### All Precalculus Resources

## Example Questions

### Example Question #47 : Fundamental Trigonometric Identities

Which expression is equivalent to

?

**Possible Answers:**

**Correct answer:**

The relevant trigonometric identity is

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

Our answer is

.

### Example Question #48 : Fundamental Trigonometric Identities

Evaluate the following.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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:

So our answer is:

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