# Precalculus : Fundamental Trigonometric Identities

## Example Questions

### Example Question #51 : Fundamental Trigonometric Identities

Evaluate the following. Possible Answers:     Correct answer: Explanation:

Here we use the double angle identity for sine, which is We can rewrite the originial expression as using the double angle identity.

From here we can calculate that .

### Example Question #52 : Fundamental Trigonometric Identities

Evaluate the following expression. Possible Answers:     Correct answer: Explanation:

One of the double angle formuals for cosine is We can use this double angle formula for cosine to rewrite the expression given as the because and .

We can then calculate that .

### Example Question #53 : Fundamental Trigonometric Identities

Evaluate the following. Possible Answers:     Correct answer: Explanation:

Here we can use another double angle formula for cosine, .

Here , and so we can use the double angle formula for cosine to rewrite the expression as .

From here we just recognize that .

### Example Question #54 : Fundamental Trigonometric Identities

Evaluate the following expression. Possible Answers:     Correct answer: Explanation:

Here we can use yet another double angle formula for cosine: .

First, realize that .

Next, plug this in to the double angle formula to find that .

Here we recognize that ### Example Question #55 : Fundamental Trigonometric Identities

Simplify the following. Leave your answer in terms of a trigonometric function. Possible Answers:     Correct answer: Explanation:

This is a quick test of being able to recall the angle sum formula for sine.

Since, , and here , we can rewrite the expression as .

### Example Question #56 : Fundamental Trigonometric Identities

Which of the following is equivalent to the expression: Possible Answers:    Correct answer: Explanation:

Which of the following is equivalent to the following expression? Recall our Pythagorean trig identity: It can be rearranged to look just like our numerator: So go ahead and change our original expression to: Then recall the definition of cosecant: So our original expression can be rewritten as: So our answer is: ### Example Question #57 : Fundamental Trigonometric Identities

Which of the following trigonometric identities is INCORRECT?

Possible Answers:     Correct answer: Explanation:

Cosine and sine are not reciprocal functions. and ### Example Question #58 : Fundamental Trigonometric Identities

Using the trigonometric identities prove whether the following is valid: Possible Answers:

False

Uncertain

True

Only in the range of: Only in the range of: Correct answer:

True

Explanation:

We begin with the left-hand side of the equation and utilize basic trigonometric identities, beginning with converting the inverse functions to their corresponding base functions: Next, we rewrite the fractional division in order to simplify the equation: In fractional division we multiply by the reciprocal as follows: If we reduce the fraction using basic identities we see that the equivalence is proven:  ### Example Question #59 : Fundamental Trigonometric Identities

Which of the following identities is incorrect?

Possible Answers:     Correct answer: Explanation:

The true identity is because cosine is an even function.

### Example Question #60 : Fundamental Trigonometric Identities

State in terms of sine and cosine.

Possible Answers:     Correct answer: Explanation:

The definition of tangent is sine divided by cosine. ### All Precalculus Resources 