1.
[latex]\sin \left(\beta + \dfrac{\pi}{4}\right)=\sin \beta + \dfrac{1}{\sqrt{2}}[/latex]
Chapter 8: More Functions and Identities
Practice each skill in the Homework Problems listed.
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.
Given that [latex]~~\sin \dfrac{7\pi}{12} = \dfrac{\sqrt{2} + \sqrt{6}}{4}~~{,}[/latex] find [latex]\sin \dfrac{-7\pi}{12}{.}[/latex] Sketch both angles.
Given that [latex]~~\cos \dfrac{7\pi}{12} = \dfrac{\sqrt{2} - \sqrt{6}}{4}~~{,}[/latex] find [latex]\cos \dfrac{-7\pi}{12}{.}[/latex] Sketch both angles.
If [latex]~~\cos(2x-0.3)=0.24~~[/latex] and [latex]~~\sin(2x-0.3) \lt 0~~{,}[/latex] find [latex]\cos(0.3-2x)[/latex] and [latex]\sin(0.3-2x){.}[/latex]
If [latex]~~\sin(1.5-\phi)=-0.28~~[/latex] and [latex]~~\cos(1.5-\phi) \gt 0~~{,}[/latex] find [latex]\sin(\phi-1.5)[/latex] and [latex]\cos(\phi-1.5){.}[/latex]
Show that [latex]\cos(45°+45°)[/latex] is not equal to [latex]\cos 45° + \cos 45°{.}[/latex]
Show that [latex]\tan(60°-30°)[/latex] is not equal to [latex]\tan 60° - \tan 30°{.}[/latex]
Use your calculator to verify that [latex]\tan (87°-29°)[/latex] is not equal to [latex]\tan 87° - \tan 29°{.}[/latex]
Use your calculator to verify that [latex]\cos (52°+64°)[/latex] is not equal to [latex]\cos 52° + \cos 64°{.}[/latex]
Use graphs to show that [latex]\sin\left(x-\dfrac{\pi}{6}\right)[/latex] is not equivalent to [latex]\sin x - \sin\dfrac{\pi}{6}{.}[/latex]
Use graphs to show that [latex]\tan\left(x+\dfrac{\pi}{4}\right)[/latex] is not equivalent to [latex]\tan x + \tan\dfrac{\pi}{4}{.}[/latex]
For Problems 13–24, find exact values for the trig ratios. (Do not use a calculator!)
Suppose [latex]\cos \alpha = \dfrac{3}{5},~ \sin \alpha = \dfrac{4}{5},~ \cos \beta = \dfrac{5}{13}{,}[/latex] and [latex]\sin \beta = \dfrac{-12}{13}{.}[/latex] Evaluate the following.
Suppose [latex]\cos \alpha = \dfrac{-2}{3},~ \sin \alpha = \dfrac{\sqrt{5}}{3},~ \cos \beta = \dfrac{\sqrt{3}}{2}{,}[/latex] and [latex]\sin \beta = \dfrac{-1}{2}{.}[/latex] Evaluate the following.
If [latex]\tan t = \dfrac{3}{4}[/latex] and [latex]\tan s = \dfrac{-7}{24}{,}[/latex] find exact values for:
If [latex]\tan x = -3[/latex] and [latex]\tan y = -5{,}[/latex] find exact values for:
Suppose [latex]\cos \theta = \dfrac{15}{17}[/latex] and [latex]\sin \phi = \dfrac{3}{5}{,}[/latex] where [latex]\theta[/latex] and [latex]\phi[/latex] are in quadrant I. Evaluate the following.
Suppose [latex]\cos \theta = \dfrac{15}{17}{,}[/latex] where [latex]\theta[/latex] is in quadrant IV, and [latex]\sin \phi = \dfrac{3}{5}{,}[/latex] where [latex]\phi[/latex] is in quadrant II. Evaluate the following.
