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# 1.24: Cofunction Identities and Reflection

Difficulty Level: At Grade Created by: CK-12
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Practice 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\begin{align*}30^\circ\end{align*}, 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\begin{align*}30^\circ\end{align*}?

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.

### 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)\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*}

#### Example A

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

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

#### Example B

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

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

#### Example C

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

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

### 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\begin{align*}\sin 45^\circ\end{align*} using a cofunction identity.

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

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

Solutions:

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

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

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

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

### Practice

1. Find a value for θ\begin{align*}\theta\end{align*} for which sinθ=cos15\begin{align*}\sin \theta=\cos 15^\circ\end{align*} is true.
2. Find a value for θ\begin{align*}\theta\end{align*} for which cosθ=sin55\begin{align*}\cos \theta=\sin 55^\circ\end{align*} is true.
3. Find a value for θ\begin{align*}\theta\end{align*} for which tanθ=cot80\begin{align*}\tan \theta=\cot 80^\circ\end{align*} is true.
4. Find a value for θ\begin{align*}\theta\end{align*} for which cotθ=tan30\begin{align*}\cot \theta=\tan 30^\circ\end{align*} is true.
5. Use cofunction identities to help you write the expression tan255\begin{align*}\tan 255^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
6. Use cofunction identities to help you write the expression sin120\begin{align*}\sin 120^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
7. Use cofunction identities to help you write the expression cos310\begin{align*}\cos 310^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
8. Use cofunction identities to help you write the expression cot260\begin{align*}\cot 260^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
9. Use cofunction identities to help you write the expression cos280\begin{align*}\cos 280^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
10. Use cofunction identities to help you write the expression tan60\begin{align*}\tan 60^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
11. Use cofunction identities to help you write the expression sin100\begin{align*}\sin 100^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
12. Use cofunction identities to help you write the expression cos70\begin{align*}\cos 70^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
13. Use cofunction identities to help you write the expression cot240\begin{align*}\cot 240^\circ\end{align*} as the function of an acute angle of measure less than 45\begin{align*}45^\circ\end{align*}.
14. Use a right triangle to prove that sinθ=cos(90θ)\begin{align*}\sin \theta=\cos (90^\circ-\theta)\end{align*}.
15. Use the sine and cosine cofunction identities to prove that tan(90θ)=cotθ\begin{align*}\tan (90^\circ-\theta)=\cot \theta\end{align*}.

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### Vocabulary Language: English

TermDefinition
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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