Precalculus : Simplify Expressions Using Trigonometric Identities

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Simplify Expressions Using Trigonometric Identities

Compute

.

Possible Answers:

Correct answer:

Explanation:

We can use the following trigonometric identity to help us in the calculation:

We plug in  to get

.

Example Question #2 : Simplify Expressions Using Trigonometric Identities

Simplify 

.

Possible Answers:

Correct answer:

Explanation:

We can use the trigonometric identity, 

along with the fact that

to compute .

We have

Example Question #3 : Simplify Expressions Using Trigonometric Identities

Which of the following is equivalent to

Possible Answers:

Correct answer:

Explanation:

When trying to identify equivalent equations that use trigonometric functions it is important to recall the general formula and understand how the terms affect the translations.

The general formula for sine is as follows.

 where  is the amplitude,  is used to find the period of the function  represents the phase shift , and  is the vertical shift.

This is also true for,

.

 

Looking at the possible answer choices lets first focus on the ones containing sine.

 has a vertical shift of  therefore it is not an equivalent function as it is moving the original function up.

 has a phase shift of  therefore it is not an equivalent function as it is moving the original function to the right.

Now lets shift our focus to the answer choices that contain cosine.

 has a vertical shift down of  units. This will create a graph that has a range that is below the -axis. It is important to remember that  has a range of . Therefore this cosine function is not an equivalent equation.

 has a phase shift to the right  units. Plugging in some values we see that,

.

Now, looking back at our original function and plugging in those same values of  and  we get,

,

.

Since the function values are the same for each of the input values, we can conclude that  is equivalent to .

Example Question #4 : Simplify Expressions Using Trigonometric Identities

Suppose:  

 

What must be the value of ?

Possible Answers:

Correct answer:

Explanation:

First, factor  into their simplified form.

The identity  equals to 1.

Factor .

Since:

 

Substitute the values of the simplified equation.

Example Question #5 : Simplify Expressions Using Trigonometric Identities

Find the exact value of each expression below without the aid of a calculator.

Possible Answers:

Correct answer:

Explanation:

In order to find the exact value of  we can use the half angle formula for sin, which is  

.

This way we can plug in a value for alpha for which we know the exact value.  is equal to  divided by two, and so we can plug in  for the alpha above.

The cosine of  is .

Therefore our final answer becomes,

.

Example Question #6 : Simplify Expressions Using Trigonometric Identities

Simplify.

Possible Answers:

None of these answers are correct.

Correct answer:

Explanation:

 

Given these identities...

Example Question #7 : Simplify Expressions Using Trigonometric Identities

Simplify  completely. 

Possible Answers:

Correct answer:

Explanation:

First simplify the fraction 

 

by multiplying it by its conjugate

.

After doing so, continue simplying:

 

Example Question #2 : Simplify Expressions Using Trigonometric Identities

Fully simplify.

Simplify:

Possible Answers:

None of these answers are correct.

Correct answer:

Explanation:

 

Given the above identities:

 

 

Example Question #9 : Simplify Expressions Using Trigonometric Identities

Simplify:

Possible Answers:

None of these answers are correct.

Correct answer:

Explanation:

and

 

Therefore...

 

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