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1.24: Cofunction Identities and Reflection

Difficulty Level: At Grade Created by: CK-12
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While toying with a triangular puzzle piece, you start practicing your math skills to see what you can find out about it. You realize one of the interior angles of the puzzle piece is 30, and decide to compute the trig functions associated with this angle. You immediately want to compute the cosine of the angle, but can only remember the values of your sine functions.

Is there a way to use this knowledge of sine functions to help you in your computation of the cosine function for 30?

Read on, and by the end of this Concept, you'll be able to apply knowledge of the sine function to help determine the value of a cosine function.

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Cofunctions

Guidance

In a right triangle, you can apply what are called "cofunction identities". These are called cofunction identities because the functions have common values. These identities are summarized below.

sinθ=cos(90θ)tanθ=cot(90θ)cosθ=sin(90θ)cotθ=tan(90θ)

Example A

Find the value of cos120.

Solution: Because this angle has a reference angle of 60, the answer is cos120=12.

Example B

Find the value of cos(120).

Solution: Because this angle has a reference angle of 60, the answer is cos(120)=cos240=12.

Example C

Find the value of sin135.

Solution: Because this angle has a reference angle of 45, the answer is sin135=22

Vocabulary

Cofunction Identity: A cofunction identity is a relationship between one trig function of an angle and another trig function of the complement of that angle.

Guided Practice

1. Find the value of sin45 using a cofunction identity.

2. Find the value of cos45 using a cofunction identity.

3. Find the value of cos60 using a cofunction identity.

Solutions:

1. The sine of 45 is equal to cos(9045)=cos45=22.

2. The cosine of 45 is equal to sin(9045)=sin45=22.

3. The cosine of 60 is equal to sin(9060)=sin30=.5.

Concept Problem Solution

Since you now know the cofunction relationships, you can use your knowledge of sine functions to help you with the cosine computation:

cos30=sin(9030)=sin(60)=32

Practice

  1. Find a value for θ for which sinθ=cos15 is true.
  2. Find a value for θ for which cosθ=sin55 is true.
  3. Find a value for θ for which tanθ=cot80 is true.
  4. Find a value for θ for which cotθ=tan30 is true.
  5. Use cofunction identities to help you write the expression tan255 as the function of an acute angle of measure less than 45.
  6. Use cofunction identities to help you write the expression sin120 as the function of an acute angle of measure less than 45.
  7. Use cofunction identities to help you write the expression cos310 as the function of an acute angle of measure less than 45.
  8. Use cofunction identities to help you write the expression cot260 as the function of an acute angle of measure less than 45.
  9. Use cofunction identities to help you write the expression cos280 as the function of an acute angle of measure less than 45.
  10. Use cofunction identities to help you write the expression tan60 as the function of an acute angle of measure less than 45.
  11. Use cofunction identities to help you write the expression sin100 as the function of an acute angle of measure less than 45.
  12. Use cofunction identities to help you write the expression cos70 as the function of an acute angle of measure less than 45.
  13. Use cofunction identities to help you write the expression cot240 as the function of an acute angle of measure less than 45.
  14. Use a right triangle to prove that sinθ=cos(90θ).
  15. Use the sine and cosine cofunction identities to prove that tan(90θ)=cotθ.

Vocabulary

Cofunction Identity

Cofunction Identity

A cofunction identity is a relationship between one trig function of an angle and another trig function of the complement of that angle.

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Date Created:
Sep 26, 2012
Last Modified:
Mar 23, 2016
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