1.7.R: Trigonometric Identities and Equations (Review)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)7.1: Solving Trigonometric Equations with Identities
For the exercises 1-6, find all solutions exactly that exist on the interval \([0,2\pi )\)
1) \(\csc ^2 t=3\)
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\(\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right ), \pi -\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right ), \pi +\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right ), 2\pi -\sin^{-1}\left ( \dfrac{\sqrt{3}}{3} \right )\)
2) \(\cos ^2 x=\dfrac{1}{4}\)
3) \(2\sin \theta =-1\)
- Answer
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\(\dfrac{7\pi }{6}, \dfrac{11\pi }{6}\)
4) \(\tan x \sin x+\sin(-x)=0\)
5) \(9\sin \omega -2=4\sin^2 \omega\)
- Answer
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\(\sin^{-1}\left ( \dfrac{1}{4} \right ), \pi -\sin^{-1}\left ( \dfrac{1}{4} \right )\)
6) \(1-2\tan(\omega )=\tan^2(\omega )\)
For the exercises 7-8, use basic identities to simplify the expression.
7) \(\sec x \cos x+\cos x-\dfrac{1}{\sec x}\)
- Answer
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\(1\)
8) \(\sin^3 x+\cos^2 x \sin x\)
For the exercises 9-10, determine if the given identities are equivalent.
9) \(\sin^2 x+\sec^2 x -1=\dfrac{(1-\cos ^2 x)(1+\cos ^2 x)}{\cos ^2 x}\)
- Answer
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Yes
10) \(\tan^3 x \csc^2 x \cot^2 x \cos x \sin x=1\)
7.2: Sum and Difference Identities
For the exercises 1-4, find the exact value.
1) \(\tan \left (\dfrac{7\pi }{12} \right )\)
- Answer
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\(-2-\sqrt{3}\)
2) \(\cos \left (\dfrac{25\pi }{12} \right )\)
3) \(\sin(70^{\circ})\cos(25^{\circ})-\cos(70^{\circ})\sin(25^{\circ})\)
- Answer
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\(\dfrac{\sqrt{2}}{2}\)
4) \(\cos(83^{\circ})\cos(23^{\circ})+\sin(83^{\circ})\sin(23^{\circ})\)
For the exercises 5-6, prove the identity.
5) \(\cos(4x)-\cos(3x)\cos x=\sin^2 x-4\cos^2 x \sin^2 x\)
- Answer
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\(\begin{align*}
\cos(4x)-\cos(3x)\cos x &= \cos(2x+2x)-\cos(x+2x)\cos x\\
&= \cos(2x)\cos(2x)-\sin(2x)\sin(2x)-\cos x \cos(2x)\cos x+\sin x \sin(2x)\cos x\\
&= (\cos ^2 x-\sin ^2 x)^2-4\cos ^2 x \sin ^2 x-\cos ^2 x(\cos ^2
x-\sin ^2 x)+\sin x(2)\sin x \cos x \cos x\\
&= (\cos ^2 x-\sin ^2 x)^2-4\cos ^2 x \sin ^2 x-\cos ^2 x(\cos ^2 x-\sin ^2 x)+2\sin ^2 x \cos ^2 x\\
&= \cos ^4x-2\cos^2x\sin^2x+\sin ^4-\cos^2x\sin^2x-\cos^4x+\cos^2x\sin^2x+2\sin^2x\cos^2x\\
&= \sin^4x-4\cos^2x\sin^2x+\cos^2x\sin^2x\\
&= \sin^2x(\sin^2x+\cos^2x)-4\cos^2x\sin^2x\\
&= \sin^2 x-4\cos^2 x \sin^2 x
\end{align*}\)
6) \(\cos(3x)-\cos^3x=-\cos x \sin^2x-\sin x \sin(2x)\)
For exercise 7, simplify the expression.
7) \(\dfrac{\tan \left ( \tfrac{1}{2}x \right )+\tan \left ( \tfrac{1}{8}x \right )}{1-\tan \left ( \tfrac{1}{8}x \right )\tan \left ( \tfrac{1}{2}x \right )}\)
- Answer
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\(\tan \left ( \dfrac{5}{8}x \right )\)
For the exercises 8-9, find the exact value.
8) \(\cos \left ( \sin^{-1}(0)-\cos^{-1}\left ( \dfrac{1}{2} \right ) \right )\)
9) \(\tan \left ( \sin^{-1}(0)-\sin^{-1}\left ( \dfrac{1}{2} \right ) \right )\)
- Answer
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\(\dfrac{\sqrt{3}}{3}\)
7.3: Double-Angle, Half-Angle, and Reduction Formulas
For the exercises 1-4, find the exact value.
