Precalculus : Trigonometric Identities

Example Questions

Example Question #4 : Product/Sum Identities

Evaluate the following.

Explanation:

Here we use the double angle identity for sine, which is

We can rewrite the originial expression as  using the double angle identity.

From here we can calculate that

.

Example Question #5 : Product/Sum Identities

Evaluate the following expression.

Explanation:

One of the double angle formuals for cosine is

We can use this double angle formula for cosine to rewrite the expression given as the  because  and .

We can then calculate that

.

Example Question #1 : Product/Sum Identities

Evaluate the following.

Explanation:

Here we can use another double angle formula for cosine,

.

Here , and so we can use the double angle formula for cosine to rewrite the expression as

.

From here we just recognize that

.

Example Question #231 : Pre Calculus

Evaluate the following expression.

Explanation:

Here we can use yet another double angle formula for cosine:

.

First, realize that .

Next, plug this in to the double angle formula to find that

.

Here we recognize that

Example Question #31 : Trigonometric Identities

Simplify the following. Leave your answer in terms of a trigonometric function.

Explanation:

This is a quick test of being able to recall the angle sum formula for sine.

Since,

, and here

, we can rewrite the expression as

.

Example Question #9 : Product/Sum Identities

Which of the following expressions best represent ?

Explanation:

Write the trigonometric product and sum identity for .

For , replace  with  and simplify the expression.

Example Question #10 : Product/Sum Identities

Explanation:

The product and sum formula can be used to solve this question.

Write the formula for cosine identity.

Split up  into two separate cosine expressions.

Substitute the 2 known angles into the formula and simplify.

Example Question #1 : Use Product/Sum Identities To Express A Product As A Sum Or Difference

Simplify the following. Leave your answer in terms of a trigonometric function.

Explanation:

This is a simple exercise to recognize the half angle formula for cosine.

The half angle formula for cosine is

.

In the expression given

With this in mind we can rewrite the expression as the , or, after dividing by two,

Example Question #1 : Use Product/Sum Identities To Express A Product As A Sum Or Difference

Solve the following over the domain  to .

Explanation:

Here we can rewrite the left side of the equation as  because of the double angle formula for sin, which is .

Now our equation is

,

and in order to get solve for  we take the  of both sides. Just divide by two from there to find

The only thing to keep in mind here is that the period of the function is half of what it normally is, which is why we have to solve for  and then add  to each answer.

Example Question #31 : Trigonometric Identities

Solve over the domain  to .