If [latex]\sin \alpha = \dfrac{12}{13},~ \dfrac{\pi}{2} \lt\alpha \lt \pi{,}[/latex] and [latex]\cos \beta = \dfrac{-3}{5},~ \pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find exact values for:
If [latex]\cos \alpha = \dfrac{3}{8},~ \dfrac{3\pi}{2} \lt\alpha \lt 2\pi{,}[/latex] and [latex]\sin \beta = \dfrac{-1}{4},~ \pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find exact values for:
Find the exact values of [latex]\cos 15°[/latex] and [latex]\tan 15°{.}[/latex]
Find the exact values of [latex]\sin 165°[/latex] and [latex]\tan 165°{.}[/latex]
If [latex]\sin \theta = 0.2[/latex] and [latex]\cos \theta \gt 0{,}[/latex] find [latex]\sin \left(\theta + \dfrac{\pi}{3}\right){.}[/latex]
If [latex]\cos \theta = 0.6[/latex] and [latex]\sin \theta \lt 0{,}[/latex] find [latex]\cos \left(\theta + \dfrac{3\pi}{4}\right){.}[/latex]
For Problems 25–30, use the sum and difference formulas to expand each expression.
[latex]\sin (\theta - 270°)[/latex]
[latex]\cos(270° + \theta)[/latex]
[latex]\cos\left(t + \dfrac{\pi}{6}\right)[/latex]
[latex]\sin \left(t - \dfrac{2\pi}{3}\right)[/latex]
[latex]\tan\left(\beta - \dfrac{\pi}{6}\right)[/latex]
[latex]\tan \left(\phi + \dfrac{\pi}{4}\right)[/latex]
For Problems 31–34, use the unit circle to estimate trig values. Then verify with your calculator.
Does [latex]\sin(2\cdot 80°) = 2\sin 80°{?}[/latex]
Does [latex]\cos(2\cdot 25°) = 2\cos 25°{?}[/latex]
Does [latex]\tan(2\cdot 70°) = 2\tan 70°{?}[/latex]
Does [latex]\tan(2\cdot 100°) = 2\tan 100°{?}[/latex]
For Problems 35–38, verify that each statement is true.
[latex]\sin 90° = 2\sin 45° \cos 45°[/latex]
[latex]\sin 60° = 2\sin 30° \cos 30°[/latex]
[latex]\cos 60° = \cos^2 30° - \sin^2 30°[/latex]
[latex]\tan 60° = \dfrac{2\tan 30°}{1 - \tan^2 30°}[/latex]
In Problems 39–42, is the statement true or false? Explain your answer.
If [latex]\cos \alpha = 0.32{,}[/latex] then [latex]\cos 2\alpha = 2(0.32) = 0.64{.}[/latex]
If [latex]\cos 2\beta = 0.86{,}[/latex] then [latex]\cos \beta = 0.43{,}[/latex] so [latex]\beta = \cos^{-1}(0.43){.}[/latex]
If [latex]\sin 2\theta = h{,}[/latex] then [latex]\sin \theta = \dfrac{h}{2}{,}[/latex] so [latex]\theta = \sin^{-1}\left(\dfrac{h}{2}\right){.}[/latex]
If [latex]\cos \phi = r{,}[/latex] then [latex]\cos 2\phi = 2r{.}[/latex]
For Problems 43–54, use the double angle identities to simplify the expression.
[latex]2\sin 34° \cos 34°[/latex]
[latex]\cos^2 \dfrac{\pi}{10} - \sin^2 \dfrac{\pi}{10}[/latex]
[latex]1 - 2\sin^2 \dfrac{\pi}{16}[/latex]
[latex]2\cos^2 18° - 1[/latex]
[latex]\cos^2 3\theta - \sin^2 3\theta[/latex]
[latex]2\sin 2\alpha \cos 2\alpha[/latex]
[latex]2\sin 5t \cos 5t[/latex]
[latex]\cos^2 4w - \sin^2 4w[/latex]
[latex]\dfrac{2\tan 64°}{1- \tan^2 64°}[/latex]
[latex]\dfrac{2\tan \dfrac{\pi}{3}}{1- \tan^2 \dfrac{\pi}{3}}[/latex]
[latex]2\cos^2 2\beta - 1[/latex]
[latex]1 - 2\sin^2 6s[/latex]
For Problems 55–58, use the figures to find the trigonometric ratios.
