# Precalculus : Fundamental Trigonometric Identities

## Example Questions

### Example Question #51 : Fundamental Trigonometric 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 #52 : Fundamental Trigonometric 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 #53 : Fundamental Trigonometric 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 #54 : Fundamental Trigonometric Identities

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 #55 : Fundamental 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 #56 : Fundamental Trigonometric Identities

Which of the following is equivalent to the expression:

Explanation:

Which of the following is equivalent to the following expression?

Recall our Pythagorean trig identity:

It can be rearranged to look just like our numerator:

So go ahead and change our original expression to:

Then recall the definition of cosecant:

So our original expression can be rewritten as:

### Example Question #57 : Fundamental Trigonometric Identities

Which of the following trigonometric identities is INCORRECT?

Explanation:

Cosine and sine are not reciprocal functions.

and

### Example Question #58 : Fundamental Trigonometric Identities

Using the trigonometric identities prove whether the following is valid:

False

Uncertain

True

Only in the range of:

Only in the range of:

True

Explanation:

We begin with the left-hand side of the equation and utilize basic trigonometric identities, beginning with converting the inverse functions to their corresponding base functions:

Next, we rewrite the fractional division in order to simplify the equation:

In fractional division we multiply by the reciprocal as follows:

If we reduce the fraction using basic identities we see that the equivalence is proven:

### Example Question #59 : Fundamental Trigonometric Identities

Which of the following identities is incorrect?

Explanation:

The true identity is  because cosine is an even function.

### Example Question #60 : Fundamental Trigonometric Identities

State  in terms of sine and cosine.