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1.24: Cofunction Identities and Reflection

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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^\circ , 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^\circ ?

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.

\sin \theta = \cos(90^\circ-\theta) && \cos \theta = \sin (90^\circ-\theta)\\\tan \theta = \cot(90^\circ-\theta) && \cot \theta = \tan (90^\circ-\theta)

Example A

Find the value of \cos 120^\circ .

Solution: Because this angle has a reference angle of 60^\circ , the answer is \cos 120^\circ = -\frac{1}{2} .

Example B

Find the value of \cos (-120^\circ) .

Solution: Because this angle has a reference angle of 60^\circ , the answer is \cos (-120^\circ) = \cos 240^\circ = -\frac{1}{2} .

Example C

Find the value of \sin 135^\circ .

Solution: Because this angle has a reference angle of 45^\circ , the answer is \sin 135^\circ = \frac{\sqrt{2}}{2}

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 \sin 45^\circ using a cofunction identity.

2. Find the value of \cos 45^\circ using a cofunction identity.

3. Find the value of \cos 60^\circ using a cofunction identity.

Solutions:

1. The sine of 45^\circ is equal to \cos (90^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2} .

2. The cosine of 45^\circ is equal to \sin (90^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2} .

3. The cosine of 60^\circ is equal to \sin (90^\circ - 60^\circ) = \sin 30^\circ = .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:

\cos 30^\circ = \sin (90^\circ - 30^\circ) = \sin (60^\circ) = \frac{\sqrt{3}}{2}

Practice

  1. Find a value for \theta for which \sin \theta=\cos 15^\circ is true.
  2. Find a value for \theta for which \cos \theta=\sin 55^\circ is true.
  3. Find a value for \theta for which \tan \theta=\cot 80^\circ is true.
  4. Find a value for \theta for which \cot \theta=\tan 30^\circ is true.
  5. Use cofunction identities to help you write the expression \tan 255^\circ as the function of an acute angle of measure less than 45^\circ .
  6. Use cofunction identities to help you write the expression \sin 120^\circ as the function of an acute angle of measure less than 45^\circ .
  7. Use cofunction identities to help you write the expression \cos 310^\circ as the function of an acute angle of measure less than 45^\circ .
  8. Use cofunction identities to help you write the expression \cot 260^\circ as the function of an acute angle of measure less than 45^\circ .
  9. Use cofunction identities to help you write the expression \cos 280^\circ as the function of an acute angle of measure less than 45^\circ .
  10. Use cofunction identities to help you write the expression \tan 60^\circ as the function of an acute angle of measure less than 45^\circ .
  11. Use cofunction identities to help you write the expression \sin 100^\circ as the function of an acute angle of measure less than 45^\circ .
  12. Use cofunction identities to help you write the expression \cos 70^\circ as the function of an acute angle of measure less than 45^\circ .
  13. Use cofunction identities to help you write the expression \cot 240^\circ as the function of an acute angle of measure less than 45^\circ .
  14. Use a right triangle to prove that \sin \theta=\cos (90^\circ-\theta) .
  15. Use the sine and cosine cofunction identities to prove that \tan (90^\circ-\theta)=\cot \theta .

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Date Created:

Sep 26, 2012

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Oct 28, 2014
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