Trigonometry : Identities of Inverse Operations

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Identities Of Inverse Operations

Simplify using identities. Leave no fractions in your answer.

Possible Answers:

Correct answer:

Explanation:

The easiest first step is to simplify our inverse identities:

Cross cancelling, we end up with

Finally, eliminate the fraction:


Thus,

Example Question #2 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions. 

1.  

Possible Answers:

Correct answer:

Explanation:

Using the quotient identities for trig functions, you can rewrite,

and

Then the fraction becomes

Example Question #2 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Identities:

and

Thus the expression becomes,

.

Example Question #4 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

Possible Answers:

Correct answer:

Explanation:

Use the distributive property (FOIL method) to simplify the expression.

Using Pythagorean Identities:

.

Example Question #5 : Identities Of Inverse Operations

Simplify each expression below. Your answer should have (at most) one trigonometric function and no fractions.

Possible Answers:

Correct answer:

Explanation:

First, simplify the first term in the expression to 1 because of the Pythagorean Identity.

Then, simplify the second term to

.

This reduces to

.

The expression is now,

 .

Distribute the negative and get,

 .

Example Question #6 : Identities Of Inverse Operations

Solve each question over the interval 

Possible Answers:

Correct answer:

Explanation:

Divide both sides by  to get .

Take the square root of both sides to get that  and .

The angles for which this is true (this is taking the arctan) are every angle when  and .

These angles are all the multiples of

Example Question #3 : Identities Of Inverse Operations

can be stated as all of the following except...

Possible Answers:

Correct answer:

Explanation:

Let's look at these individually:

is true by definition, as is .

is also true because of a co-function identity.

This leaves two - and we can tell which of these does not work using the fact that , which means that is our answer.

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