = π/3, 2π/3, 7π/3, 8π/3M1 A1

x = π/6, π/3, 7π/6, 4π/3A1

WORKED EXAMPLE 8 Shifted angle (exam level)

Solve cos(x − π/4) = ½ for 0 ≤ x ≤ 2π.

Solution

u = x − π/4, −π/4 ≤ u ≤ 7π/4: u = π/3, 5π/3M1 A1

x = 7π/12, 23π/12A1

Common mistake

Solving sin 2x = k for x in [0, 2π] by only finding two solutions. The interval for 2x is [0, 4π], giving four.

Practice exercise 6

1. Solve cos 3x = 0 for 0 ≤ x ≤ π.

2. Solve tan(x/2) = 1 for 0 ≤ x ≤ 4π.

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

7 Quadratic trigonometric equations

Factorize as a quadratic.

Equations such as 2sin2x − sin x − 1 = 0 are quadratics in sin x. Let s = sin x, factorize or use the formula, then solve each resulting basic equation. Reject any value outside the range: sin x and cos x must lie between −1 and 1.

WORKED EXAMPLE 9 Quadratic in sine (standard)

Solve 2sin2x − sin x − 1 = 0 for 0 ≤ x ≤ 2π.

Solution

(2 sin x + 1)(sin x − 1) = 0M1

sin x = 1: x = π/2A1

sin x = −½: x = 7π/6, 11π/6A1

Practice exercise 7

1. Solve tan2x = 3 for 0 ≤ x ≤ 2π.

2. Solve 2cos2x + 3 cos x − 2 = 0 for 0° ≤ x ≤ 360°.

3. Solve sin2x = sin x for 0 ≤ x ≤ 2π.

8 Equations using identities

Reduce to a single function.

If an equation contains more than one trigonometric function or angle, use an identity to rewrite it in terms of a single function of a single angle. Typical moves: replace cos2x by 1 − sin2x; replace cos 2x by the form that matches the other term; replace sin 2x by 2 sin x cos x and factorize. Never divide by sin x or cos x, since this loses the solutions where they are zero; factorize instead.

WORKED EXAMPLE 10 Using cos 2x (exam level)

Solve cos 2x = sin x for 0 ≤ x ≤ 2π.

Solution

1 − 2sin2x = sin x, so 2sin2x + sin x − 1 = 0M1 A1

(2 sin x − 1)(sin x + 1) = 0A1

x = π/6, 5π/6, 3π/2A1

WORKED EXAMPLE 11 Using sin 2x and factorizing (exam level)

Solve sin 2x = sin x for 0 ≤ x ≤ 2π.

Solution

2 sin x cos x − sin x = 0, so sin x(2 cos x − 1) = 0M1 A1

sin x = 0: x = 0, π, 2πA1

cos x = ½: x = π/3, 5π/3A1

Common mistake

Dividing both sides of sin 2x = sin x by sin x, which loses x = 0, π and 2π.

Practice exercise 8

1. Solve 2cos2x = 3 sin x for 0 ≤ x ≤ 2π.

2. Solve cos 2x + cos x = 0 for 0 ≤ x ≤ 2π.

3. Show that 3 sin2x + cos 2x = 2 can be written as sin2x = 1, and solve for 0 ≤ x ≤ 2π.

9 Solving graphically

When there is no exact method.

Equations that mix trigonometric and other functions, such as sin x = x/2 or 3 cos x = x2, have no exact solution method. Graph both sides on a GDC (in the correct angle mode) and find the intersections. The number of solutions in an interval is the number of intersections.

WORKED EXAMPLE 12 Graphical solution (standard)

Solve 2 sin x = x − 1 for 0 ≤ x ≤ 2π.

Solution

Graph y = 2 sin x and y = x − 1M1

x = 2.38A1

Figure 1. One intersection in [0, 2π].

Practice exercise 9

1. Solve cos x = x for x ∈ ℝ.

2. How many solutions does sin x = x/5 have for x ≥ 0?

10 Using the Pythagorean identity to simplify

Replace sin2 or cos2.

Expressions containing both sin2θ and cos2θ can often be simplified with cos2θ + sin2θ = 1, or rewritten entirely in terms of one function. Factorizing (difference of squares, common factors) is often the key.

WORKED EXAMPLE 13 Simplifying (standard)

Simplify (1 − cos2x)/(1 − sin2x).

Solution

sin2x/cos2xM1

= tan2xA1

WORKED EXAMPLE 14 Factorizing (standard)

Simplify cos4x − sin4x.

