Precalculus : Double- and Half-Angle Identities

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Double And Half Angle Identities

Find the exact value

.

Possible Answers:

Correct answer:

Explanation:

By the double angle formula

Example Question #1 : Double Angle Identities

Find the exact value

.

Possible Answers:

Correct answer:

Explanation:

By the double angle formula

Example Question #3 : Double And Half Angle Identities

Find the exact value

.

Possible Answers:

Correct answer:

Explanation:

By the double-angula formula for cosine

For this problem

Example Question #41 : Trigonometric Identities

Find the exact value

.

Possible Answers:

Correct answer:

Explanation:

By the double-angle formula for the sine function

we have

thus the double angle formula becomes,

Example Question #1 : Double Angle Identities

If , which of the following best represents ?

Possible Answers:

Correct answer:

Explanation:

The expression  is a double angle identity that can also be rewritten as:

Replace the value of theta for .

The correct answer is: 

Example Question #6 : Double And Half Angle Identities

Which expression is equivalent to  ?

Possible Answers:

Correct answer:

Explanation:

The relevant trigonometric identity is:

In this case, "u" is since .

The only one that actually follows this is

 

Example Question #7 : Double And Half Angle Identities

Compute

Possible Answers:

Correct answer:

Explanation:

A useful trigonometric identity to remember for this problem is 

or equivalently,

If we substitute  for , we get

Example Question #8 : Double And Half Angle Identities

Compute 

Possible Answers:

Correct answer:

Explanation:

A useful trigonometric identity to remember is 

If we plug in  into this equation, we get

We can divide the equation by 2 to get

Example Question #9 : Double And Half Angle Identities

Using the half-angle identities, which of the following answers best resembles ?

Possible Answers:

Correct answer:

Explanation:

Write the half angle identity for sine.

Since we are given , the angle is equal to .  Set these two angles equal to each other and solve for .

Substitute this value into the formula.

 

Example Question #10 : Double And Half Angle Identities

Let  and  two reals. Given that:

What is the value of:

?

Possible Answers:

Correct answer:

Explanation:

We have:

 and : 

 

(1)-(2) gives:

Knowing from the above formula that:( take a=b in the formula above)

This gives:

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