In Exercises 7174, powers of trigonometric functions are rewritten to be useful in

In Exercises 1–8, fill in the blank to complete the trigonometric identity. \(\frac{1}{\tan u}=\) ___________ Text Transcription: 1/tan u =

In Exercises 1–8, fill in the blank to complete the trigonometric identity. \(\frac{1}{\csc u}=\) ___________ Text Transcription: 1/csc u =

In Exercises 1–8, fill in the blank to complete the trigonometric identity. \(\frac{\sin u}{\cos u}=\) ___________ Text Transcription: sin u/cos u =

In Exercises 1–8, fill in the blank to complete the trigonometric identity. \(\frac{1}{\sec u}=\) ___________ Text Transcription: 1/sec u =

In Exercises 1–8, fill in the blank to complete the trigonometric identity. \(\sin ^{2} u+\) ___________ = 1 Text Transcription: sin^2 u +

In Exercises 1–8, fill in the blank to complete the trigonometric identity. \(\tan \left(\frac{\pi}{2}-u\right)=\) ___________ Text Transcription: tan(pi/2 - u) =

In Exercises 1–8, fill in the blank to complete the trigonometric identity. cos( - u) = ________

In Exercises 1–8, fill in the blank to complete the trigonometric identity. csc( - u) = ________

Is a graphical solution sufficient to verify a trigonometric identity?

Is a conditional equation true for all real values in its domain?

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. sin t csc t = 1

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. sec y cos y = 1

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(\frac{\csc ^{2} x}{\cot x}=\csc x \sec x\) Text Transcription: csc^2 x / cot x = csc x sec x

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(\frac{\sin ^{2} t}{\tan ^{2} t}=\cos ^{2} t\) Text Transcription: sin^2 t / tan^2 t = cos^2 t

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(\cos ^{2} \beta-\sin ^{2} \beta=1-2 \sin ^{2} \beta\) Text Transcription: cos^2 beta - sin^2 beta = 1 - 2 sin^2 beta

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(\cot ^{2} \beta+\csc ^{2} \beta=2 \csc ^{2} \beta-1\) Text Transcription: cot^2 beta + csc^2 beta = 2 csc^2 beta - 1

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(\tan ^{2} \theta+6=\sec ^{2} \theta+5\) Text Transcription: tan^2 theta + 6 = sec^2 theta + 5

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(3+\sin ^{2} z=4-\cos ^{2} z\) Text Transcription: 3 + sin^2 z = 4 - cos^2 z

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \((1+\sin x)(1-\sin x)=\cos ^{2} x\) Text Transcription: (1 + sin x)(1 - sin x) = cos^2 x

Verifying a Trigonometric Identity In Exercises 11-20, verify the identity. \(\tan ^{2} y\left(\csc ^{2} y-1\right)=1\) Text Transcription: tan^2 y(csc^2 y - 1) = 1

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\frac{\sec\theta-1}{1-\cos\theta},\quad\ \ y_2=\sec\theta\) Text Transcription: y_1 = y_2 y_1 = sec theta - 1 / 1 - cos theta, y_2 = sec theta

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\frac{\csc x-1}{1-\sin x},\quad\ \ y_2=\csc x\) Text Transcription: y_1 = y_2 y_1 = csc x - 1 / 1 - sin x, y_2 = csc x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\left(1+\cot^{2}x\right)\cos^{2}x,\quad\ \ y_2=\cot^{2}x\) Text Transcription: y_1 = y_2 y_1 = (1 + cot^2 x)cos^2 x, y_2 = cot^2 x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\left(1+\tan^2x\right)\sin^2x,\quad\ \ y_2=\tan^2x\) Text Transcription: y_1 = y_2 y_1 = (1 + tan^2 x)sin^2 x, y_2 = tan^2 x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\csc x-\sin x,\quad\ \ y_2=\cos x\cot x\) Text Transcription: y_1 = y_2 y_1 = csc x - sin x, y_2 = cos x cot x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\sec x-\cos x,\quad\ \ y_2=\sin x\tan x\) Text Transcription: y_1 = y_2 y_1 = sec x - cos x, y_2 = sin x tan x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\sin x+\cos x\cot x,\quad\ \ y_2=\csc x\) Text Transcription: y_1 = y_2 y_1 = sin x + cos x cot x, y_2 = csc x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\cos x+\sin x\tan x,\quad\ \ y_2=\sec x\) Text Transcription: y_1 = y_2 y_1 = cos x + sin x tan x, y_2 = sec x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\frac{1}{\tan x}+\frac{1}{\cot x},\quad\ \ y_2=\tan x+\cot x\) Text Transcription: y_1 = y_2 y_1 = 1/tan x + 1/cot x, y_2 = tan x + cot x

Algebraic-Graphical-Numerical In Exercises 21-30, use a graphing utility to complete the table and graph the functions in the same viewing window. Use both the table and the graph as evidence that \(y_{1}=y_{2}\). Then verify the identity algebraically. \(y_1=\frac{1}{\sin x}-\frac{1}{\csc x},\quad\ \ y_2=\csc x-\sin x\) Text Transcription: y_1 = y_2 y_1 = 1/sin x - 1/csc x, y_2 = csc x - sin x

