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TRIGONOMETRY AND COMPOUND ANGLESLearning Outcomes and Assessment Standards

Learning Outcome 3: Space, shape and measurementAssessment Standard 12.3.5 Derive and use the following compound angle identities:

cos(α ± β) = cos α cos β ∓ sin α sin βsin (α ± β) = sin α sin ± cos α cos β

sin 2α = 2 sin α cos αcos 2α = cos2α − sin2 α

= 1 − 2 sin2α= 2 cos2 α − 1

OverviewIn this lesson you will:

establish formulae for double angles

use the double angle formulae to solve problems.

Lesson Formula you will use from previous lessons

cos (α ± β) = cos α cos β ∓ sin α sin β

sin (α ± β) = sin α sin β ± cos α cos β

sin2 α + cos2α = 1

Formula for sin 2A

sin 2A = sin (A + A)

= sin A cos A + sin A cos A

= 2 sin A cos A

Formula for cos 2A

cos 2A = cos (A + A)

= cos A cos A + sin A sin A

= cos2 A − sin2 A

In terms of cos A

cos 2A = cos2A − sin2A

= cos2A − (1 − cos2A)

= 2 cos2A − 1

In terms of sin A

cos 2A = cos2A − sin2A

= 1 − sin2A − sin2A

= 1 − 1 sin2A

Two identities to remember

cos2A + sin2A = 1 → variation: –cos2A – sin2A = –1

cos2A − sin2A = cos 2A → variation: sin2A − cos2A = −cos 2A

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Using identities to solve problems

Example 1

Prove that 1 + sin 2A __ cos 2A = cos A + sin A __ cos A – sin A

Solution:

LHS 1 + 2 sin A cos A ___ cos2A = sin2A

Change 1 to cos2 A + sin2 A so you can factorise

= sin2A + 2 sin A cos A + cos2 A ____ (cos A – sin A)(cos A + sin A)

= (sin A + cos A)2

____ (sin A + cos A)(cos A – sin A)

= sin A + cos A __ cos A – sin A

= RHS

Example 2

Express sin 3α in terms of sinα only

Solution:

sin 3α = sin(2α + α)

= sin 2α + cos α + cos 2α sin α = 2 sin α cos α . cos α + sin α (1 − 2 sin2 α)

= 2 sin α cos2α + sin α − 2 sin3 α = 2 sin α(1 − sin2α) + sin α − 2 sin3 α = 2 sin α − 2 sin3α + sin α − 2 sin3α = 3 sin α − 4 sin3α

Example 3

If cos 22° = p find in terms of p

a) sin 44° b) cos 224° c) sin 52°

Solution:

cos 22° = p then sin 22° = √ _ 1 – p2

a) sin 44° = sin 2(22°)

= 2 sin 22° cos 22°

= 2p √ _ 1 – p2

b) cos 224° = −cos 44°

= −cos2(22°)

= −(2 cos222° − 1)

= −(2p2 − 1)

= 1 − 2p2

c) sin 52° = sin (22° + 30°)

= sin 22° cos 30° + cos 22° sin 30°

= √ _ 1 – p2 √

_ 3 _ 2 + p· 1 _ 2

= √ _ 3 – 3p2 _ 2 + p _ 2

Remember:

If cos 22 = p, then sin 68 = p

sin222 = 1 – cos2 22°

Remember:

If cos 22 = p, then sin 68 = p

sin222 = 1 – cos2 22°

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Activity 11. If sin x = 2 _ 5 find the value of cos 2x with no calculator.

2. If cos 2x = 0,28 find the value of cos x with no calculator.

3. Find sin θ + 1 if cos 2θ = 0,02 (no calculator).

4. If cos 2B = 0.96 find tan B with no calculator.

5. Express cos 3β in terms of cos β.

Activity 21. Prove that: tan x = 1 – cos 2x + sin 2x ___

1 + cos 2x + sin 2x

2. If sin 2θ = – √ _ 11 _ 6 ; 180° ≤ 2θ ≤ 270° , without a calculator, calculate

a) cos 2θ b) cos θ c) sin θ

3. If sin 12° = p find in terms of p

a) cos 204° b) cos 66° c) sin 42°

4. Prove that: 2 sin 5α cos 4α − sin 9α = sin α

5. Prove that: cos 4x _ sin 2x

+ sin 4x _ cos 2x = 1 _

sin 2x

6. If A + B = 90°, show that

cos2 (A + B) + cos2 (A − B) = 4 cos2 B sin2 B

Activity 31. Prove that: 1 – cos 2x – sin x ___

sin 2x – cos x = tan x

2. Prove that: 2cos 5α cos 3α − cos 8α + 2sin2 α = 1

3. a) Prove that: 2 cos2 (45° − A) = 1 + sin 2A

b) Use this identify to calculate the value of cos 15°, without a calculator.

4. If sin 40° = a express the following in terms of a

a) sin 140° b) cos 140° c) sin 100°

5. Without a calculator evaluate

a) 2 cos 15° __ sin 255°

b) sin (x + 300°)

__ cos (150° – x)

6. If sin A = p and cos B = – √ _ 1 – p2 where p > 0; 90° < A < 360° and

180° < B < 360°, prove that A + B = 360°.

7. Simplify (no calculator): ( 1 _ cos 15° + 1 _ sin 15° ) 2

Activity 41. If cos2 25° = C, calculate in terms of C, with no calculator the value of

cos 20° cos 70° + sin 20° sin 70°.

2. If 1 − sin2 17° = k find the value of cos 51° in terms of k.

3. If sin 28° = m find in terms of m sin2 14°.

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4. Answer this question without a calculator.

In �ABC, C is an obtuse angle.

Calculate the value of sin A, if sin B = 3 _ 5 and sin C = 12 _ 13

5. Show that cos 6x + cos 2x = 2cos 4x cos 2x

6. Simplify without a calculator tan 15°·tan 75° __ sin 15° sin 75°

7. Simplify without a calculator:

a) (sin 75° − cos 75°)(sin 75° + cos 75°)

b) (sin 75° − cos 75°)2

Activity 5No calculator can be used.

1. Evaluate

a) cos 22,5° (b) sin 22,5°

2. Prove that 8 sin A cos A·cos 2A cos 4A = sin 8A

3. Prove that:

a) sin 2x + 2sin2 (45 − x) = 1 b) Hence show that sin2 15° = 2 – √ _ 3 _ 4

4. a) Prove that cos 2A + cos A __ sin2 A – sin A

= cos A + 1 _ sin A tan 15° = √ _ 3 – 1 _

√ _ 3 + 1

b) Hence show that cos 4θ + cos2θ __ sin 4θ – sin 2θ = 1 _ tan θ

c) Hence show that tan 15° = √ _ 3 – 1 _

√ _ 3 + 1

5. Prove that:

a) sin 3x − sinx = 2 cos 2x sin x b) sin 3x + sin x __ 1 + cos 2x = 2 sin x

c) cos 8x + cos 4x = 2 cos 6x cos 2x d) sin 42 + sin 18 = cos 12°

e) sin 70° − sin 50° = sin 10°

6. If A and B are both acute angles and sin A = 1 _ √

_ 5 and sin B = 1 _

√ _ 10 , prove that

A + B = 45°

7. a) Prove that cos 5A + cos A = 2 cos 3A cos 2A

b) Prove that cos 80° + cos 40° ___ cos 130° – cos 10° = – 1 _ √

_ 3

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