1.
[latex]\sin \left(\beta + \dfrac{\pi}{4}\right)=\sin \beta + \dfrac{1}{\sqrt{2}}[/latex]
Chapter 8: More Functions and Identities
Problems: #6, 14, 18, 24, 26, 30, 36, 42, 48, 56, 62, 64, 72, 76, 80
For Problems 1–8, answer true or false.
[latex]\sin \left(\beta + \dfrac{\pi}{4}\right)=\sin \beta + \dfrac{1}{\sqrt{2}}[/latex]
[latex]\cos \left(\dfrac{\pi}{3} - t\right)=\dfrac{1}{2}-\cos t[/latex]
[latex]\tan (z-w)=\dfrac{\sin (z-w)}{\cos (z-w)}[/latex]
[latex]\sin 2\phi = 1 - \cos 2\phi[/latex]
[latex]\sin \left(\dfrac{\pi}{2}-x\right)=1-\sin x[/latex]
[latex]\sin (\pi - x)=\sin x[/latex]
[latex]\cos^2 \alpha - \sin^2 \alpha = -1[/latex]
[latex]\tan^{-1} s = \dfrac{1}{\tan s}[/latex]
If [latex]\sin x = -0.4[/latex] and [latex]\cos x \gt 0,[/latex] find an exact value for [latex]\cos \left(x + \dfrac{3\pi}{4}\right)\text{.}[/latex]
If [latex]\cos x = -0.75[/latex] and [latex]\sin x \lt 0,[/latex] find an exact value for [latex]\cos \left(x - \dfrac{4\pi}{3}\right)\text{.}[/latex]
If [latex]\cos \theta = \dfrac{-3}{8},~ \pi \lt \theta \lt \dfrac{3\pi}{2},[/latex] and [latex]\sin \phi = \dfrac{1}{4},~ \dfrac{\pi}{2} \lt \phi \lt \pi,[/latex] find exact values for
If [latex]\sin \rho = \dfrac{5}{6},~ \dfrac{\pi}{2} \lt \rho \lt \pi,[/latex] and [latex]\cos \mu = \dfrac{-1}{3},~ \dfrac{\pi}{2} \lt \mu \lt \pi,[/latex] find exact values for
If [latex]\tan (x + y) =2[/latex] and [latex]\tan y = \dfrac{1}{3},[/latex] find [latex]\tan x\text{.}[/latex]
If [latex]\tan (x - y) =\dfrac{1}{4}[/latex] and [latex]\tan x = 4,[/latex] find [latex]\tan y\text{.}[/latex]
For Problems 15-16, use the sum and difference formulas to expand each expression.
[latex]\tan \left(t - \dfrac{5\pi}{3}\right)[/latex]
[latex]\cos \left(s+ \dfrac{7\pi}{4}\right)[/latex]
For Problems 17–18, use the figure to find the trigonometric ratios.
For Problems 19–24, use identities to simplify each expression.
[latex]\sin 4x \cos 5x + \cos 4x \sin 5x[/latex]
[latex]\cos 3\beta \cos 1.5 - \sin 3\beta \sin 1.5[/latex]
[latex]\dfrac{\tan 2\phi - \tan 2}{1 + \tan 2\phi \tan 2}[/latex]
[latex]\dfrac{\tan \dfrac{5\pi}{9} - \tan \dfrac{2\pi}{9}}{1 + \tan \dfrac{5\pi}{9} \tan \dfrac{2\pi}{9}}[/latex]
[latex]2\sin 4\theta \cos 4\theta[/latex]
[latex]1-2\sin^2 3\phi[/latex]
For Problems 25–26,
[latex]\cos 2\theta - \sin \theta = 1[/latex]
[latex]\tan 2z + \tan z = 0[/latex]
For Problems 27–28, graph the function and decide if it has an inverse function.
[latex]f(x) = 4x - x^3[/latex]
[latex]g(x) = 5 + \sqrt[3]{x - 2}[/latex]
For Problems 29–30, give exact values in radians.
An IMAX movie screen is 52.8 feet high.
Rembrandt’s painting The Night Watch measures 13 feet high by 16 feet wide.
For Problems 33–34, solve for [latex]\theta\text{.}[/latex]
[latex]v_y = v_0\sin \theta - gt[/latex]
[latex]\Delta W = -q_0 E\cos (\pi - \theta)\Delta l[/latex]
For Problems 35–36, find exact values without using a calculator.
[latex]\cos\left[\tan^{-1}\left(\dfrac{-\sqrt{5}}{2}\right)\right][/latex]
[latex]\tan\left[\sin^{-1}\left(\dfrac{2}{7}\right)\right][/latex]
For Problems 37–38, simplify the expression.
[latex]\sin(\cos^{-1}2t)[/latex]
[latex]\tan(\cos^{-1}m)[/latex]
Explain why one of the expressions [latex]\sin^{-1}x[/latex] or [latex]\sin^{-1}\left(\dfrac{1}{x}\right)[/latex] must be undefined.
