3.1.6: Cofunction Identities
- Page ID
- 4176
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A cofunction identity is a relationship between one trig function of an angle and another trig function of the complement of that angle.
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 \(30^{\circ}\), 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 \(30^{\circ}\)?
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.
\(\begin{array}{rr}
\sin \theta=\cos \left(90^{\circ}-\theta\right) & \cos \theta=\sin \left(90^{\circ}-\theta\right) \\
\tan \theta=\cot \left(90^{\circ}-\theta\right) & \cot \theta=\tan \left(90^{\circ}-\theta\right)
\end{array}\)
Let's take a look at some problems involving cofunction identities and reflection.
1. Find the value of \(\cos 120^{\circ}\).
Because this angle has a reference angle of \(60^{\circ}\), the answer is \(\cos 120^{\circ} =−12\).
2. Find the value of \(\ cos(−120^{\circ} )\).
Because this angle has a reference angle of \(60^{\circ}\), the answer is \(\cos(−120^{\circ} )= \cos 240^{\circ} =−\dfrac{1}{2}\).
3. Find the value of \(\sin 135^{\circ}\).
Because this angle has a reference angle of \(45^{\circ}\), the answer is \(\sin 135^{\circ} =\dfrac{\sqrt{2}}{2}\)
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.
Solution
Since you now know the cofunction relationships, you can use your knowledge of sine functions to help you with the cosine computation:
\(\cos 30^{\circ} =\sin\left(90^{\circ} −30^{\circ}\right)=\sin(60^{\circ} )=\dfrac{\sqrt{3}}{2}\)
Find the value of \(\sin 45^{\circ}\) using a cofunction identity.
Solution
The sine of \(45^{\circ}\) is equal to \(\cos\left(90^{\circ} −45^{\circ} \right)=\cos 45^{\circ} =\dfrac{\sqrt{2}}{2}\).
Find the value of \(\cos 45^{\circ}\) using a cofunction identity.
Solution
The cosine of \(45^{\circ}\) is equal to \(\sin\left(90^{\circ} −45^{\circ} \right)=\sin 45^{\circ} =\dfrac{\sqrt{2}}{2}\).
Find the value of \(\cos 60^{\circ}\) using a cofunction identity.
Solution
The cosine of \(60^{\circ}\) is equal to \(\sin\left(90^{\circ} −60^{\circ} \right)=\sin 30^{\circ} =.5\).
Review
- Find a value for \(\theta\) for which \(\sin\theta =\cos15^{\circ}\) is true.
- Find a value for \(\theta\) for which \(\cos\theta =\sin55^{\circ}\) is true.
- Find a value for \(\theta\) for which \(\tan\theta =\cot80^{\circ}\) is true.
- Find a value for \(\theta\) for which \(\cot\theta =\tan30^{\circ}\) is true.
- Use cofunction identities to help you write the expression \(\tan 255^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\sin 120^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\cos 310^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\cot 260^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\cos 280^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\tan 60^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\sin 100^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\cos 70^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use cofunction identities to help you write the expression \(\cot 240^{\circ}\) as the function of an acute angle of measure less than \(45^{\circ}\).
- Use a right triangle to prove that \(\sin \theta =\cos(90^{\circ} −\theta )\).
- Use the sine and cosine cofunction identities to prove that \(\tan(90^{\circ} −\theta )=\cot\theta\).
Review (Answers)
To see the Review answers, open this PDF file and look for section 1.24.
Vocabulary
| Term | Definition |
|---|---|
| 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. |
Additional Resources
Video: Cofunctions