scholar study guide cfe higher mathematics ...2014/01/08 · scholar study guide cfe higher...

SCHOLAR Study Guide

CfE Higher MathematicsAssessment Practice 6:Trigonometry

Authored by:Margaret Ferguson

Reviewed by:Jillian Hornby

Previously authored by:Jane S Paterson

Dorothy A Watson

Heriot-Watt University

Edinburgh EH14 4AS, United Kingdom.

First published 2014 by Heriot-Watt University.

This edition published in 2017 by Heriot-Watt University SCHOLAR.

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SCHOLAR Study Guide Assessment Practice: CfE Higher Mathematics

1. CfE Higher Mathematics Course Code: C747 76

AcknowledgementsThanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created thesematerials, and to the many colleagues who reviewed the content.

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The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum.

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1

Topic 7

Trigonometry

Contents7.1 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

7.2 Assessment practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 TOPIC 7. TRIGONOMETRY

Learning objective

You should be able to:

• convert between degrees and radians or between radians and degrees;

• sketch trigonometric graphs in degrees and radians;

• solve trigonometric equations in:

◦ degrees or radians with a calculator;

◦ degrees or radians as exact values;

• expand the

◦ addition formulae for sin or cos;

◦ double angle formulae for sin or cos;

• prove complex trigonometric identities;

• express the sum of two trig functions as a single trig function of the formk sin(x± a) or k cos(x± a)

• solve equations using the wave formula;

• find maximum and minimum values using the wave function.

By the end of this topic, you should have identified your strengths and areas for furtherrevision.

© HERIOT-WATT UNIVERSITY

TOPIC 7. TRIGONOMETRY 3

7.1 Learning points

TrigonometryDegrees and radians

• π radians = 180◦ and 2π radians = 360◦.

• To change degrees to radians multiply the degrees by π and divide by 180.

• To change radians to degrees replace the π symbol with multiply by 180 or multiply radians by180 and divide by π.

• It is much easier to work in degrees then change your final answer into radians.

• Learn the table of exact values or remember how to sketch the two triangles.

Sketching trigonometric graphs

• To sketch trigonometric graphs of the form y = a sin(bx + c) + d or y = a cos(bx + c) + d:

1. identify the amplitude a and vertical shift d.

2. identify the maximum and minimum values a ± d.

3. identify the number of cycles b and the period 360 ÷ b.

4. factorise the bracket b(x + c

b

).

5. identify the horizontal shift cb .

6. evaluate the y-intercept let x = 0.

7. remember to keep the basic shape of sin or cos.

Solving trigonometric equations

• When solving trigonometric equations make a quadrant chart to determine where solutions lie.

• Identify the period and make sure that you find all the solutions. The domain is usually 0 ≤x ≤ 360◦ or 0 ≤ x ≤ 2π but not always so be careful.

Trigonometric identities

• cos2x + sin2x = 1

• tanA = sinAcosA

Addition formula

• sin(A + B) = sin A cos B + cos A sin B

• sin(A − B) = sin A cos B − cos A sin B

• cos(A + B) = cos A cos B − sin A sin B

• cos(A − B) = cos A cos B + sin A sin B



Double angle formula

• sin 2x = 2sin x cos x

•cos 2A = cos2A − sin2A

= 2cos2 − 1

= 1 − 2sin2A

The wave function

• Expand the addition formula and write your expression underneath it.

• Identify the value of k sin α and k cos α, you may have to swap terms around or change signs.

• Find the value of k.

• Find the value of the phase angle α◦.

• Express the answer to the question.

• The maximum and minimum values are ± k.



7.2 Assessment practice

Make sure that you have read through the learning points and completed some revision beforeattempting these questions.

Go onlineAssessment practice: Trigonometry

SQA Past Paper: 2003 Paper 2

The diagram shows a sketch of part of the graph of a trigonometric function whose equationis of the form y = a sin bx + c.

Q1: What is the equation of this trigonometric function?

3 marks


Q2: Solve the equation sin 2x◦ − cos x◦ = 0 in the interval 0 ≤ x ≤ 180.

4 marks



The diagram shows part of two trigonometric graphs, y = sin 2x◦ and y = cos x◦. Use yoursolutions to the previous parts of this question to find P .

Q3: What are the coordinates of the point P?

1 mark


Q4: Express 8cos x◦ − 6sin x◦ in the form kcos(x + a)◦ where k > 0 and 0 < a < 360.

4 marks


In triangle ABC, find the exact value of sin(a + b).

Q5: What is the exact value of sin(a + b)?

4 marks



Functions f(x) = sin x, g(x) = cos x and h(x) = x +π4 are defined on a suitable set of

real numbers.

Q6: What are suitable expressions for f(h(x)) and g(h(x))?

2 marks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q7: Show that f (h (x)) = 1√2sinx + 1√

2cos x and give a similar expression for g(h(x)).

2 marks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q8: Hence solve the equation f(h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π;.

