Page 29 - Ebook Trigonometri_Krisna Hasugian_MESP 2019

Sum and Difference Formulas

In this section we will use formulas that involve the sum or difference of two
angles, call the sum and difference formulas.

Sum and difference formulas for sines and cosines

sin( + ) = sin cos + cos sin

sin( − ) = sin cos − cos sin

cos( + ) = cos cos − sin sin

cos( − ) = cos cos + sin sin

How do we use these formulas?

Example 27 Find the exact value of cos 105°

Well we can break 105° into 60° and 45° since those values are relatively easy to
find the cosine of.

Therefore cos 105° = cos(60° + 45°)= cos 60° cos 45° − sin 60° sin 45°

Using the unit circle we obtain,

1 √2 √3 √2
= ∙ − ∙
2 2 2 2

√2
= − = (√2 − √6)
√6
1
4 4 4

Example 28 Find the exact value of sin 15°
= sin(45° − 30°)

= sin 45° cos 30° − cos 45° sin 30°

√2 √3 √2 1
= ∙ − ∙
2 2 2 2

√2
1
√6
= − = (√6 − √2)
4 4 4

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