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Amy Ferguson Moncure

IB AA · Unit 2 · 18B Trigonometric Functions

Tangent Relationships

Write tangent as sine over cosine when you need a sign. Tangent repeats every π.

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Key Ideas Easy Medium Hard

tan(θ + π) = tan θ

tan(−θ) = −tan θ

tan(π2 − θ) = 1tan θ

What to Remember

Tangent repeats every half turn

tan θ = sin θcos θ

tan(θ + π) = tan θ

tan(−θ) = −tan θ

The complementary angle

tan(π2 − θ) = 1tan θ

tan(π2 + θ) = −1tan θ

Easy

Build the identity

Replace sine and cosine, then simplify the quotient.

Example
Show that tan(θ − π) = tan θ.

tan(θ − π) = sin(θ − π)cos(θ − π)

sin(θ − π) = −sin θ

cos(θ − π) = −cos θ

tan(θ − π) = −sin θ−cos θ

tan(θ − π) = sin θcos θ

tan(θ − π) = tan θ

Answer: tan(θ − π) = tan θ

1
Show that tan(θ + π) = tan θ.
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tan(θ + π) = sin(θ + π)cos(θ + π)

sin(θ + π) = −sin θ

cos(θ + π) = −cos θ

tan(θ + π) = sin θcos θ

tan(θ + π) = tan θ

Answer: tan(θ + π) = tan θ

2
Show that tan(−θ) = −tan θ.
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tan(−θ) = sin(−θ)cos(−θ)

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −sin θcos θ

tan(−θ) = −tan θ

Answer: tan(−θ) = −tan θ

3
Show that tan(π − θ) = −tan θ.
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tan(π − θ) = sin(π − θ)cos(π − θ)

sin(π − θ) = sin θ

cos(π − θ) = −cos θ

tan(π − θ) = sin θ−cos θ

tan(π − θ) = −tan θ

Answer: tan(π − θ) = −tan θ

4
Show that tan(2π − θ) = −tan θ.
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tan(2π − θ) = sin(2π − θ)cos(2π − θ)

sin(2π − θ) = −sin θ

cos(2π − θ) = cos θ

tan(2π − θ) = −sin θcos θ

tan(2π − θ) = −tan θ

Answer: tan(2π − θ) = −tan θ

5
Show that tan(θ + 2π) = tan θ.
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sin(θ + 2π) = sin θ

cos(θ + 2π) = cos θ

tan(θ + 2π) = sin θcos θ

tan(θ + 2π) = tan θ

Answer: tan(θ + 2π) = tan θ

Medium

Use a rounded value

The given tangent is positive. A half turn keeps it. An opposite angle changes the sign.

Example
Given that tan π8 = 0.414, find tan 9π8.

9π8 = π8 + π

tan(α + π) = tan α

tan 9π8 = tan π8

tan 9π8 = 0.414

Answer: 0.414

1
Given that tan π8 = 0.414, find tan(−π8).
Show Answer

tan(−θ) = −tan θ

tan(−π8) = −0.414

Answer: −0.414

2
Given that tan π8 = 0.414, find tan 7π8.
Show Answer

7π8 = π − π8

tan(π − θ) = −tan θ

tan 7π8 = −0.414

Answer: −0.414

3
Given that tan π8 = 0.414, find tan 17π8.
Show Answer

17π8 = π8 + 2π

tan(θ + 2π) = tan θ

tan 17π8 = 0.414

Answer: 0.414

4
Given that tan π8 = 0.414, find tan 15π8.
Show Answer

15π8 = 2π − π8

tan(2π − θ) = −tan θ

tan 15π8 = −0.414

Answer: −0.414

5
Given that tan π8 = 0.414, find tan(−7π8).
Show Answer

−7π8 + π = π8

tan(−7π8) = tan π8

tan(−7π8) = 0.414

Answer: 0.414

Hard

The complementary angle

tan(π/2 − θ) is the reciprocal. The given 0.414 is already rounded.

Example
Show that tan(π2 − θ) = 1tan θ.

tan(π2 − θ) = sin(π2 − θ)cos(π2 − θ)

sin(π2 − θ) = cos θ

cos(π2 − θ) = sin θ

tan(π2 − θ) = cos θsin θ

tan(π2 − θ) = 1tan θ

Answer: tan(π2 − θ) = 1tan θ

1
Show that tan(π2 + θ) = −1tan θ.
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tan(π2 + θ) = sin(π2 + θ)cos(π2 + θ)

sin(π2 + θ) = cos θ

cos(π2 + θ) = −sin θ

tan(π2 + θ) = cos θ−sin θ

tan(π2 + θ) = −1tan θ

Answer: tan(π2 + θ) = −1tan θ

2
Given that tan π8 = 0.414, find tan 3π8 to 3 significant figures.
Show Answer

3π8 = π2 − π8

tan(π2 − θ) = 1tan θ

tan 3π8 = 10.414

tan 3π8 ≈ 2.42

Answer: 2.42

3
Given that tan π8 = 0.414, find tan 5π8 to 3 significant figures.
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5π8 = π − 3π8

tan(π − α) = −tan α

tan 3π8 ≈ 2.42

tan 5π8 ≈ −2.42

Answer: −2.42

4
Given that tan π8 = 0.414, find tan 11π8 to 3 significant figures.
Show Answer

11π8 = 3π8 + π

tan(α + π) = tan α

tan 11π8 ≈ 2.42

Answer: 2.42

5
Given that tan π8 = 0.414, find tan(−3π8) to 3 significant figures.
Show Answer

tan(−α) = −tan α

tan 3π8 ≈ 2.42

tan(−3π8) ≈ −2.42

Answer: −2.42

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