3 marks


The diagram shows the graph of a cosine function from 0 to π.

Q9: What is the equation of the graph?

1 mark

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q10: The line y = −√3 intersects the graph at the points A and B. What are the

coordinates of B?

3 marks




Q11: Express sin x◦ − cos x in the form ksin(x − α) where k > 0 and 0 < a < 2π.

4 marks


A is the point (8,4).The line OA is inclined at an angle of p radians to the x-axis.

Q12: What is the exact value of sin 2p?

3 marks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q13: What is the exact value of cos 2p?

2 marks

The line OB is inclined at an angle 2p radians to the x-axis.

Q14: What is the exact value of the gradient of OB?

1 mark


Q15: f(x) = cos 2x − 3sin 4x What is the exact value of f ′ (π6

)?

4 marks




Part of the graph of y = 2sin x◦ + 5cos x◦ is shown in the diagram.

Q16: Express y = 2sin x◦ + 5cos x◦ in the form ksin(x + a)◦ where k > 0 and 0 ≤ a < 360.

4 marks

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q17: What are the coordinates of the minimum turning point P?

3 marks


Q18: Solve the equation 3 cos 2x + 10cos x − 1 = 0 for 0 ≤ x ≤ π, correct to 2 decimalplaces.

5 marks


In the diagram ∠DEC = ∠CEB = x◦ and ∠CDE = ∠BEA = 90◦. CD = 1 unit andDE = 3 units.

Q19: By writing ∠DEA in terms of x◦, what is the exact value of cos(DEA)?

7 marks


10 ANSWERS: UNIT 2 TOPIC 6

Answers to questions and activities

Topic 6: Trigonometry

Assessment practice: Trigonometry (page 5)

Q1:

Hints:

• Find the amplitude.• The graph has a period of π radians or 180◦.• The amplitude tells us the maximum and minimum values. This graph has a maximum of 5 and

a minimum of -3.

Steps:

• What is the amplitude? 4• How many repeats of the basic sin curve are there in 2π? 2• By how many units has the graph of y = 4 sin 2x been moved up or down? 1

Answer: y = 4 sin 2x + 1

Q2:

Hints:

• sin2x is a double angle formula, go to the formula sheet.• Factorise your equation.

Steps:

• From the formula sheet what is sin 2x? 2sin x cos x

• Substitute this into the equation, factorise and solve.

Answer:2 sinx cos x − cos x = 0

cos x (2 sin x − 1) = 0

cos x = 0 or 2 sinx = 1

x = 90◦ sinx =1

2x = 30◦, 150◦

x = 30, 90, 150 Notice all solutions are in the interval 0 ≤ x ≤ 180

Q3:

Hints:

• This is the largest solution for x.

Steps:

• What is the x-coordinate of P? 150• Substitute this value into one of the equations to find the exact value of the y-coordinate.


ANSWERS: UNIT 2 TOPIC 6 11

Answer:y = cos 150◦ remember cos is negative in the sin quadrant

y = - cos 30◦ the exact value of cos 30◦ =√3

2so P =

(150 , -

√32

)You need to know your exact values!

Q4:

Hints:

• Expand the addition formula for cos.• Use Pythagoras to find k.• Use the identity tan a = sin a

cos a to find the angle a.

Steps:

• What is the value of kcos a? 8• What is the value of ksin a? 6• Use these values and Pythagorus to find the value of k.• In which quadrant is a, A or S or T or C? A• What is tan a? 6

8

• Use this to find the value of a.

Answer: 10cos(x + 36 · 9)◦

Q5:

Hints:

• Go to the formula sheet and find the expansion for sin(a + b).

Steps:

• What is the length of AC?• What is the length of BC?

• What is the exact value of sin a? 1√2

• What is the exact value of sin b? 1√10

• What is the exact value of cos a? 1√2

• What is the exact value of cos b? 3√10

Answer:sin(a + b) = sin a cos b + cos a sin b

=1√2

× 3√10

+1√2

× 1√10

=4√20

=2√5



Q6:

f (h (x)) = sin(x + π

4

)and g (h (x)) = cos

(x + π

4

)Q7:

Hints:

• Use the addition formula rule for sin(A + B) with sin(x + π

4

).

• Use the addition formula rule for cos(A + B) with cos(x + π

4

).

Steps:

• Expand sin(x + π

4

). sinx cos π

4 + sin π4 cos x

• What is π4? 45◦

• What are cos 45◦ and sin 45◦? 1√2

• Substitute in your answers.• Simplify cos

(x + π

4

)in the same way and find a similar expression.

Answer:f (h (x)) = sin

(x +

π

4

)= sinx cos

π

4+ sin

π

4cos x

= sinx cos 45 + sin 45 cos x

=1√2sinx +

1√2cos x

g (h (x)) = cos(x +

π

4

)= cos x cos

π

4− sinx sin

π

4= cos x cos 45 − sinx sin 45

=1√2cos x − 1√

2sinx

Q8:

Hints:

• Substitute in the composite functions.• Remember to change solutions to radians by × π ÷ 180.

