Precalculus : Fundamental Trigonometric Identities

Study concepts, example questions & explanations for Precalculus

varsity tutors app store varsity tutors android store

Example Questions

Example Question #41 : Fundamental Trigonometric Identities

Given that:

Compute

 in function of .

Possible Answers:

Correct answer:

Explanation:

We will use the following formulas for calculating the tangent:

 

We have then:

 

This gives us :

Simplifying a bit we get :

Writing

We now have :

 

Using the fact that , we have finally:

Example Question #42 : Fundamental Trigonometric Identities

Which of the following expressions best represent ?

Possible Answers:

Correct answer:

Explanation:

Write the trigonometric product and sum identity for .

For , replace  with  and simplify the expression.

Example Question #41 : Fundamental Trigonometric Identities

Find the exact answer of:  

Possible Answers:

Correct answer:

Explanation:

The product and sum formula can be used to solve this question.

Write the formula for cosine identity.

Split up  into two separate cosine expressions.

Substitute the 2 known angles into the formula and simplify.

 

Example Question #44 : Fundamental Trigonometric Identities

Simplify the following. Leave your answer in terms of a trigonometric function. 

Possible Answers:

Correct answer:

Explanation:

This is a simple exercise to recognize the half angle formula for cosine.

The half angle formula for cosine is 

.

In the expression given

With this in mind we can rewrite the expression as the , or, after dividing by two, 

Example Question #45 : Fundamental Trigonometric Identities

Solve the following over the domain  to .

Possible Answers:

Correct answer:

Explanation:

Here we can rewrite the left side of the equation as  because of the double angle formula for sin, which is .

Now our equation is

,

and in order to get solve for  we take the  of both sides. Just divide by two from there to find

The only thing to keep in mind here is that the period of the function is half of what it normally is, which is why we have to solve for  and then add  to each answer. 

Example Question #46 : Fundamental Trigonometric Identities

Solve over the domain  to .

Possible Answers:

Correct answer:

Explanation:

We can rewrite the left side of the equation using the angle difference formula for cosine

 as

.

From here we just take the  of both sides and then add  to get .

Example Question #47 : Fundamental Trigonometric Identities

Which expression is equivalent to

?

Possible Answers:

Correct answer:

Explanation:

The relevant trigonometric identity is

In this case, "u" is and "v" is .

Our answer is

.

Example Question #48 : Fundamental Trigonometric Identities

Evaluate the following.

Possible Answers:

Correct answer:

Explanation:

We can use the angle sum formula for sine here.

If we recall that, 

,

we can see that the equation presented is equal to 

 because .

We can simplify this to , which is simply

Example Question #49 : Fundamental Trigonometric Identities

Evaluate the following.

Possible Answers:

Correct answer:

Explanation:

The angle sum formula for cosine is, 

.

First, we see that . We can then rewrite the expression as,

All that is left to do is to recall the unit circle to evaluate,

.

Example Question #50 : Fundamental Trigonometric Identities

Evaluate the following.

Possible Answers:

Correct answer:

Explanation:

This one is another angle sum/difference problem, except it is using the trickier tangent identity.

The angle sum formula for tangent is

We can see that .

We can then rewrite the expression as , which is .

Learning Tools by Varsity Tutors