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Cofunction Identities and Reflection

Based on complements of angles.

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Cofunction Identities and Reflection

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 \begin{align*}30^\circ\end{align*}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 \begin{align*}30^\circ\end{align*}30?

Cofunction Identities and Reflection

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.

\begin{align*}\sin \theta = \cos(90^\circ-\theta) && \cos \theta = \sin (90^\circ-\theta)\\ \tan \theta = \cot(90^\circ-\theta) && \cot \theta = \tan (90^\circ-\theta)\end{align*}sinθ=cos(90θ)tanθ=cot(90θ)cosθ=sin(90θ)cotθ=tan(90θ)

 

 

 

 

Find the value of \begin{align*}\cos 120^\circ\end{align*}cos120.

Because this angle has a reference angle of \begin{align*}60^\circ\end{align*}60, the answer is \begin{align*}\cos 120^\circ = -\frac{1}{2}\end{align*}cos120=12.

Find the value of \begin{align*}\cos (-120^\circ)\end{align*}cos(120).

Because this angle has a reference angle of \begin{align*}60^\circ\end{align*}60, the answer is \begin{align*}\cos (-120^\circ) = \cos 240^\circ = -\frac{1}{2}\end{align*}cos(120)=cos240=12.

Find the value of \begin{align*}\sin 135^\circ\end{align*}sin135.

Because this angle has a reference angle of \begin{align*}45^\circ\end{align*}45, the answer is \begin{align*}\sin 135^\circ = \frac{\sqrt{2}}{2}\end{align*}sin135=22

Examples

Example 1

Earlier, you were asked if there is a way to use your knowledge of sine functions to help you in your computation of the cosine function.

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

\begin{align*}\cos 30^\circ = \sin (90^\circ - 30^\circ) = \sin (60^\circ) = \frac{\sqrt{3}}{2}\end{align*}cos30=sin(9030)=sin(60)=32

Example 2

 Find the value of \begin{align*}\sin 45^\circ\end{align*}sin45 using a cofunction identity.

The sine of \begin{align*}45^\circ\end{align*}45 is equal to \begin{align*}\cos (90^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2}\end{align*}cos(9045)=cos45=22.

Example 3

Find the value of \begin{align*}\cos 45^\circ\end{align*}cos45 using a cofunction identity.

The cosine of \begin{align*}45^\circ\end{align*}45 is equal to \begin{align*}\sin (90^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}\end{align*}sin(9045)=sin45=22.

Example 4

Find the value of \begin{align*}\cos 60^\circ\end{align*}cos60 using a cofunction identity.

The cosine of \begin{align*}60^\circ\end{align*}60 is equal to \begin{align*}\sin (90^\circ - 60^\circ) = \sin 30^\circ = .5\end{align*}sin(9060)=sin30=.5.

Review

  1. Find a value for \begin{align*}\theta\end{align*}θ for which \begin{align*}\sin \theta=\cos 15^\circ\end{align*}sinθ=cos15 is true.
  2. Find a value for \begin{align*}\theta\end{align*}θ for which \begin{align*}\cos \theta=\sin 55^\circ\end{align*}cosθ=sin55 is true.
  3. Find a value for \begin{align*}\theta\end{align*}θ for which \begin{align*}\tan \theta=\cot 80^\circ\end{align*}tanθ=cot80 is true.
  4. Find a value for \begin{align*}\theta\end{align*}θ for which \begin{align*}\cot \theta=\tan 30^\circ\end{align*}cotθ=tan30 is true.
  5. Use cofunction identities to help you write the expression \begin{align*}\tan 255^\circ\end{align*}tan255 as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}45.
  6. Use cofunction identities to help you write the expression \begin{align*}\sin 120^\circ\end{align*}sin120 as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}45.
  7. Use cofunction identities to help you write the expression \begin{align*}\cos 310^\circ\end{align*}cos310 as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}45.
  8. Use cofunction identities to help you write the expression \begin{align*}\cot 260^\circ\end{align*}cot260 as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}45.
  9. Use cofunction identities to help you write the expression \begin{align*}\cos 280^\circ\end{align*} as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}.
  10. Use cofunction identities to help you write the expression \begin{align*}\tan 60^\circ\end{align*} as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}.
  11. Use cofunction identities to help you write the expression \begin{align*}\sin 100^\circ\end{align*} as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}.
  12. Use cofunction identities to help you write the expression \begin{align*}\cos 70^\circ\end{align*} as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}.
  13. Use cofunction identities to help you write the expression \begin{align*}\cot 240^\circ\end{align*} as the function of an acute angle of measure less than \begin{align*}45^\circ\end{align*}.
  14. Use a right triangle to prove that \begin{align*}\sin \theta=\cos (90^\circ-\theta)\end{align*}.
  15. Use the sine and cosine cofunction identities to prove that \begin{align*}\tan (90^\circ-\theta)=\cot \theta\end{align*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.24. 

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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.

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