1) Find \(\sin (2\theta )\), \(\cos (2\theta )\)
2) Find \(\sin (2\theta )\), \(\cos (2\theta )\), and \(\tan (2\theta )\) given \(\sec \theta =-\dfrac{5}{3}\) and \(\theta \) is in the interval \(\left [\dfrac{\pi }{2} , \pi \right ]\).
- Answer
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\(-\dfrac{24}{25}, -\dfrac{7}{25}, \dfrac{24}{7}\)
3) \(\sin \left (\dfrac{7\pi }{8} \right )\)
4) \(\sec \left (\dfrac{3\pi }{8} \right )\)
- Answer
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\(\sqrt{2(2+\sqrt{2})}\)
For the exercises 5-6, use Figure below to find the desired quantities.
5) \(\sin(2\beta ),\cos(2\beta ),\tan(2\beta ),\sin(2\alpha ),\cos(2\alpha ),\tan(2\alpha )\)
6) \(\sin \left (\frac{\beta }{2} \right ) ,\cos\left (\frac{\beta }{2} \right ),\tan\left (\frac{\beta }{2} \right ),\sin\left (\frac{\alpha }{2} \right ),\cos\left (\frac{\alpha }{2} \right ),\tan\left (\frac{\alpha }{2} \right )\)
- Answer
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\(\dfrac{\sqrt{2}}{10},\dfrac{7\sqrt{2}}{10},\dfrac{1}{7},\dfrac{3}{5},\dfrac{4}{5},\dfrac{3}{4}\)
For the exercises 7-8, prove the identity.
7) \(\dfrac{2\cos (2x)}{\sin (2x)}=\cot x-\tan x\)
8) \(\cot x \cos (2x) = -\sin (2x)+\cot x\)
- Answer
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\(\begin{align*} \cot x \cos (2x) &= \cot x(1-2\sin ^2 x)\\ &= \cot x-\dfrac{\cos x}{\sin x}(2)\sin ^2 x\\ &= -2\sin x \cos \\ &= -\sin (2x)+\cot x \end{align*}\)
For the exercises 9-10, rewrite the expression with no powers.
9) \(\cos ^2 x \sin ^4 (2x)\)
10) \(\tan ^2 x \sin ^3 x\)
- Answer
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\(\dfrac{10\sin x-5\sin (3x)+\sin (5x)}{8(\cos (2x)+1)}\)
7.4: Sum-to-Product and Product-to-Sum Formulas
For the exercises 1-3, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
1) \(\cos \left ( \dfrac{\pi }{3} \right )\sin \left ( \dfrac{\pi }{4} \right )\)
2) \(2\sin \left ( \dfrac{2\pi }{3} \right )\sin \left ( \dfrac{5\pi }{6} \right )\)
- Answer
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\(\dfrac{\sqrt{3}}{2}\)
3) \(2\cos \left ( \dfrac{\pi }{5} \right )\cos \left ( \dfrac{\pi }{3} \right )\)
For the exercises 4-5, evaluate the sum by using a product formula. Write the exact answer.
4) \(\sin \left ( \dfrac{\pi }{12} \right )-\sin \left ( \dfrac{7\pi }{12} \right )\)
- Answer
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\(-\dfrac{\sqrt{2}}{2}\)
5) \(\cos \left ( \dfrac{5\pi }{12} \right )+\cos \left ( \dfrac{7\pi }{12} \right )\)
For the exercises 6-9, change the functions from a product to a sum or a sum to a product.
6) \(\sin(9x)\cos(3x)\)
- Answer
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\(\dfrac{1}{2}(\sin(6x)+\sin(12x))\)
7) \(\cos(7x)\cos(12x)\)
8) \(\sin(11x)+\sin(2x)\)
- Answer
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\(2\sin \left (\dfrac{13}{2}x \right )\cos \left (\dfrac{9}{2}x \right )\)
9) \(\cos(6x)+\cos(5x)\)
7.5: Solving Trigonometric Equations
For the exercises 1-2, find all exact solutions on the interval \([0,2\pi )\).
1) \(\tan x+1=0\)
- Answer
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\(\dfrac{3\pi }{4}, \dfrac{7\pi }{4}\)
2) \(2\sin(2x)+\sqrt{2}=0\)
For the exercises 3-7, find all exact solutions on the interval \([0,2\pi )\).
