Trigonometry : Pythagorean Identities

Example Questions

Example Question #1 : Trigonometric Identities

Simplify .

Explanation:

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 .

Explanation:

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

1

-1

0

1

Explanation:

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:

1

Explanation:

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:

The equation cannot be further simplified.

Explanation:

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 ?

Explanation:

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?

Explanation:

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

Explanation:

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?

Explanation:

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 ?

Explanation:

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

.