27.4: Review of trigonometric functions
- Page ID
- 68485
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\[\begin{array}{c||c|c|c|c|c}
& 0 & \dfrac{\pi}{6} & \dfrac{\pi}{4} & \dfrac{\pi}{3} & \dfrac{\pi}{2} \\
\hline \hline \sin (x) & & & & & \\
\hline \cos (x) & & & & & \\
\hline \tan (x) & & & & &
\end{array} \nonumber \]
- Answer
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\(\begin{array}{c||c|c|c|c|c}
x & 0=0^{\circ} & \dfrac{\pi}{6}=30^{\circ} & \dfrac{\pi}{4}=45^{\circ} & \dfrac{\pi}{3}=60^{\circ} & \dfrac{\pi}{2}=90^{\circ} \\
\hline \hline \sin (x) & 0 & \dfrac{1}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{3}}{2} & 1 \\
\hline \cos (x) & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{1}{2} & 0 \\
\hline \tan (x) & 0 & \dfrac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text { undef. }
\end{array}\)
Find the exact values of
- \(\cos\left(-\dfrac{\pi}{6}\right)\)
- \(\sin\left(-\dfrac{\pi}{4}\right)\)
- \(\tan\left(-\dfrac{\pi}{3}\right)\)
- Answer
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- \(\dfrac{\sqrt{3}}{2}\)
- \(\dfrac{-\sqrt{2}}{2}\)
- \(-\sqrt{3}\)
Find the exact value of the trigonometric function.
- \(\sin\left(\dfrac{5\pi}{4}\right)\)
- \(\cos\left(\dfrac{11\pi}{6}\right)\)
[Hint: Use the special \(45^\circ-45^\circ-90^\circ\) or \(30^\circ-60^\circ-90^\circ\) triangles to find the solution.]
- Answer
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- \(\dfrac{-\sqrt{2}}{2}\)
- \(\dfrac{\sqrt{3}}{2}\)
Find the amplitude, period, and the phase shift of the given function. Draw the graph over a one-period interval. Label all maxima, minima and intercepts.
- \(y=3 \cos\left(4 x-\pi\right)\)
- \(y=-5\sin\left(x+\dfrac{\pi}{2}\right)\)
- Answer
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- amplitude \(3\), period \(\dfrac{\pi}{2}\), phase-shift \(\dfrac{\pi}{4}\)
- amplitude \(5\), period \(2\pi \), phase-shift \(\dfrac{-\pi}{2}\)
- amplitude \(3\), period \(\dfrac{\pi}{2}\), phase-shift \(\dfrac{\pi}{4}\)
Find the exact trigonometric function value.
- \(\cos\left(\dfrac{\pi}{12}\right)\) [Hint: Use the addition and subtraction of angles formulas.]
- \(\cos\left(\dfrac{3\pi}{8}\right)\) [Hint: Use the half-angles formulas.]
- Answer
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- \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
- \(\dfrac{\sqrt{2-\sqrt{2}}}{2}\)
Let \(\sin(\alpha)=-\dfrac{4}{5}\) and let \(\alpha\) be in quadrant III. Find \(\sin(2\alpha)\), \(\cos(2\alpha)\), and \(\tan(2\alpha)\).
- Answer
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\(\sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}\)
Find the exact value of:
- \(\sin^{-1}\left(-\dfrac{1}{2}\right)\)
- \(\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)\)
- \(\tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)\)
- Answer
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- \(\dfrac{-\pi}{6}\)
- \(\dfrac{5 \pi}{6}\)
- \(\dfrac{-\pi}{6}\)
Solve for \(x\): \(2\sin(x)+\sqrt{3}=0\)
- Answer
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\(x=(-1)^{n+1} \dfrac{\pi}{3}+n \pi\), where \(n=0, \pm 1, \ldots \)
Solve for \(x\): \(\tan^2(x)-1=0\)
- Answer
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\(x=\pm \dfrac{\pi}{4}+n \pi \) where \(n=0, \pm 1, \ldots \)
Solve for \(x\).
- \(2\cos^2(x)-1=0\)
- \(2\sin^2(x)+15\sin(x)+7=0\)
- Answer
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- \(x=\pm \dfrac{\pi}{4}+2 n \pi\) or \(x=\pm \dfrac{3 \pi}{4}+2 n \pi\) where \(n=0, \pm 1, \ldots\)
- \((-1)^{n+1} \dfrac{\pi}{6}+n \pi\) where \(n=0, \pm 1, \ldots\)