Solution

(cos2x − sin2x)(cos2x + sin2x)M1

= cos 2xA1

Practice exercise 10

1. Simplify sin3x + sin x cos2x.

2. Simplify (sin x + cos x)2 − 1.

3. Simplify (1 − sin x)(1 + sin x)/cos x.

11 Double angle identities in practice

Choosing the right form of cos 2x.

cos 2x = cos2x − sin2x = 2cos2x − 1 = 1 − 2sin2x. Choose the form that matches the other terms in the expression or equation. sin 2x = 2 sin x cos x often produces a common factor.

WORKED EXAMPLE 15 Exact values (standard)

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

Solution

sin x = 4/5A1

sin 2x = 2(4/5)(−3/5) = −24/25A1

cos 2x = 2(9/25) − 1 = −7/25A1

WORKED EXAMPLE 16 Half-angle (exam level)

Find the exact value of cos(π/8).

Solution

cos(π/4) = 2cos2(π/8) − 1M1

cos2(π/8) = (2 + √2)/4A1

cos(π/8) = √(2 + √2)/2A1

Practice exercise 11

1. Write sin2x in terms of cos 2x.

2. Find the exact value of sin2(π/12).

3. Simplify (sin 2x)/(1 + cos 2x).

12 Proofs of identities: strategies

A toolkit.

Useful moves: write everything in sin and cos; use a common denominator; factorize; use cos2 + sin2 = 1; multiply by a conjugate such as (1 + sin x). Work on one side only, and write each step clearly.

WORKED EXAMPLE 17 Conjugate (exam level)

Prove that cos x/(1 − sin x) ≡ (1 + sin x)/cos x.

Solution

Multiply by (1 + sin x)/(1 + sin x): cos x(1 + sin x)/(1 − sin2x)M1 A1

= cos x(1 + sin x)/cos2x = (1 + sin x)/cos xA1 AG

WORKED EXAMPLE 18 Double angle proof (exam level)

Prove that (cos 2x + 1)/sin 2x ≡ 1/tan x.

Solution

(2cos2x)/(2 sin x cos x)M1 A1

= cos x/sin x = 1/tan xAG

Practice exercise 12

1. Prove that tan x + cos x/(1 + sin x) ≡ 1/cos x.

2. Prove that (sin 2x)/(1 + cos 2x) ≡ tan x.

13 Equations with multiple angles

Extend the interval.

For sin(kx) = c with x in [a, b], solve for u = kx in [ka, kb], then divide by k. The number of solutions increases by a factor of about k.

WORKED EXAMPLE 19 Triple angle (standard)

Solve sin 3x = ½ for 0 ≤ x ≤ π.

Solution

u = 3x ∈ [0, 3π]: u = π/6, 5π/6, 13π/6, 17π/6M1 A1

x = π/18, 5π/18, 13π/18, 17π/18A1

Practice exercise 13

1. Solve cos 2x = −√3/2 for 0 ≤ x ≤ 2π.

2. Solve tan(x/2) = √3 for 0 ≤ x ≤ 4π.

14 Equations reducing to quadratics

Use an identity, then factorize.

Replace cos2x by 1 − sin2x (or vice versa), or cos 2x by the appropriate form, to obtain a quadratic in a single trigonometric function. Factorize, reject impossible values (|sin x| > 1), and solve the basic equations.

WORKED EXAMPLE 20 Quadratic in cosine (exam level)

Solve 2sin2x + 3cos x = 3 for 0 ≤ x ≤ 2π.

Solution

2(1 − cos2x) + 3cos x − 3 = 0: 2cos2x − 3cos x + 1 = 0M1 A1

cos x = 1 or ½A1

x = 0, π/3, 5π/3, 2πA1

Practice exercise 14

1. Solve cos 2x = 3 sin x + 2 for 0 ≤ x ≤ 2π.

2. Solve 2cos2x = sin x + 1 for 0 ≤ x ≤ 2π.

15 Equations with tan

Divide by cos only when safe.

Equations of the form a sin x = b cos x become tan x = b/a, provided cos x ≠ 0 (check whether cos x = 0 could be a solution). tan has period π, so solutions repeat every π.

WORKED EXAMPLE 21 Tan equation (standard)

Solve √3 sin x = cos x for 0 ≤ x ≤ 2π.

Solution

tan x = 1/√3 (cos x = 0 is not a solution)M1

x = π/6, 7π/6A1

Practice exercise 15

1. Solve sin 2x = cos 2x for 0 ≤ x ≤ π.

2. Solve tan2x = 3 for 0 ≤ x ≤ π.

16 Graphical solutions and counting solutions

Intersections.

Sketch both sides to count the solutions before solving. For equations like sin x = x/3 or cos x = 0.5x, technology gives the numerical solutions.

Figure 2. sin x = x/3 has three solutions.

WORKED EXAMPLE 22 Counting (standard)

Solve sin x = x/3.

