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MAT 151: College Algebra with Trigonometry

Subsection 5.2 More Trigonometric Identities Exercises

  1. Prove each trigonometric identity.
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    1. \(\displaystyle \frac{\sin x}{1-\cos (-x)}=\frac{1+ \cos (-x)}{\sin x}\)
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    2. \(\displaystyle \frac{\tan (-x)}{\sin (-x)}=\sec x\)
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    3. \(\displaystyle (2\cos^2x-1)^2+(2\cos x \sin x)^2=1\)
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    4. \(\displaystyle \sin (2x) = \frac{2 \tan x}{1+\tan^2x}\)
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    5. \(\displaystyle (\cos x - \sin x)^2 = 1 - \sin (2x)\)
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    6. \(\displaystyle (\cos x + \sin x)^2 = 1 + \sin (2x)\)
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    7. \(\displaystyle \csc (2x)-\cot(2x) = \tan x\)
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    8. \(\displaystyle \sin(\frac{\pi}{2}+x)=\cos x\)
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    9. \(\displaystyle \csc 2 \theta=\frac{\csc \theta}{2 \cos \theta}\)
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  2. The graph of \(f(x)\) is shown below. Let \(g(x)=f(x-1)+3\text{.}\)
    A graph of a function on a coordinate plane composed of multiple pieces. The graph begins at (-2,-2) and increases at a constant rate up to the origin. At x = 0, the graph transitions smoothly into a curved segment that increases and levels off until (2,4). At (2,4) the graph transitions to a horizontal line at y=2 and ends at (6,2).
    Figure 5.2.7.
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    1. Describe the transformations that would be required to transform \(f(x)\) in to \(g(x)\text{.}\)
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    2. On the same set of axes, sketch the graph of \(g(x)\text{.}\)
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    3. What are the domain and range of \(f(x)\text{.}\)
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    4. What are the domain and range of \(g(x)\text{.}\)
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  3. Perform the indicated operation and simplify if possible.
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    1. \(\displaystyle \frac{s^2+2s-8}{2s^2-8}\)
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    2. \(\displaystyle \frac{b^2+5cb-6c^2}{2b^2} \cdot \frac{12b}{4b-4c}\)
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    3. \(\displaystyle \frac{5}{v}+\frac{3}{v-1}\)
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  4. For each angle \(\theta\) below, find \(\sin \theta\text{,}\) \(\cos \theta\text{,}\) \(\tan \theta\text{,}\) \(\sec \theta\text{,}\) \(\csc \theta\) and \(\cot \theta\text{.}\)
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    1. \(\displaystyle \theta=-\frac{\pi}{2}\)
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    2. \(\displaystyle \theta=18\pi\)
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    3. \(\displaystyle \theta=-\frac{15 \pi}{4}\)
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  5. Suppose that in \(\triangle ABC\text{,}\) \(m\angle B =\frac{3\pi}{5}\text{,}\) \(m\angle C = \frac{\pi}{18}\text{,}\) and \(AC=10\text{.}\) Find the length \(AB\text{.}\)
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  6. Let \(z(t)=3 \cos (4t)\text{.}\)
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    1. On the domain \(0 \leq t \leq 2\pi\text{,}\) list all intervals on which \(z\) is decreasing.
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    2. What is the average rate of change of \(z\) on the interval \(\frac{\pi}{6} \leq t \leq \frac{\pi}{3}\text{?}\)
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    3. Find the inverse function \(z^{-1}\) and state the domain and range of \(z^{-1}\text{.}\)
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  7. A platform is in the shape of a circle, and has 12 wedge-shaped tiles of equal size. The circle has a radius of 8 feet.
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    1. What is the length of the arc along the edge of one tile?
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    2. Draw a diagram and impose coordinates so that the center of the circle is at \((0,0)\text{,}\) and the edge of one tile lines up along the positive \(x\)-axis.
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    3. A bug is crawling clockwise along the edge of the circle, and begins at the point \((8,0)\text{.}\) Suppose he crawls at a rate of 1 foot/second. Write parametric equations for the bug’s position at time \(t\) seconds after he begins his journey.
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  8. Refer to \(f(x)\) in Figure 5.2.8.
    A graph of a sinusoidal function on a coordinate plane. The curve oscillates smoothly between a maximum value at y = 1 and a minimum value at y = −5, with a midline at y = −2. The graph has a peak at x = 0 and another x = 8, and troughs at x = 4 and x = 12, indicating a repeating wave pattern.
    Figure 5.2.8.
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    1. Write an equation for \(f(x)\text{.}\)
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    2. Draw a graph of \(f(2x)\) on the interval \(0 \leq x \leq 12\text{.}\)
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  9. Refer to the function \(a(x)=10e^x\text{.}\) Let \(b(x)=4x+3\text{.}\)
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    1. Graph \(a(x)\) on the domain \(-2 \leq x \leq 5\text{.}\)
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    2. Solve \(a(x) > 50\text{.}\)
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    3. Write a function equation for the inverse function, \(a^{-1}(x)\text{.}\)
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    4. Let \(c(x)=a(b(x))\text{.}\) Write a function equation for \(c(x)\text{.}\)
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    5. Let \(d(x)=a(x)+b(x)\text{.}\) Write a function equation for \(d(x)\text{.}\)
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