CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 6. Analytic Trigonometry

sin(θ+β) 6 =sinθ+sinβ
First look at the derivation of the cosine difference identity:
cos(α−β) =cosαcosβ+sinαsinβ

Start by drawing two arbitrary anglesαandβ. In the image aboveαis the angle in red andβis the angle in
blue. The differenceα−βis noted in black asθ. The reason why there are two pictures is because the image on
the right has the same angleθin a rotated position. This will be useful to work with because the length ofABwill
be the same as the length ofCD.

√ AB=CD

(cosα−cosβ)^2 +(sinα−sinβ)^2 =

√

(cosθ− 1 )^2 +(sinθ− 0 )^2

(cosα−cosβ)^2 +(sinα−sinβ)^2 = (cosθ− 1 )^2 +(sinθ− 0 )^2

(cosα)^2 −2 cosαcosβ+(cosβ)^2 +(sinα)^2 −2 sinαsinβ+(sinβ)^2 = (cosθ− 1 )^2 +(sinθ)^2

2 −2 cosαcosβ−2 sinαsinβ= (cosθ)^2 −2 cosθ+ 1 +(sinθ)^2

2 −2 cosαcosβ−2 sinαsinβ= 1 −2 cosθ+ 1

−2 cosαcosβ−2 sinαsinβ=−2 cosθ

cosαcosβ+sinαsinβ=cosθ

=cos(α−β)

The proofs for sine and tangent are left to examples and exercises. They are listed here for your reference. Cotan-
gent, secant and cosecant are excluded because you can use reciprocal identities to get those once you have sine,
cosine and tangent.
Summary:

• cos(α±β) =cosαcosβ∓sinαsinβ


• sin(α±β) =sinαcosβ±cosαsinβ


• tan(α±β) =cossin((αα±±ββ))= 1 tan∓tanα±αtantanββ

Example A