Suppose [latex]\cos \theta = \dfrac{12}{13}[/latex] and [latex]\dfrac{3\pi}{2} \lt \theta \lt 2\pi{.}[/latex] Compute exact values for:
Suppose [latex]\sin \phi = \dfrac{5}{6}[/latex] and [latex]\dfrac{\pi}{2} \lt \phi \lt \pi{.}[/latex] Compute exact values for:
If [latex]\tan u = -4[/latex] and [latex]270° \lt u \lt 360°{,}[/latex] find exact values for:
If [latex]\tan v = \dfrac{2}{3}[/latex] and [latex]180° \lt u \lt 270°{,}[/latex] find exact values for:
For Problems 63–72,
[latex]\sin 2\theta + \sqrt{2} \cos \theta = 0[/latex]
[latex]\sin 2\alpha \sin \alpha =\cos \alpha[/latex]
[latex]\cos 2t -5\cos t + 3 = 0[/latex]
[latex]\cos 2x + 3\sin x = 2[/latex]
[latex]\tan 2\beta + 2\sin \beta = 0[/latex]
[latex]\tan 2z - 2\cos z = 0[/latex]
[latex]3\cos \phi - \sin \left(\dfrac{\pi}{2} - \phi\right) = \sqrt{3}[/latex]
[latex]\sin w + \cos\left(\dfrac{\pi}{2} - w\right) = 1[/latex]
[latex]\sin 2\phi \cos \phi + \cos 2\phi \sin \phi = 1[/latex]
[latex]\cos \theta \cos 3\theta + \sin \theta \sin 3\theta = \dfrac{\sqrt{2}}{2}[/latex]
Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula.
Use the sum of angles formulas for sine and cosine to derive a formula for each expression. Then use graphs to verify your formula.
Use the difference of angles formulas for sine and cosine to prove that:
Use the difference of angles formulas for sine and cosine to derive formulas for:
Prove the double angle identity [latex]\sin 2\theta = 2\sin \theta \cos\theta{.}[/latex] (Hint: Start with the sum of angles formula for sine and replace both [latex]\alpha[/latex] and [latex]\beta[/latex] by [latex]\theta{.}[/latex])
Prove the double angle identity [latex]\cos 2\theta = \cos^2 \theta - \sin^2 \theta{.}[/latex] (Hint: Start with the sum of angles formula for sine and replace both [latex]\alpha[/latex] and [latex]\beta[/latex] by [latex]\theta{.}[/latex])
For Problems 79–88,
[latex]\sin \left(\dfrac{\pi}{2} + \beta\right) = 1 + \sin \beta[/latex]
[latex]\cos \left(\dfrac{\pi}{3} - \beta\right) =\cos \left(\beta - \dfrac{\pi}{3}\right)[/latex]
[latex]\sin(A+180°) = -\sin A[/latex]
[latex]\tan \theta + \tan(-\theta) = 0[/latex]
[latex]\cos 4\theta = 4\cos \theta[/latex]
[latex]\cos\left(\phi + \dfrac{\pi}{3}\right)= \dfrac{1}{2} + \cos \phi[/latex]
[latex]\sin\left(x + \dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2}(\sin x + \cos x)[/latex]
[latex]2\cos\left(x - \dfrac{\pi}{6}\right) = \sin x + \sqrt{3}\cos x[/latex]
[latex]\sin\left(x - \dfrac{\pi}{3}\right) + \cos \left(x + \dfrac{\pi}{6}\right)= 0[/latex]
[latex]\cos 2x = (\cos x + \sin x)(\cos x - \sin x)[/latex]
Problems 89 and 90 verify the addition and subtraction formulas for acute angles.
The figure below shows a right triangle inscribed in a rectangle.
The figure below shows a right triangle inscribed in a rectangle.
Follow the steps to prove the difference of angles formula for cosine,
[latex]\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta[/latex]