1.
[latex]\csc 27°[/latex]
Chapter 8: More Functions and Identities
Practice each skill in the Homework Problems listed.
For Problems 1–8, evaluate. Round answers to 3 decimal places.
[latex]\csc 27°[/latex]
[latex]\sec 8°[/latex]
[latex]\cot 65°[/latex]
[latex]\csc 11°[/latex]
[latex]\sec 1.4[/latex]
[latex]\cot 4.3[/latex]
[latex]\csc \dfrac{5\pi}{16}[/latex]
[latex]\sec \dfrac{7\pi}{20}[/latex]
For Problems 9–16, evaluate. Give exact values.
[latex]\csc 30°[/latex]
[latex]\sec 0°[/latex]
[latex]\cot 45[/latex]
[latex]\csc 60°[/latex]
[latex]\sec 150°[/latex]
[latex]\cot 120°[/latex]
[latex]\csc 135°[/latex]
[latex]\sec 270°[/latex]
For Problems 17–18, complete the tables with exact values.
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{6}[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{3}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{2\pi}{3}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\dfrac{5\pi}{6}[/latex] | [latex]\pi[/latex] |
| [latex]\sec \theta[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\csc \theta[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\cot \theta[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\theta[/latex] | [latex]\pi[/latex] | [latex]\dfrac{7\pi}{6}[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{4\pi}{3}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{5\pi}{3}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]\dfrac{11\pi}{6}[/latex] | [latex]2\pi[/latex] |
| [latex]\sec \theta[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\csc \theta[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\cot \theta[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
Evaluate. Round answers to three decimal places.
Evaluate. Round answers to three decimal places.
For Problems 21–28, find exact values for the six trigonometric ratios of the angle [latex]\theta{.}[/latex]
The distance that sunlight must travel to pass through a layer of Earth’s atmosphere depends on both the thickness of the atmosphere and the angle of the sun.
In railroad design, the degree of curvature of a section of track is the angle subtended by a chord 100 feet long.
When a plane is tilted by an angle [latex]\theta[/latex] from the horizontal, the time required for a ball starting from rest to roll a horizontal distance of [latex]l[/latex] feet on the plane is
[latex]t=\sqrt{\dfrac{l}{8}\csc(2\theta)}~~ {seconds}[/latex]
After a heavy rainfall, the depth, [latex]D{,}[/latex] of the runoff flow at a distance [latex]x[/latex] feet from the watershed down a slope at angle [latex]\alpha[/latex] is given by
[latex]D=(kx)^{0.6}(\cot \alpha)^{0.3}~~{inches}[/latex]
where [latex]k[/latex] is a constant determined by the surface roughness and the intensity of the runoff.
For Problems 33–38, write algebraic expressions for the six trigonometric ratios of the angle [latex]\theta{.}[/latex]
The diagram shows a unit circle. Find six line segments whose lengths are, respectively, [latex]\sin t,~ \cos t,~ \tan t,~ \sec t,~ \csc t,[/latex] and [latex]\cot t{.}[/latex]
Use the figure in Problem 39 to find each area in terms of the angle [latex]t{.}[/latex]
For Problems 41–46, sketch the reference angle and find exact values for all six trigonometric functions of the angle.
[latex]\sec \theta = 2,~~\theta[/latex] in Quadrant IV
[latex]\csc \phi = 4,~~\phi[/latex] in Quadrant II
[latex]\csc \alpha = 3,~~\alpha[/latex] in Quadrant I
[latex]\sec \beta = 4,~~\beta[/latex] in Quadrant IV
[latex]\cot \gamma = \dfrac{1}{4},~~\gamma[/latex] in Quadrant III
[latex]\tan \theta = 6,~~\theta[/latex] in Quadrant I
For Problems 47–52, evaluate.