3) \(2\sin^2 x-\sin x=0\)
- Answer
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\(0, \dfrac{\pi }{6}, \dfrac{5\pi }{6}, \pi \)
4) \(\cos^2 x-\cos x -1=0\)
5) \(2\sin^2 x+5\sin x +3=0\)
- Answer
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\(\dfrac{3\pi }{2}\)
6) \(\cos x - 5\sin(2x)=0\)
7) \(\dfrac{1}{\sec ^2 x}+2+\sin^2 x+4\cos ^2 x=0\)
- Answer
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No solution.
For the exercises 8-9, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval \([0,2\pi )\)
8) \(\sqrt{3}\cot ^2 x+\cot x=1\)
9) \(\csc ^2 x-3\csc x-4=0\)
- Answer
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\(0.2527,2.8889,4.7124\)
For the exercises 10-11, graph each side of the equation to find the zeroes on the interval \([0,2\pi )\).
10) \(20\cos^2x+21\cos x+1=0\)
11) \(\sec^2x-2\sec x=15\)
- Answer
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\(1.3694, 1.9106, 4.3726, 4.9137\)
7.6: Modeling with Trigonometric Equations
For the exercises 1-3, graph the points and find a possible formula for the trigonometric values in the given table.
1)
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \(y\) | 1 | 6 | 11 | 6 | 1 | 6 |
2)
| \(x\) | \(y\) |
|---|---|
| 0 | -2 |
| 1 | 1 |
| 2 | -2 |
| 3 | -5 |
| 4 | -2 |
| 5 | 1 |
- Answer
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\(3\sin \left ( \dfrac{x\pi }{2} \right )-2\)
3)
| \(x\) | \(y\) |
|---|---|
| -3 | \(3+2\sqrt{2}\) |
| -2 | 3 |
| -1 | \(2\sqrt{2}-1\) |
| 0 | 1 |
| 1 | \(3-2\sqrt{2}\) |
| 2 | -1 |
| 3 | \(-1-2\sqrt{2}\) |
4) A man with his eye level \(6\) feet above the ground is standing \(3\) feet away from the base of a \(15\)-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?
- Answer
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\(71.6^{\circ}\)
5) Using the ladder from the previous exercise, if a \(6\)-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?
For the exercises 6-7, construct functions that model the described behavior.
6) A population of lemmings varies with a yearly low of \(500\) in March. If the average yearly population of lemmings is \(950\), write a function that models the population with respect to \(t\)
- Answer
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\(P(t)=950-450\sin \left ( \dfrac{\pi }{6}t \right )\)
7) Daily temperatures in the desert can be very extreme. If the temperature varies from \(90^{\circ}\)F to \(30^{\circ}\)F and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.
For the exercises 8-9, find the amplitude, frequency, and period of the given equations.
8) \(y=3\cos(x\pi )\)
- Answer
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Amplitude: \(3\), period: \(2\), frequency: \(\dfrac{1}{2}\)Hz
9) \(y=-2\sin(16x\pi )\)
For the exercises 10-11, model the described behavior and find requested values.
10) An invasive species of carp is introduced to Lake Freshwater. Initially there are \(100\) carp in the lake and the population varies by \(20\) fish seasonally. If by year \(5\), there are \(625\) carp, find a function modeling the population of carp with respect to \(t\)
- Answer
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\(C(t)=20\sin (2\pi t)+100(1.4427)^t\)
11) The native fish population of Lake Freshwater averages \(2500\) fish, varying by \(100\) fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by \(5\%\) each year. Find a function modeling the population of native fish with respect to \(t\)
Practice Test
For the exercises 1-2, simplify the given expression.
1) \(\cos(-x)\sin x \cot x+\sin^2x\)
- Answer
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\(1\)
2) \(\sin(-x)\cos(-2x)-\sin(-x)\cos(-2x)\)
For the exercises 3-6, find the exact value.
3) \(\cos \left ( \dfrac{7\pi }{12} \right )\)
- Answer
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\(\dfrac{\sqrt{2}-\sqrt{6}}{4}\)
4) \(\tan \left ( \dfrac{3\pi }{8} \right )\)
5) \(\tan \left (\sin^{-1}\left (\dfrac{\sqrt{2}}{2} \right )+\tan^{-1}\sqrt{3} \right )\)
- Answer
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\(-\sqrt{2}-\sqrt{3}\)
6) \(2\sin \left (\dfrac{\pi }{4} \right )\sin \left (\dfrac{\pi }{6} \right )\)
For the exercises 7-16, find all exact solutions to the equation on \([0,2\pi )\).
