### All Trigonometry Resources

## Example Questions

### Example Question #1 : Trigonometric Identities

Simplify .

**Possible Answers:**

**Correct answer:**

To simplify, recognize that is a reworking on , meaning that .

Plug that into our given equation:

Remember that , so .

### Example Question #1 : Pythagorean Identities

Simplify .

**Possible Answers:**

**Correct answer:**

Recognize that is a reworking on , meaning that .

Plug that in to our given equation:

Notice that one of the 's cancel out.

.

### Example Question #2 : Pythagorean Identities

**Possible Answers:**

1

-1

0

**Correct answer:**

1

Recall the Pythagorean Identity:

We can rearrange the terms:

This is exactly what our original equation looks like, so the answer is 1.

### Example Question #1 : Pythagorean Identities

Simplify the equation using identities:

**Possible Answers:**

1

**Correct answer:**

There are a couple valid strategies for solving this problem. The simplest is to first factor out from both sides. This leaves us with:

Next, substitute with the known identity to get:

From here, we can eliminate the quadratic by converting:

giving us

Thus,

### Example Question #1 : Trigonometric Identities

Simplify the expression:

**Possible Answers:**

The equation cannot be further simplified.

**Correct answer:**

The expression represents a *difference of squares. *In this case, the product is (remember that 1 is also a perfect square).

One Pythagoran identity for trigonometric functions is:

Thus, we can say that the most simplified version of the expression is .

### Example Question #5 : Trigonometric Identities

If theta is in the second quadrant, and , what is ?

**Possible Answers:**

**Correct answer:**

Write the Pythagorean Identity.

Substitute the value of and solve for .

Since the cosine is in the second quadrant, the correct answer is:

### Example Question #2 : Trigonometric Identities

For which values of is the following equation true?

**Possible Answers:**

**Correct answer:**

According to the Pythagorean identity

,

the right hand side of this equation can be rewritten as . This yields the equation

.

Dividing both sides by yields:

.

Dividing both sides by yields:

.

This is precisely the definition of the tangent function; since the domain of consists of all real numbers, the values of which satisfy the original equation also consist of all real numbers. Hence, the correct answer is

.

### Example Question #6 : Trigonometric Identities

**Possible Answers:**

**Correct answer:**

By the Pythagorean identity, the first two terms simplify to 1:

.

Dividing the Pythagorean identity by allows us to simplify the right-hand side.

### Example Question #7 : Trigonometric Identities

What is equal to?

**Possible Answers:**

**Correct answer:**

Step 1: Recall the trigonometric identity that has sine and cosine in it...

The sum is equal to 1.

### Example Question #6 : Pythagorean Identities

Given , what is ?

**Possible Answers:**

**Correct answer:**

Using the Pythagorean Identity

,

one can solve for by plugging in for .

Solving for , you get it equal to .

Taking the square root of both sides will get the correct answer of

.

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