[latex]4 \cot \dfrac{\pi}{3} + 2\sec \dfrac{\pi}{4}[/latex]
[latex]\dfrac{1}{2} \csc \dfrac{\pi}{6} - \dfrac{1}{4}\cot \dfrac{\pi}{6}[/latex]
[latex]\dfrac{1}{2} \csc \dfrac{5\pi}{3}\cot \dfrac{3\pi}{4}[/latex]
[latex]6 \cot \dfrac{7\pi}{6} \sec \dfrac{5\pi}{4}[/latex]
[latex]\left(\csc \dfrac{2\pi}{3} - \sec \dfrac{3\pi}{4}\right)^2[/latex]
[latex]\sec^2 \dfrac{5\pi}{6} \csc^2 \dfrac{4\pi}{3}[/latex]
Complete the table and sketch a graph of [latex]y=\sec x{.}[/latex]
| [latex]x[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]\sec x[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
Complete the table and sketch a graph of [latex]y=\csc x{.}[/latex]
| [latex]x[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]\csc x[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
Use the graph of [latex]y=\sin x[/latex] to sketch a graph of its reciprocal, [latex]y=\csc x{.}[/latex]
Use the graph of [latex]y=\cos x[/latex] to sketch a graph of its reciprocal, [latex]y=\sec x{.}[/latex]
Complete the table and sketch a graph of [latex]y=\cot x{.}[/latex]
| [latex]x[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]\cot x[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
Use the graphs of [latex]y=\cos x[/latex] and [latex]y=\sin x[/latex] to sketch a graph of [latex]y=\cot x = \dfrac{\cos x}{\sin x}{.}[/latex]
For Problems 59-64,
[latex]y=\dfrac{\csc x}{\cot x}[/latex]
[latex]y=\dfrac{\sec x}{\tan x}[/latex]
[latex]y=\dfrac{\sec x \cot x}{\csc x}[/latex]
[latex]y=\dfrac{\csc x \tan x}{\sec x}[/latex]
[latex]y=\tan x \csc x[/latex]
[latex]y=\sin x \sec x[/latex]
For Problems 65–70, find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
[latex]3\csc \theta + 2=8[/latex]
[latex]-2\sec \theta + 7=3[/latex]
[latex]\sqrt{2}\sec \theta =-2[/latex]
[latex]8+\csc \theta =6[/latex]
[latex]2\cot \theta = -\sqrt{12}[/latex]
[latex]\sqrt{3}\cot \theta =1[/latex]
For Problems 71–76, use identities to find exact values or to write algebraic expressions.
If [latex]\tan \alpha = -2[/latex] and [latex]\dfrac{\pi}{2} \lt \alpha \lt \pi{,}[/latex] find [latex]\cos \alpha{.}[/latex]
If [latex]\cot \beta = \dfrac{5}{4}[/latex] and [latex]\pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find [latex]\sin \beta{.}[/latex]
If [latex]\sec x = \dfrac{a}{2}[/latex] and [latex]0 \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\tan x{.}[/latex]
If [latex]\csc y = \dfrac{1}{b}[/latex] and [latex]\dfrac{\pi}{2} \lt y \lt \pi{,}[/latex] find [latex]\cot y{.}[/latex]
If [latex]\csc \phi = w[/latex] and [latex]\dfrac{3\pi}{2} \lt \alpha \lt 2\pi{,}[/latex] find [latex]\cos \phi{.}[/latex]
If [latex]\sec \theta = \dfrac{3}{z}[/latex] and [latex]\pi \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\sin \theta{.}[/latex]
For Problems 77–80, find exact values for [latex]\sec s,~ \csc s,[/latex] and [latex]\cot s{.}[/latex]
For Problems 81–88, write the expression in terms of sine and cosine, and simplify.
[latex]\sec \theta \tan \theta[/latex]
[latex]\csc \phi \cot \phi[/latex]
[latex]\dfrac{\csc t}{cot t}[/latex]
[latex]\dfrac{\tan v}{\sec v}[/latex]
[latex]\sec \beta - \tan \beta[/latex]
[latex]\cot \alpha + \csc \alpha[/latex]
[latex]\sin x \tan x - \sec x[/latex]
[latex]\csc y - \cos y \cot y[/latex]
Prove the Pythagorean identity [latex]1 + \tan^2 \theta = \sec^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\cos^2 \theta{.}[/latex])
Prove the Pythagorean identity [latex]1 + \cot^2 \theta = \csc^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\sin^2 \theta{.}[/latex])
Suppose that [latex]\cot \theta = 5[/latex] and [latex]\theta[/latex] lies in the third quadrant.
Suppose that [latex]\tan \theta = -2[/latex] and [latex]\theta[/latex] lies in the second quadrant.
Write each of the other five trig functions in terms of [latex]\sin t[/latex] only.
Write each of the other five trig functions in terms of [latex]\cos t[/latex] only.
Show that if the angles of a triangle are [latex]A,~B,[/latex] and [latex]C[/latex] and the opposite sides are respectively [latex]a,~b,[/latex] and [latex]c,[/latex] then
[latex]a \csc A = b \csc B = c \csc C[/latex]
The figure shows a unit circle and an angle [latex]\theta[/latex] in standard position. Each of the six trigonometric ratios for [latex]\theta[/latex] is represented by the length of a line segment in the figure. Find the line segment for each ratio and explain your choice.