Solution

Three intersectionsR1

x = 0, ±2.28A1 A1

Practice exercise 16

1. How many solutions has cos x = x/10 for x ≥ 0?

2. Solve 2 cos x = x.

17 Trigonometric equations in context

Interpreting solutions.

Modelling questions often require solving a trigonometric equation and interpreting the solutions as times. Convert to clock times or dates, and keep only solutions in the stated interval.

WORKED EXAMPLE 23 Tide (exam level)

Depth d(t) = 5 + 2 sin(πt/6), t in hours after midnight. Find when the depth is 6 m in the first 12 hours.

Solution

sin(πt/6) = ½: πt/6 = π/6, 5π/6M1 A1

t = 1 and t = 5 (01:00 and 05:00)A1

Practice exercise 17

1. For the tide, find when the depth is below 4 m in [0, 12].

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 24 Mixed problem (exam level)

(a) Show that sin 3x ≡ 3 sin x − 4sin3x (given sin 3x = sin(2x + x)). (b) Hence solve sin 3x = 2 sin x for 0 ≤ x ≤ π.

Solution

(a) sin 2x cos x + cos 2x sin x = 2 sin x cos2x + sin x − 2sin3x = 3 sin x − 4sin3xM1 A1

(b) sin x(1 − 4sin2x) = 0M1

x = 0, π, π/6, 5π/6A1

Practice exercise 18

1. Solve sin x + sin 2x = 0 for 0 ≤ x ≤ 2π.

2. Prove that sin x/(1 + cos x) ≡ (1 − cos x)/sin x.

19 Additional exam-style practice

Paper 1 questions.

WORKED EXAMPLE 25 Paper 1 question (exam level)

(a) Show that 3 cos 2x + 7 sin x = 5 can be written as 6sin2x − 7 sin x + 2 = 0. (b) Hence solve for 0 ≤ x ≤ π, giving answers to 3 s.f.

Solution

(a) 3(1 − 2sin2x) + 7 sin x − 5 = 0M1 AG

(b) (2 sin x − 1)(3 sin x − 2) = 0A1

x = π/6, 5π/6, 0.730, 2.41A1 A1

Practice exercise 19

1. Solve 4sin2x = 3 for 0 ≤ x ≤ 2π.

20 Exact values and identities together

Combining skills.

Exam questions often require using one identity to find an exact value, then substituting into another: for example, find cos x from sin x, then sin 2x and tan 2x.

WORKED EXAMPLE 26 Combined (exam level)

Given tan x = 2 with 0 < x < π/2, find sin 2x and cos 2x exactly.

Solution

sin x = 2/√5, cos x = 1/√5A1

sin 2x = 4/5; cos 2x = 1/5 − 4/5 = −3/5A1 A1

Practice exercise 20

1. Given sin x = 1/3 (x acute), find tan 2x exactly.

21 Checking with technology

Graph both sides.

An identity is true for all x, so the graphs of both sides coincide exactly. An equation is true only at the intersections. Graphing both sides is a quick check of either.

WORKED EXAMPLE 27 Check (standard)

Use technology to check that sin 2x = 2 sin x cos x.

Solution

The graphs coincide for all xR1

GDC tip

Graph y = (1 − cos 2x)/sin 2x and y = tan x on the same axes: if they overlap everywhere they are identical (except where undefined).

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

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

(a) Show that cos 2x + 3 sin x = 2 can be written as 2sin2x − 3 sin x + 1 = 0. [2]

(b) Hence solve cos 2x + 3 sin x = 2 for 0 ≤ x ≤ 2π. [5]

Markscheme

Part	Answer	Marks
(a)	1 − 2sin2x + 3 sin x = 2	M1
	2sin2x − 3 sin x + 1 = 0	AG A1
(b)	(2 sin x − 1)(sin x − 1) = 0	M1 A1
	sin x = ½: π/6, 5π/6	A1 A1
	sin x = 1: π/2	A1

Examiner's comments. Finding all three solutions was essential for full marks.

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

Let θ be an angle with cos θ = 2/3 and 3π/2 < θ < 2π.

(a) Find the exact value of sin θ. [2]

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

Markscheme

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

Examiner's comments. The fourth-quadrant sign of sin θ carries through to sin 2θ.

