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Amy Ferguson Moncure

IB AA · Unit 2 · 18B Trigonometric Functions

Related Angles: Sine

Rewrite the angle by a full turn, a half turn, or a right angle. The value is sin θ or its opposite.

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Key Ideas Easy Medium Hard

sin(θ + 2π) = sin θ

sin(θ + π) = −sin θ

cos(π2 − θ) = sin θ

What to Remember

Full turns and half turns

sin(θ + 2π) = sin θ

sin(θ − 2π) = sin θ

sin(θ + π) = −sin θ

sin(θ − π) = −sin θ

The angle on the other side

sin(π − θ) = sin θ

sin(2π − θ) = −sin θ

sin(−θ) = −sin θ

A right-angle shift

cos(π2 − θ) = sin θ

cos(θ + π2) = −sin θ

Cosine of a right-angle shift is sine, and the sign follows the quadrant.

Easy

Add or subtract a half turn

A full turn keeps the value. A half turn changes the sign.

Example
Given that sin θ = 0.42, find sin(θ + 2π).

sin(θ + 2π) = sin θ

sin(θ + 2π) = 0.42

Answer: 0.42

1
Given that sin θ = 0.42, find sin(θ − 2π).
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sin(θ − 2π) = sin θ

sin(θ − 2π) = 0.42

Answer: 0.42

2
Given that sin θ = 0.42, find sin(θ + 4π).
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4π = 2 × 2π

sin(θ + 4π) = sin θ

sin(θ + 4π) = 0.42

Answer: 0.42

3
Given that sin θ = 0.42, find sin(θ + π).
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sin(θ + π) = −sin θ

sin(θ + π) = −0.42

Answer: −0.42

4
Given that sin θ = 0.42, find sin(θ − π).
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sin(θ − π) = −sin θ

sin(θ − π) = −0.42

Answer: −0.42

5
Given that sin θ = 0.42, find sin(θ + 3π).
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3π = π + 2π

sin(θ + 3π) = sin(θ + π)

sin(θ + π) = −sin θ

sin(θ + 3π) = −0.42

Answer: −0.42

Medium

Reflect the angle

sin(π − θ) matches sin θ. A clockwise turn changes the sign.

Example
Given that sin θ = 0.42, find sin(2π − θ).

sin(2π − θ) = −sin θ

sin(2π − θ) = −0.42

Answer: −0.42

1
Given that sin θ = 0.42, find sin(π − θ).
Show Answer

sin(π − θ) = sin θ

sin(π − θ) = 0.42

Answer: 0.42

2
Given that sin θ = 0.42, find sin(−θ).
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sin(−θ) = −sin θ

sin(−θ) = −0.42

Answer: −0.42

3
Given that sin θ = 0.42, find sin(3π − θ).
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3π − θ = (π − θ) + 2π

sin(3π − θ) = sin(π − θ)

sin(π − θ) = sin θ

sin(3π − θ) = 0.42

Answer: 0.42

4
Given that sin θ = 0.42, find sin(−π − θ).
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sin(−π − θ) = −sin(π + θ)

sin(π + θ) = −sin θ

sin(−π − θ) = sin θ

sin(−π − θ) = 0.42

Answer: 0.42

5
Given that sin θ = 0.42, find sin(4π − θ).
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4π − θ = −θ + 2 × 2π

sin(4π − θ) = sin(−θ)

sin(−θ) = −sin θ

sin(4π − θ) = −0.42

Answer: −0.42

Hard

Switch to cosine

A shift of π/2 turns sine into cosine.

Example
Given that sin θ = 0.42, find cos(π2 − θ).

cos(π2 − θ) = sin θ

cos(π2 − θ) = 0.42

Answer: 0.42

1
Given that sin θ = 0.42, find cos(θ + π2).
Show Answer

cos(θ + π2) = −sin θ

cos(θ + π2) = −0.42

Answer: −0.42

2
Given that sin θ = 0.42, find cos(θ − π2).
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cos(θ − π2) = sin θ

cos(θ − π2) = 0.42

Answer: 0.42

3
Given that sin θ = 0.42, find cos(3π2 − θ).
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3π2 − θ = π + (π2 − θ)

cos(π + α) = −cos α

cos(3π2 − θ) = −cos(π2 − θ)

cos(π2 − θ) = sin θ

cos(3π2 − θ) = −0.42

Answer: −0.42

4
Given that sin θ = 0.42, find cos(θ + 3π2).
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θ + 3π2 = (θ + π2) + π

cos(θ + 3π2) = −cos(θ + π2)

cos(θ + π2) = −sin θ

cos(θ + 3π2) = sin θ

cos(θ + 3π2) = 0.42

Answer: 0.42

5
Given that sin θ = 0.42, find cos(5π2 − θ).
Show Answer

5π2 − θ = (π2 − θ) + 2π

cos(5π2 − θ) = cos(π2 − θ)

cos(π2 − θ) = sin θ

cos(5π2 − θ) = 0.42

Answer: 0.42

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