# Precalculus : Fundamental Trigonometric Identities

## Example Questions

### Example Question #41 : Fundamental Trigonometric Identities

Given that:

Compute

in function of .

Explanation:

We will use the following formulas for calculating the tangent:

We have then:

This gives us :

Simplifying a bit we get :

Writing

We now have :

Using the fact that , we have finally:

### Example Question #42 : Fundamental Trigonometric 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 #41 : Fundamental Trigonometric 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 #44 : Fundamental Trigonometric Identities

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 #45 : Fundamental Trigonometric Identities

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 #46 : Fundamental Trigonometric Identities

Solve over the domain  to .

Explanation:

We can rewrite the left side of the equation using the angle difference formula for cosine

as

.

From here we just take the  of both sides and then add  to get .

### Example Question #47 : Fundamental Trigonometric Identities

Which expression is equivalent to

?

Explanation:

The relevant trigonometric identity is

In this case, "u" is and "v" is .

.

### Example Question #48 : Fundamental Trigonometric Identities

Evaluate the following.

Explanation:

We can use the angle sum formula for sine here.

If we recall that,

,

we can see that the equation presented is equal to

because .

We can simplify this to , which is simply

### Example Question #49 : Fundamental Trigonometric Identities

Evaluate the following.

Explanation:

The angle sum formula for cosine is,

.

First, we see that . We can then rewrite the expression as,

All that is left to do is to recall the unit circle to evaluate,

.

### Example Question #50 : Fundamental Trigonometric Identities

Evaluate the following.