23 Common misconceptions and links

Students often think…	Actually…
sin 2x = 2 sin x	sin 2x = 2 sin x cos x
divide by sin x to simplify	factorize to keep solutions
sin x = k has one solution	usually two per period
cos2x = cos x2	cos2x = (cos x)2

How 3.6 connects to the rest of the course

Topic	Connection
3.4 Unit circle	exact values and quadrants
3.7 Trig functions	graphs and periods
3.9 Further trigonometry (HL)	compound angles
5.6 Calculus	differentiating trig functions

24 Check your understanding

Section A: multiple choice

1. sin240° + cos240° =

A. 1 B. 0 C. 80 D. 2

2. cos 2θ can be written as

A. 2 sin θ cos θ B. 1 + 2sin2θ C. 1 − 2sin2θ D. 2 cos θ

3. sin x = 0.5 in [0, 2π] has

A. 1 solution B. 2 solutions C. 4 solutions D. no solutions

4. sin 2x = 0.5 in [0, 2π] has

A. 2 solutions B. 1 solution C. 8 solutions D. 4 solutions

5. tan θ =

A. cos θ/sin θ B. sin θ/cos θ C. sin θ cos θ D. 1/sin θ

Section B: structured

6. Solve 2 cos2x − cos x = 0 for 0 ≤ x ≤ 2π. [4]

7. Prove that sin 2x/(1 + cos 2x) ≡ tan x. [3]

Answers

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

6. cos x(2 cos x − 1) = 0: π/2, 3π/2, π/3, 5π/3.

7. 2 sin x cos x/(2cos2x) = tan x.

25 Summary and topic map

Idea	Remember
Pythagorean	cos2θ + sin2θ = 1
Tangent	tan θ = sin θ/cos θ
Double angle	sin 2θ = 2 sin θ cos θ; cos 2θ three forms
Basic equations	principal value, symmetry, period
Multiple angles	extend the interval
Quadratics	factorize in sin or cos
Identities in equations	reduce to one function; factorize

Before the exam, make sure you…

•find every solution in the interval
•never divide by a trig function
•choose the right form of cos 2θ
•check calculator mode

Trigonometric Identities and Equations 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 3. Topic map for Trigonometric Identities and Equations.

26 Answers to practice exercises

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

Practice exercise 1

1. −0.8

2. sin22x

3. 2

Practice exercise 2

1. π/4, 5π/4

2. sin θ = 2/√5, cos θ = 1/√5

Practice exercise 3

1. −7/25

2. 2 sin 2x cos 2x

3. sin 6x

4. (1 + cos 2x)/2

Practice exercise 4

1. expand: sin2 + cos2 + 2 sin cos

2. (sin2 + cos2)/(sin cos) = 1/(½ sin 2θ)

3. difference of squares, then cos2 − sin2

Practice exercise 5

1. 3π/4, 7π/4

2. −π/3, −2π/3

3. 66.4°, 293.6°

Practice exercise 6

1. π/6, π/2, 5π/6

2. π/2, 5π/2

3. u = 2x + π/3 ∈ [π/3, 7π/3]: u = 5π/6, 13π/6: x = π/4, 11π/12

Practice exercise 7

1. π/3, 2π/3, 4π/3, 5π/3

2. cos x = ½ (cos x = −2 rejected): 60°, 300°

3. 0, π/2, π, 2π

Practice exercise 8

1. 2 − 2sin2x = 3 sin x: sin x = ½: π/6, 5π/6

2. 2cos2x + cos x − 1 = 0: cos x = ½ or −1: π/3, π, 5π/3

3. 3s2 + 1 − 2s2 = 2 gives s2 = 1: π/2, 3π/2

Practice exercise 9

1. x = 0.739

2. 2 (x = 0 and x = 2.60), since x/5 > 1 for x > 5

Practice exercise 10

1. sin x

2. sin 2x

3. cos x

Practice exercise 11

1. (1 − cos 2x)/2

2. (2 − √3)/4

3. tan x

Practice exercise 12

1. common denominator; sin x(1 + sin x) + cos2x = 1 + sin x

2. (2 sin x cos x)/(2cos2x) = tan x

Practice exercise 13

1. 5π/12, 7π/12, 17π/12, 19π/12

2. 2π/3, 8π/3

Practice exercise 14

1. 1 − 2s2 = 3s + 2: 2s2 + 3s + 1 = 0: s = −1/2, −1: 7π/6, 3π/2, 11π/6

2. 2 − 2s2 = s + 1: s = ½ or −1: π/6, 5π/6, 3π/2

Practice exercise 15

1. tan 2x = 1: π/8, 5π/8

2. π/3, 2π/3

Practice exercise 16

1. 3 (x = 1.43, 5.27, 7.07); for x > 10 the line is above 1

2. 1.03

Practice exercise 17

1. sin(πt/6) < −½: 7 < t < 11

Practice exercise 18

1. sin x(1 + 2cos x) = 0: 0, 2π/3, π, 4π/3, 2π

2. cross-multiply: sin2x = 1 − cos2x

Practice exercise 19

1. π/3, 2π/3, 4π/3, 5π/3

Practice exercise 20

1. cos x = 2√2/3: sin 2x = 4√2/9, cos 2x = 7/9: tan 2x = 4√2/7

revisionprep.comIB-style notes · question bank · mock papers