Does [latex]\sin^{-1}(-x) = -\sin^{-1}(x)\text{?}[/latex] Does [latex]\cos^{-1}(-x) = -\cos^{-1}(x)\text{?}[/latex]
For Problems 41–42, evaluate. Round answers to 3 decimal places if necessary.
For Problems 43–50, find all six trigonometric ratios for the angle [latex]\theta\text{.}[/latex]
[latex]6\cos \alpha = -5\text{,}[/latex] [latex]~ 180° \lt \alpha \lt 270°[/latex]
[latex]4\sin \theta = 3,~ \theta[/latex] is obtuse
For Problems 51–56, write algebraic expressions for the six trigonometric ratios of the angle.
[latex]2\sin \alpha - k = 0,~\dfrac{\pi}{2} \lt \alpha \lt \pi[/latex]
[latex]h\cos \beta - 3 = 0,~\dfrac{3\pi}{2} \lt \beta \lt 2\pi[/latex]
For Problems 57–58, find all six trigonometric ratios of the arc [latex]\theta\text{.}[/latex] Round to two places.
For Problems 59–62, evaluate exactly.
[latex]4\cot \dfrac{3\pi}{4} - \sec^2 \dfrac{\pi}{3}[/latex]
[latex]\dfrac{1}{2}\csc \dfrac{2\pi}{3} + \tan^2 \dfrac{5\pi}{6}[/latex]
[latex]\csc \dfrac{7\pi}{6}\cos \dfrac{5\pi}{4}[/latex]
[latex]\sec \dfrac{7\pi}{4}\cot \dfrac{4\pi}{3}[/latex]
For Problems 63–64, find all solutions between [latex]0[/latex] and [latex]2\pi\text{.}[/latex] Round your solutions to tenths.
[latex]3\csc \theta + 2 = 12[/latex]
[latex]5\cot \theta + 15 = -3[/latex]
For Problems 65–70, sketch a graph of each function. Then choose the function or functions described by each statement.
[latex]y = \sec x ~~~~~~~~~~~ y = \csc x ~~~~~~~~~~~ y = \cot x[/latex]
[latex]y = \cos^{-1} x ~~~~~~~~ y = \sin^{-1} x ~~~~~~ y = \tan^{-1} x[/latex]
The graph has vertical asymptotes at multiples of [latex]\pi\text{.}[/latex]
The graph has a horizontal asymptote at [latex]\dfrac{\pi}{2}\text{.}[/latex]
The function values are the reciprocals of [latex]y = \cos x\text{.}[/latex]
The function is defined only for [latex]x[/latex]-values between [latex]-1[/latex] and [latex]1\text{,}[/latex] inclusive.
None of the function values lie between [latex]-1[/latex] and [latex]1\text{.}[/latex]
The graph includes the origin.
For Problems 71–74,
[latex]f(x)=\tan x(\cos x - \cot x)[/latex]
[latex]g(x)=\csc x - \cot x\cos x[/latex]
[latex]G(x) = \sin x(\sec x - \csc x)[/latex]
[latex]F(x) = \dfrac{1}{2}\left(\dfrac{\cos x}{1+\sin x} + \dfrac{1+ \sin x}{\cos x}\right)[/latex]
For Problems 75–78, simplify the expression.
[latex]1-\dfrac{\sin x}{\csc x}[/latex]
[latex]\dfrac{\sin x}{\csc x}+\dfrac{\cos x}{\sec x}[/latex]
[latex]\dfrac{2+\tan^2 B}{\sec^2 B} - 1[/latex]
[latex]\dfrac{\csc t}{\tan t + \cot t}[/latex]
For Problems 79–82, use the suggested substitution to simplify the expression.
[latex]\dfrac{\sqrt{16+x^2}}{x},~~x = 4\tan \theta[/latex]
[latex]x\sqrt{4-x^2},~~x=2\sin \theta[/latex]
[latex]\dfrac{x^2 - 3}{x},~~x=\sqrt{3}\sec \theta[/latex]
[latex]\dfrac{x}{\sqrt{x^2+2}},~~x=\sqrt{2}\tan \theta[/latex]
This problem outlines a geometric proof of difference of angles formula for tangent.
Let [latex]L_1[/latex] and [latex]L_2[/latex] be two lines with slopes [latex]m_1[/latex] and [latex]m_2\text{,}[/latex] respectively, and let [latex]\theta[/latex] be the acute angle formed between the two lines. Use an identity to show that
[latex]\tan \theta = \dfrac{m_2-m_1}{1+m_1m_2}[/latex]
For Problems 85–86, use the fact that if [latex]\theta[/latex] is one angle of a triangle and [latex]s[/latex] is the length of the opposite side, then the diameter of the circumscribing circle is
[latex]d=s \csc \theta[/latex]
Round your answers to the nearest hundredth.
In the figure above, find the diameter of the circumscribing circle, the angle [latex]\alpha\text{,}[/latex] and the sides [latex]a[/latex] and [latex]b\text{.}[/latex]
A triangle has one side of length 17 cm and the angle opposite is [latex]26°\text{.}[/latex] Find the diameter of the circle that circumscribes the triangle.