Trigonometric Proof (Edexcel International A Level Maths: Pure 3)

Exam Questions

3 hours40 questions
1
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2 marks

Show that

cot space thetaidentical to fraction numerator cos space theta over denominator sin space theta end fraction

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2a
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2 marks

Use the identity

cosopen parentheses A plus B close parentheses identical tocos Acos B minussin Asin B

to show that

cos 2 A identical tocos2 A minussin2 A

2b
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3 marks

Show by counter-example that

cos 2 theta not identical tocos theta pluscos theta

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3a
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3 marks

Given that is theta small and measured in radians, use an appropriate approximation to show that

3 sin theta minus 2cos theta almost equal to theta squared plus 3 theta minus 2

3b
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1 mark

Use the result in part (a) to find an approximation to 3 sinopen parentheses 0.2 close parentheses minus 2 cosopen parentheses 0.2 close parentheses.

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4
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2 marks

Prove the identity

fraction numerator sin space 2 theta over denominator 2 space sin space theta end fraction identical to cos space theta comma space space space space space space space space space space space space theta not equal to kπ

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5
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4 marks

Show that

sin squared space theta open parentheses sec squared space theta plus cosec squared space theta close parentheses identical to sec squared space theta

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6
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5 marks
(i)
Use the quotient rule to show that

fraction numerator d over denominator d x end fraction open square brackets cosec space x close square brackets equals fraction numerator negative cos space x over denominator sin squared space x end fraction

(ii)
Hence show that

fraction numerator d over denominator d x end fraction open square brackets cosec space x close square brackets equals negative cot space x space cosec space x

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7
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3 marks

Show that

3 sin 2 theta minus 2 sin theta identical to 2 sin theta space open parentheses 3 space cos space theta minus 1 close parentheses

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8
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5 marks

Prove the identity

2 cosec 2 xcot x identical tocosec2 x comma space space space space space space space space space space space x not equal to fraction numerator k straight pi over denominator 2 end fraction

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9a
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2 marks

Find the value of

(i)
arccos open parentheses cos open parentheses 150 degree close parentheses close parentheses

(ii)
arcsin open parentheses sin open parentheses 210 degree close parentheses close parentheses
9b
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2 marks

Explain why the answer to part (a) (ii) is not 210 degree.

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10
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3 marks

Use the identity

R sinopen parentheses theta plus straight alpha close parentheses identical to R spacecos alpha sin theta plus R spacesin alpha cos theta

to show that

4 sinopen parentheses theta plus straight pi over 4 close parentheses identical to 2 square root of 2 open parentheses sin space theta plus cos space theta close parentheses

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1
Sme Calculator
4 marks

Given the identity

sin2 theta pluscos2 theta identical to 1

prove the following identities:

(i)

sec2 theta identical to 1 plustan2 theta

(ii)
cosec2 theta identical to 1 pluscot2 theta

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2
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5 marks
(i)
By using the double angle formula for cosine, prove the identity

cos 4 theta identical to 8 cos4 theta minus 8 cos2 theta plus 1

(ii)
Show by counter-example that

sin 4 theta not identical to 8 sin4 theta minus 8 sin2 theta plus 1

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3a
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3 marks

Given that theta is small, and that terms involving theta cubed or higher powers of theta can be ignored, use an appropriate approximation to show that

4 spacecos 4 theta minus 2 spacecos2 2 theta almost equal to 2 minus 24 theta squared

3b
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3 marks

Show that the result in part (a) gives a percentage error of 0.583%, to 3 significant figures, when used to approximate

4 spacecosstraight pi over 6 minus 2cos2 straight pi over 12

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4
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3 marks

Prove the identity

fraction numerator 4 space sin to the power of 4 space theta over denominator sin squared space 2 theta end fraction identical to tan squared space theta space space space space space space space space space space theta not equal to kπ

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5
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4 marks

Show that

sin theta open parentheses cosec squared space theta minus 2 close parentheses identical to fraction numerator cos space 2 theta over denominator sin space theta end fraction

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6
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5 marks

Use the quotient rule to show that

fraction numerator d over denominator d x end fraction open square brackets tan space x close square brackets equals sec squared space x

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7
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5 marks

Show that

sin 3 theta plussin theta identical to 4 sin theta minus 4 spacesin3 theta

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8
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5 marks

Prove the identity

fraction numerator 4 space cot space x space cos space 2 x over denominator sin space 4 x end fraction identical to cosec squared space x italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space x not equal to kπ over 4

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9a
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2 marks

Show that

sinopen parentheses arccos open parentheses negative 1 half close parentheses close parentheses equals square root of 3 space sin open parentheses straight pi over 6 close parentheses

9b
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2 marks

Show that

arcsin open parentheses cos fraction numerator 3 straight pi over denominator 4 end fraction close parentheses equals negative arcsin open parentheses cos straight pi over 4 close parentheses

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10
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3 marks

Show that

square root of 2 sin open parentheses theta minus straight pi over 4 close parentheses identical to sin space theta minus cos space theta

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1
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4 marks

Given the identity

cosopen parentheses A plus B close parentheses equalscos A cos Bminussin A sin B

prove the following identities:

(i)

cos 2 theta identical tocos2 theta minussin2 theta

(ii)

cos 2 theta identical to 1 minus 2 spacesin2 theta

(iii)
cos 2 theta identical to 2 spacecos2 theta minus 1

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2
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5 marks
(i)
Prove the identity

sin 3 theta identical to 3 sin theta minus 4 sin3 theta

(ii)
Show by counter-example that

cos 3 theta not identical to 3 cos theta minus4 cos3 theta

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3a
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3 marks

Given that theta is small, and that terms involving theta cubed or higher powers of theta can be ignored, show that

fraction numerator 1 over denominator cosec squared space open parentheses theta over 2 close parentheses end fraction plus fraction numerator 1 over denominator sec squared space open parentheses theta over 4 close parentheses end fraction almost equal to 1 plus 3 over 16 theta squared

3b
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3 marks

Determine the percentage error when the result in part (a) is used to approximate

fraction numerator 1 over denominator cosec squared space open parentheses 7 over 20 close parentheses end fraction plus fraction numerator 1 over denominator sec squared open parentheses 7 over 40 close parentheses end fraction

giving your answer correct to 3 significant figures.

