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Amy Ferguson Moncure

IB AA · Unit 2 · 18B Trigonometric Functions

Exact Lengths and Simplifying

Use special-triangle values, a turn around the circle, or the cosine rule with cos 60° and cos 120°.

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Key Ideas Easy Medium Hard

c² = a² + b² − 2ab cos C

cos 60° = 12

y = x tan θ

What to Remember

Exact sine and cosine

sin 45° = cos 45° = square root of 22

sin 30° = 12, and cos 30° = square root of 32

tan π3 = square root of 3

A sum around the circle

sin(2π − x) = −sin x

cos(x + π2) = −sin x

cos(x + π) = −cos x

cos(x + 3π2) = sin x

An exact side

c² = a² + b² − 2ab cos C

cos 60° = 12, and cos 120° = −12.

A line through the origin at angle θ has equation y = x tan θ.

Easy

Special values

Substitute the exact sine, cosine, or tangent, then simplify.

Example
Find the exact value of (tan π3 + 1)².

tan π3 = square root of 3

(tan π3 + 1)² = (square root of 3 + 1)²

(square root of 3 + 1)² = (square root of 3)² + 2(square root of 3)(1) + 1²

= 3 + 2square root of 3 + 1

= 4 + 2square root of 3

Answer: 4 + 2square root of 3

1
Write tan π3 + sin π3 in the form ksquare root of 3.
Show Answer

tan π3 = square root of 3

sin π3 = square root of 32

tan π3 + sin π3 = square root of 3 + square root of 32

= 2square root of 32 + square root of 32

= 3square root of 32

Answer: 3square root of 32

2
Show that sin 45° cos 45° + cos 45° sin 45° = 1.
Show Answer

sin 45° = square root of 22

cos 45° = square root of 22

sin 45° cos 45° = square root of 22 × square root of 22

= 24

= 12

The sum is 12 + 12

= 1

Answer: 1

3
Find the exact value of sin 30° cos 30° + cos 60° sin 60°.
Show Answer

sin 30° = 12

cos 30° = square root of 32

sin 30° cos 30° = 12 × square root of 32

= square root of 34

cos 60° = 12

sin 60° = square root of 32

cos 60° sin 60° = square root of 34

The sum is square root of 34 + square root of 34

= square root of 32

Answer: square root of 32

4
Write sin π3 + cos π6 in the form ksquare root of 3.
Show Answer

sin π3 = square root of 32

cos π6 = square root of 32

sin π3 + cos π6 = square root of 32 + square root of 32

= square root of 3

Answer: square root of 3

5
Write tan π6 + tan π3 in the form ksquare root of 3.
Show Answer

tan π6 = 1square root of 3

tan π6 = square root of 33

tan π3 = square root of 3

tan π6 + tan π3 = square root of 33 + square root of 3

= square root of 33 + 3square root of 33

= 4square root of 33

Answer: 4square root of 33

Medium

A sum of shifts

Replace each shifted sine or cosine, then collect like terms.

Example
Show that sin x + sin(2π − x) = 0.

sin(2π − x) = −sin x

sin x + sin(2π − x) = sin x + (−sin x)

= 0

Answer: 0

1
Show that sin x + sin(π − x) = 2 sin x.
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sin(π − x) = sin x

sin x + sin(π − x) = sin x + sin x

= 2 sin x

Answer: 2 sin x

2
Show that cos x + cos(2π − x) = 2 cos x.
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cos(2π − x) = cos x

cos x + cos(2π − x) = cos x + cos x

= 2 cos x

Answer: 2 cos x

3
Show that cos x + cos(π − x) = 0.
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cos(π − x) = −cos x

cos x + cos(π − x) = cos x + (−cos x)

= 0

Answer: 0

4
Simplify cos x + cos(x + π2) + cos(x + π) + cos(x + 3π2) + cos(x + 2π).
Show Answer

cos(x + π2) = −sin x

cos(x + π) = −cos x

cos(x + 3π2) = sin x

cos(x + 2π) = cos x

The sum is cos x + (−sin x) + (−cos x) + sin x + cos x

The sin x terms add to 0.

cos x − cos x + cos x = cos x

The sum is cos x.

Answer: cos x

5
Show that sin x + sin(π + x) = 0.
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sin(π + x) = −sin x

sin x + sin(π + x) = sin x + (−sin x)

= 0

Answer: 0

Hard

An exact side, or a line

cos 120° is negative, so the middle term of the cosine rule adds. A 60° line has slope √3.

Example
In triangle ABC, a = 5 cm, b = 8 cm, and the included angle C = 60°. Find the exact length of side c.

c² = a² + b² − 2ab cos C

c² = 5² + 8² − 2(5)(8) cos 60°

cos 60° = 12

c² = 25 + 64 − 80 × 12

c² = 89 − 40

c² = 49

c = 7 cm

Answer: 7 cm

1
In triangle ABC, a = 3 cm, b = 5 cm, and the included angle C = 120°. Find the exact length of side c.
Show Answer

c² = a² + b² − 2ab cos C

c² = 3² + 5² − 2(3)(5) cos 120°

cos 120° = −12

c² = 9 + 25 − 30 × (−12)

30 × (−12) = −15

c² = 34 − (−15)

c² = 34 + 15

c² = 49

c = 7 cm

Answer: 7 cm

2
In triangle ABC, a = 6 cm, b = 3 cm, and the included angle C = 120°. Find the exact length of side c.
Show Answer

c² = a² + b² − 2ab cos C

c² = 6² + 3² − 2(6)(3) cos 120°

cos 120° = −12

c² = 36 + 9 − 36 × (−12)

36 × (−12) = −18

c² = 45 − (−18)

c² = 45 + 18

c² = 63

c = square root of 63

63 = 9 × 7

c = 3square root of 7 cm

Answer: 3square root of 7 cm

3
In triangle ABC, a = 5 cm, b = 5 cm, and the included angle C = 120°. Find the exact length of side c.
Show Answer

c² = a² + b² − 2ab cos C

c² = 5² + 5² − 2(5)(5) cos 120°

cos 120° = −12

c² = 25 + 25 − 50 × (−12)

50 × (−12) = −25

c² = 50 − (−25)

c² = 50 + 25

c² = 75

c = square root of 75

75 = 25 × 3

c = 5square root of 3 cm

Answer: 5square root of 3 cm

4
In triangle ABC, a = 4 cm, b = 6 cm, and the included angle C = 120°. Find the exact length of side c.
Show Answer

c² = a² + b² − 2ab cos C

c² = 4² + 6² − 2(4)(6) cos 120°

cos 120° = −12

c² = 16 + 36 − 48 × (−12)

48 × (−12) = −24

c² = 52 − (−24)

c² = 52 + 24

c² = 76

c = square root of 76

76 = 4 × 19

c = 2square root of 19 cm

Answer: 2square root of 19 cm

5
A line through the origin makes a 60° angle with the positive x-axis. Find its equation. Then show that tan 60° = tan 240°, and simplify tan 60° × tan 150°.
Show Answer

The slope of a line through the origin is the tangent of the angle it makes with the positive x-axis.

m = tan θ

m = tan 60°

tan 60° = square root of 3

y = square root of 3 x

240° = 60° + 180°

tan(θ + 180°) = tan θ

tan 240° = tan 60°

150° = 180° − 30°

tan(180° − θ) = −tan θ

tan 30° = 1square root of 3

tan 30° = square root of 33

tan 150° = −square root of 33

tan 60° × tan 150° = square root of 3 × (−square root of 33)

= −33

= −1

Answer: y = square root of 3 x; tan 60° = tan 240°; the product is −1

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