Answer:1√2sinx +

1√2cos x −

(1√2cosx − 1√

2sinx

)= 1

1√2sinx +

1√2cosx − 1√

2cos x +

1√2sinx = 1

2 × 1√2sinx = 1

√2 sinx = 1

sinx =1√2

∴ x =π

4,3π

4



Q9: y = 2cos 2x

Q10:

Hints:

• Solve 2 cos 2x = −√3 to find the x-coordinate of B.

Steps:

• What is cos 2x equal to?√32

• In which quadrant is x, A or S or T or C? S and T

• What is the first solution for x?

cos−1

(√3

2

)= 30◦

2x = 180 − 30 = 150

x = 75◦

• What is the second solution for x?2x = 180 + 30 = 210

x = 105◦

Answer:x =

105 × π

180

=7π

12Hence B has coordinates

(7π12 , −√

3).

Q11:

Hints:

• Expand ksin(x − α)◦.

Steps:

• What is the value of k?k cosα = 1

k sinα = 1

k =√

12 + 12

k =√2

• What is the value of a?k sinαk cosα = 1

1 ∴ tanα = 1

Both ksin α and kcos α are positive so there are two ticks in the ALL quadrant.α = 45◦ = π

4

Answer:√2 sin

(x − π

4

)



Q12:

Steps:

• What is the length of OA?√80

• What is the exact value of sin p? 4√80

• What is the exact value of cos p? 8√80

• Use the trig formula for sin 2A and your answers to find the exact value of sin 2p.

Answer:sin 2p = 2 sin p cos p

= 2 × 4√80

× 8√80

=64

80

=4

5

Q13:

Hints:

• Use the trig formula for cos 2A and your answers to find the exact value of cos 2p.

Steps:

• What is cos 2A? cos2 A − sin2 A

• Use this to solve cos 2p

Answer:cos 2p = cos2p − sin2p

=

(8√80

)2

−(

4√80

)2

=64

80− 16

80

=3

5

Q14:

Hints:

• Use the trig formula for tan 2A and your answers to find the exact value of mOB

Answer:mOB = tan 2p

tan 2p =sin 2p

cos 2p

=4

5÷ 3

5

=4

3



Q15:

Hints:

• Find the derivative of f(x) = cos 2x − 3sin 4x and solve for f ′ (π6

).

Steps:

• What is the derivative of cos 2x? −2sin 2x

• What is the derivative of − 3sin 4x? −12cos 4x

• What is −2 sin(2 × π

6

)? −√

3

• What is −12 cos(4 × π

6

)? 6

Answer:f ′(π6

)= −2 sin 2x − 12 cos 4x

= −2 sin(2 × π

6

)− 12 cos

(4 × π

6

)= −2 sin

π

3− 12 cos

2π

3= −2 × sin 60◦ − 12 cos 120◦

= −2 ×√3

2− 12 × −1

2

= 6 −√3

Q16:

Hints:

• Expand ksin(x + α)◦.

Steps:

• What is the value of k?k cosα = 2

k sinα = 5

k =√

22 + 52

=√29

• What is the value of a?k sinαk cosα = 5

2 ∴ tanα = 2 · 5

Both ksin α and kcos α are positive so there are two ticks in the ALL quadrant.

Answer:√29 sin (x + 68 · 2)◦



Q17:

Hints:

• y =√29 sin (x + 68 · 2)◦

• We know that the maximum value is√29 and the minimum value is −√

29.

Steps:

• What is the value of y at point P? y = −√29

• Substitute this value into the equation to solve for x.

Answer:−√29 =

√29 sin (x + 68 · 2)◦

sin (x + 68 · 2) = −1

x + 68 · 2 = 270

x = 201 · 8

Q18:

Hints:

• Decide what the best formula to us for cos 2x is.

• Substitute the appropriate formula into the equation, factorise and solve.

• Remember to give your answer in radians.

Steps:

• What is the best formula to use for cos 2x? 2cos2x − 1

• What is the factorised equation when the substitution has been made?3 cos 2x + 10 cos x − 1 = 0

3(2cos2x − 1

)+ 10 cos x − 1 = 0

6cos2x − 3 + 10 cos x − 1 = 0

6cos2x + 10 cos x − 4 = 0

2(3cos2x + 5cos x − 2

)= 0

2 (3 cos x − 1) (cos x + 2) = 0

Answer:



Q19:

Steps:

• What is ∠DEA in terms of x? 2x + 90

• What is an expression for cos(DEA)? cos (2x + 90)

• What is the expansion for your answer? cos 2x cos 90 − sin 2x sin 90

• What is the exact value of cos 90? 0

• What is the exact value of sin 90? 1

• What is an expression for cos(DEA) in its simplest form? −2sin x cos x

• What is the exact value of cos x? 3√10

• What is the exact value of sin x? 1√10

Answer: − 35


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