7) \(\cos^2x-\sin^2x-1=0\)
- Answer
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\(0, \pi \)
8) \(\cos^2x=\cos x\)
- Answer
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\(\sin^{-1}\left (\dfrac {1}{4}\left(\sqrt{13}-1\right) \right ), \pi - \sin^{-1}\left (\dfrac {1}{4}\left(\sqrt{13}-1\right) \right )\)
9) \(\cos (2x)+\sin ^2 x = 0\)
10) \(2\sin ^2 x - \sin x = 0\)
- Answer
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\(0, \dfrac{\pi }{6}, \dfrac{5\pi }{6}, \pi\)
11) Rewrite the expression as a product instead of a sum: \(\cos (2x)+\cos (-8x)\)
12) Find all solutions of \(\tan (x)-\sqrt{3}=0\).
- Answer
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\(\dfrac{\pi }{3}+k\pi\)
13) Find the solutions of \(\sec ^2x -2\sec x=15\) on the interval \([0,2\pi )\) algebraically; then graph both sides of the equation to determine the answer.
14) Find \(\sin (2\theta )\), \(\cos (2\theta )\), and \(\tan (2\theta )\) given \(\cot \theta =-\dfrac{3}{4}\) and \(\theta \) is on the interval \(\left [ \dfrac{\pi }{2}, \pi \right ]\).
- Answer
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\(-\dfrac{24}{25}, -\dfrac{7}{25}, \dfrac{24}{7}\)
15) Find \(\sin \left (\dfrac{\theta }{2} \right )\), \(\cos \left (\dfrac{\theta }{2} \right )\), and \(\tan \left (\dfrac{\theta }{2} \right )\) given \(\cos \theta =-\dfrac{7}{25}\) and \(\theta \) is in quadrant \(\mathrm{IV}\).
16) Rewrite the expression \(\sin ^4 x\) with no powers greater than \(1\).
- Answer
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\(\dfrac{1}{8}(3+\cos (4x)-4\cos (2x))\)
For the exercises 17-19, prove the identity.
17) \(\tan^3x-\tan x \sec^2x=\tan(-x)\)
18) \(\sin(3x)-\cos x \sin(2x)=\cos^2x \sin x-\sin^3x\)
- Answer
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\(\begin{align*} \sin(3x)-\cos x \sin(2x) &= \\ \sin(x+2x)-\cos x(2\sin x \cos x) &= \\ \sin x \cos(2x)+\sin(2x)\cos x -2\sin x \cos ^2x &= \\ \sin x(\cos ^2x - \sin ^2x)+2\sin x \cos x \cos x - 2\sin x \cos ^2x &= \\ \sin x \cos ^2x - \sin ^3x +0 &= \\ \cos^2x \sin x - \sin ^3x &= \cos^2x \sin x-\sin^3x \end{align*}\)
19) \(\dfrac{\sin (2x)}{\sin x}-\dfrac{\cos (2x)}{\cos x}=\sec x\)
20) Plot the points and find a function of the form \(y=A\cos(Bx+C)+D\) that fits the given data.
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \(y\) | -2 | 2 | -2 | 2 | -2 | 2 |
- Answer
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\(y=2\cos(\pi x+\pi )\)
21) The displacement \(h(t)\) in centimeters of a mass suspended by a spring is modeled by the function \(h(t)=\dfrac{1}{4}\sin (120\pi t)\)
22) A woman is standing \(300\) feet away from a \(2000\)-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15th floor (\(1500\) feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.
- Answer
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\(81.5^{\circ}, 78.7^{\circ}\)
23) Two frequencies of sound are played on an instrument governed by the equation \(n(t)=8\cos(20\pi t)\cos(1000\pi t)\)
24) The average monthly snowfall in a small village in the Himalayas is \(6\) inches, with the low of \(1\) inch occurring in July. Construct a function that models this behavior. During what period is there more than \(10\) inches of snowfall?
- Answer
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\(6+5\cos \left ( \dfrac{\pi }{6}(1-x) \right )\). From November 23 to February 6.
25) A spring attached to a ceiling is pulled down \(20\) cm. After \(3\) seconds, wherein it completes \(6\) full periods, the amplitude is only \(15\) cm. Find the function modeling the position of the spring \(t\) seconds after being released. At what time will the spring come to rest? In this case, use \(1\) cm amplitude as rest.
26) Water levels near a glacier currently average \(9\) feet, varying seasonally by \(2\) inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady \(3\) inches. Find a function modeling the depth of the water \(t\) months from now. If the docks are \(2\) feet above current water levels, at what point will the water first rise above the docks?
- Answer
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\(D(t)=2\cos \left ( \dfrac{\pi }{6}t \right )+108+\dfrac{1}{4}t\), \(93.5855\) months (or \(7.8\) years) from now
Contributors and Attributions
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.