Precalculus : Trigonometric Functions

Example Questions

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.

Explanation:

This one is another angle sum/difference problem, except it is using the trickier tangent identity.

The angle sum formula for tangent is

We can see that .

We can then rewrite the expression as , which is .

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: