Exercise 4 Page 477

We are asked to explain how to evaluate trigonometric expressions of the sum or difference of two angles. Let's begin with something simpler. We already know the values of the sine and cosine for certain angles. These can be summarized in the following table.

	sin θ	cos θ
θ=0^(∘)	0	1
θ=30^(∘)	1/2	sqrt(3)/2
θ=45^(∘)	sqrt(2)/2	sqrt(2)/2
θ=60^(∘)	sqrt(3)/2	1/2
θ=90^(∘)	1	0

In order to use these values we need to write the requested angle as a sum or difference of the known angles. For example, consider the following expression. sin15^(∘) We do not know the value of the sine of 15^(∘), but we can notice that it can be written as the difference of two of our known angles. 15^(∘)=45^(∘)-30^(∘) Next, we identify the corresponding difference formula. Since we want to evaluate sin15^(∘), we will use the difference formula for sine. sin(a-b)=sinacosb-cosasinb Let's use this formula!

sin15^(∘)

Rewrite

Rewrite 15^(∘) as 45^(∘)-30^(∘)

sin(45^(∘)-30^(∘))

sin(α-β)=sin(α)cos(β)-cos(α)sin(β)

sin45^(∘)cos30^(∘)-cos45^(∘)sin30^(∘)

▼

Evaluate

SubstituteII

sin45^(∘)= sqrt(2)/2, cos45^(∘)= sqrt(2)/2

sqrt(2)/2*cos30^(∘)- sqrt(2)/2*sin30^(∘)

Substitute