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4
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5 marks

Show that

cos 4 theta pluscos straight pi over 3 identical to 8 sin4 theta minus 8 sin2 theta plus 3 over 2

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5
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5 marks

Prove that

cot2 theta minustan2 theta identical to 4 cot 2 theta cosec 2 theta.

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6
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5 marks

Prove the identity

fraction numerator 1 minus tan squared space x over denominator cos space 2 x end fraction identical to sec squared space x space space space space space space space space space space space space x not equal to fraction numerator 2 straight k plus 1 over denominator 4 end fraction straight pi

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7
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4 marks

Prove the identity

cosec x identical to fraction numerator 1 half space sec squared x over 2 over denominator tan x over 2 end fraction

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8
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5 marks

Show that

tanx over 2 identical to fraction numerator 1 over denominator cosec space x plus cot space x end fraction space space space space space space space space space space space space space x not equal to 2 kπ

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9a
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3 marks

Given that y equals arcsin open parentheses k x close parentheses, where k is a constant, show that   x equals 1 over k cos open parentheses straight pi over 2 minus y close parentheses .

9b
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3 marks

Hence show that the value of arcsin space k x plus arccos space k x is constant and independent of k.
Find the value of this constant.

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10
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5 marks

Show that

fraction numerator 10 over denominator 4 space cos space theta plus 3 space sin space theta end fraction identical to 2 space sec open parentheses theta minus straight alpha close parentheses

where

alpha equals arctan open parentheses 3 over 4 close parentheses

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1
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6 marks

Consider the three triangles, all of height 1, as shown below.

q1-5-8-trignometric-proof-a-level-only-edexcel-a-level-pure-maths-veryhard

By applying the area of a triangle formula A equals 1 half ab space sin space C  to each one, prove that,

sinopen parentheses A plus B close parentheses identical tosin A cos B plussin B cos A

Briefly explain why this only proves the result for A and B being acute angles.

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2
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4 marks

Prove the identity

tan space 4 theta identical to fraction numerator 4 space tan space theta open parentheses 1 minus tan squared space theta close parentheses over denominator 1 minus 6 space tan squared space theta plus tan to the power of 4 space theta end fraction

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3a
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4 marks

Use the small angle approximations for sine and cosine to confirm the following two limit results:

limit as straight h rightwards arrow 0 of fraction numerator sin space h over denominator h end fraction equals 1 space space space space space space space space space space limit as straight h rightwards arrow 0 of fraction numerator cos space h minus 1 over denominator h end fraction equals 0

Be sure to explain why use of the small angle approximations is justified here.

3b
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5 marks

Hence prove from first principles that

fraction numerator straight d over denominator d x end fraction open square brackets sin space x close square brackets equals cos space x

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4
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5 marks

Prove the identity

negative 16 cot 2 theta cosec3 2 theta identical tosec4 theta minuscosec4 theta

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5
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4 marks

Show that

fraction numerator square root of 2 cos open parentheses theta plus straight pi over 4 close parentheses over denominator sin open parentheses theta minus straight pi over 2 close parentheses end fraction identical to tan space theta minus 1

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6a
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4 marks

Show that

sin 3 theta identical to 3 sin straight theta cos2 theta minussin3 theta

6b
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5 marks

Hence, or otherwise, show that

fraction numerator cos space 3 theta minus cos space theta over denominator sin space 3 theta space sin space theta end fraction identical to fraction numerator 4 space cos space theta over denominator 1 minus 4 space cos squared space theta end fraction space space space space space space space space space space theta not equal to kπ

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7
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5 marks

Show that

4cos2 open parentheses x minus straight pi over 6 close parentheses identical to 3 minus 2sin2 x plus square root of 3 sin 2 x

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8
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6 marks

Show that

tanopen parentheses fraction numerator 2 x plus straight pi over denominator 4 end fraction close parentheses identical tosec x plustan x

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9a
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4 marks

On the axes below sketch the graphs of

y equals arccos open parentheses negative x close parentheses  and  y equals open vertical bar arcsin space x close vertical bar

q9a-5-8-trignometric-proof-a-level-only-edexcel-a-level-pure-maths-veryhard

9b
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2 marks

With the help of your sketch, determine the exact solution(s) to the equation

arccos open parentheses negative x close parentheses equals open vertical bar arcsin space x close vertical bar

9c
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2 marks

What can you say about the solution(s) to the equation

open vertical bar arccos space x close vertical bar equals arcsin open parentheses negative x close parentheses ?

Justify your answer.

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10
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9 marks

Show that

1 over open parentheses fraction numerator square root of 3 over denominator 2 end fraction cos space theta minus 1 half sin space theta close parentheses squared plus 1 over open parentheses fraction numerator square root of 3 over denominator 2 end fraction sin space theta plus 1 half cos space theta close parentheses squared identical to 4 space cosec squared open parentheses 2 theta plus straight pi over 3 close parentheses

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