Precalculus : Trigonometric Identities

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Sum And Difference Identities For Sine

Evaluate

.

Possible Answers:

Correct answer:

Explanation:

is equivalent to or more simplified .

We can use the sum identity to evaluate this sine:

From the unit circle, we can determine these measures:

Example Question #1 : Sum And Difference Identities For Sine

Evaluate

.

Possible Answers:

Correct answer:

Explanation:

The angle  or .

Using the first one: 

We can find these values in the unit circle:

Example Question #1 : Sum And Difference Identities For Cosine

Evaluate the exact value of:  

Possible Answers:

Correct answer:

Explanation:

In order to solve , two special angles will need to be used to solve for the exact values.

The angles chosen are  and  degrees, since:

Write the formula for the cosine additive identity.

Substitute the known variables.

Example Question #2 : Sum And Difference Identities For Cosine

Find the exact value of .

Possible Answers:

Correct answer:

Explanation:

To solve , we will need to use both the sum and difference identities for cosine.

Write the formula for these identities.

To solve for  and , find two special angles whose difference  and sum equals to the angle 15 and 75, respectively.  The two special angles are 45 and 30.

Substitute the special angles in the formula.

Evaluate both conditions.

Solve for .

Example Question #3 : Sum And Difference Identities For Cosine

In the problem below,  and .

Find

.

Possible Answers:

Correct answer:

Explanation:

Since and is in quadrant I, we can say that and and therefore:

So .

Since  and  is in quadrant I, we can say that  and  and therefore: 

.  So .

Using the cosine sum formula, we then see:

.

 

 

Example Question #4 : Sum And Difference Identities For Cosine

In the problem below, and .

Find

.

Possible Answers:

Correct answer:

Explanation:

Since and is in quadrant I, we can say that and and therefore: 

So .

Since and is in quadrant I, we can say that and and therefore: 

So .

Using the cosine difference formula, we see:

Example Question #1 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve the following equation for :

Possible Answers:

Correct answer:

Explanation:

Use the trigonometric identities to switch sec into terms of tan:

hence,

 

So we have , making 

Therefore the solution is  for n being any integer.

Example Question #2 : Solving Trigonometric Equations And Inequalities

Which of the following is not a solution to   for 

Possible Answers:

Correct answer:

Explanation:

We begin by setting the right side of the equation equal to 0.

The equation might be easier to factor using the following substitution.

This gives the following

This can be factored as follows

Therefore

Replacing our substitution therefore gives

Within our designated domain, we get three answers between our two equations.

       when 

        when 

Therefore, the only choice that isn't correct is 

 

Example Question #1 : Solving Trigonometric Equations And Inequalities

Find one possible value of .

Possible Answers:

Correct answer:

Explanation:

Begin by isolating the tangent side of the equation:

Next, take the inverse tangent of both sides:

Divide by five to get the final answer:

Example Question #1 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve for the angle value.

Possible Answers:

Correct answer:

Explanation:

There are two ways to solve this problem. The first involves two trigonometric identities:

The second method allows us to only use the first trigonometric identity:

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