Precalculus : Use Reciprocal and Ratio Identities

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #7 : 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 expression:

Begin by recalling the following identity:

Next, recall the relationship between cotangent and tangent:

As well as the relationship between tangent, sine and cosine

So to put it all together, we can pull out the negative sign from our original expression:

Next, we can rewrite our cotangent as tangent

Finally, we can change our tangent to sine and cosine, but because we are dealing with the reciprocal of tangent, we will need the reciprocal of our identity.

Making our answer:

Beware trap answer:

This may look good on the surface, but recall

Example Question #3 : Fundamental Trigonometric Identities

Simplify:  

Possible Answers:

Correct answer:

Explanation:

Write the reciprocal identity for cosecant.

Rewrite the expression and use the double angle identities for sine to simplify.

Example Question #9 : Fundamental Trigonometric Identities

Determine which of the following is equivalent to .

Possible Answers:

Correct answer:

Explanation:

Rewirte  using the reciprocal identity of cosine.

Example Question #10 : Fundamental Trigonometric Identities

Which of the following is similar to 

?

Possible Answers:

Correct answer:

Explanation:

Write the reciprocal/ratio identity for cosecant.

Replace cosecant with sine.

Example Question #11 : Fundamental Trigonometric Identities

Evaluate:  

Possible Answers:

Correct answer:

Explanation:

Rewrite  in terms of sine and cosine.

Dividing fractions is the same as multiplying the numerator by the reciprocal of the denominator.

Multiply the second term by sine to get a common denominator. Then after subtracting the second term from the first you can see that a Pythagorean Identity is in the numerator.

Reducing further we arrive at the final answer.

Example Question #12 : Fundamental Trigonometric Identities

Which of the following is equal to:

Possible Answers:

Correct answer:

Explanation:

Recall that , and that 

Therefore:

Since that term is eliminated, we have left:

Recall that

 

Therefore:

 

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