Error Analysis In Exercises 31 and 32, describe the error. Latex Transcription: \((1+\tan x)[1+\cot (-x)]\) \(=(1+\tan x)(1+\cot x)\) \(=1+\cot x + \tan x+\tan x \cot x\) \(=1+\cot x+\tan x+1\) \(=2+\cot x+\tan x\) Text Transcription: (1 + tan x)[1 + cot(-x)] = (1 + tan x)(1 + cot x) = 1 + cot x + tan x + tan x cot x = 1 + cot x + tan x + 1 = 2 + cot x + tan x

Error Analysis In Exercises 31 and 32, describe the error. Latex Transcription: \(\frac{1+\sec (-\theta)}{\sin (-\theta)+\tan (-\theta)}=\frac{1-\sec \theta}{\sin \theta-\tan \theta}\) \(=\frac{1-\sec\theta}{\sin\theta-\frac{\sin\theta}{\cos\theta}}\) \(=\frac{1-\sec \theta}{(\sin \theta)\left[1-\left(\frac{1}{\cos \theta}\right)\right]}\) \(=\frac{1-\sec \theta}{\sin \theta(1-\sec \theta)}\) \(=\frac{1}{\sin \theta}\) \(=\csc \theta\) Text Transcription: 1+sec(-theta)/sin(-theta)+tan(-theta) = 1-sec theta/sin theta - tan theta = 1-sec theta/sin theta - sin theta/cos theta = 1-sec theta/(sin theta)[1-(1/cos theta)] = 1-sec theta/sin theta(1-sec theta) = 1/sin theta =csc theta

Verifying a Trigonometric Identity In Exercises 33 and 34, fill in the missing step(s). \(\sec ^{4} x-2 \sec ^{2} x+1=\left(\sec ^{2} x-1\right)^{2}\) = __________ \(=\tan ^{4} x\) Text Transcription: sec^4 x - 2 sec^2 x + 1 = sec^2 x - 1)^2 = tan^4 x

Verifying a Trigonometric Identity In Exercises 33 and 34, fill in the missing step(s). \(\frac{\tan x-\cot x}{\tan x+\cot x}=\frac{\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}}\) = __________ \(=\frac{\sin ^{2} x-\cos ^{2} x}{1}\) \(=\sin ^{2} x-\cos ^{2} x\) = __________ \(=1-2 \cos ^{2} x) Text Transcription: tan x - cot x/tan x + cot x = sin x/cos x - cos x/sin x / sin x/cos x + cos x/sin x = sin^2 x - cos^2 x / 1 = sin^2 x - cos^2 x = 1 - 2 cos^2 x

Verifying a Trigonometric Identity In Exercises 35-40, verify the identity. \(\cot \left(\frac{\pi}{2}-x\right) \csc x=\sec x\) Text Transcription: cot(pi/2 - x)csc x = sec x

Verifying a Trigonometric Identity In Exercises 35-40, verify the identity. \(\frac{\cos \left[\left(\frac{\pi}{2}\right)-x\right]}{\sin \left[\left(\frac{\pi}{2}\right)-x\right]}=\tan x\) Text Transcription: cos[(pi/2) - x)] / sin[(pi/2) - x] = tan x

Verifying a Trigonometric Identity In Exercises 35-40, verify the identity. \(\frac{\csc (-x)}{\sec (-x)}=-\cot x\) Text Transcription: csc(-x)/sec(-x) = - cot x

Verifying a Trigonometric Identity In Exercises 35-40, verify the identity. \((1+\sin y)[1+\sin (-y)]=\cos ^{2} y\) Text Transcription: (1 + sin y)[1 + sin(-y)] = cos^2 y

Verifying a Trigonometric Identity In Exercises 35-40, verify the identity. \(\sin ^{1 / 2} x \cos x-\sin ^{5 / 2} x \cos x=\cos ^{3} x \sqrt{\sin x}\) Text Transcription: sin^1/2 x cos x - sin^5/2 x cos x = cos^3 x sqrt sin x

Verifying a Trigonometric Identity In Exercises 35-40, verify the identity. \(\sec ^{6} x(\sec x \tan x)-\sec ^{4} x(\sec x \tan x)=\sec ^{5} x \tan ^{3} x\) Text Transcription: sec^6 x(sec x tan x) - sec^4 x(sec x tan x) = sec^5 x tan^3 x

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\frac{\cos x-\cos y}{\sin x+\sin y}+\frac{\sin x-\sin y}{\cos x+\cos y}=0\) Text Transcription: cos x - cos y/sin x + sin y + sin x - sin y/cos x + cos y = 0

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\frac{\tan x+\cot y}{\tan x \cot y}=\tan y+\cot x\) Text Transcription: tan x + cot y/tan x cot y = tan y + cot x

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta\) Text Transcription: cos theta/1 - sin theta = sec theta + tan theta

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \((\sec \theta-\tan \theta)(\csc \theta+1)=\cot \theta\) Text Transcription: (sec theta - tan theta)(csc theta + 1) = cot theta

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\sin ^{2}\left(\frac{\pi}{2}-x\right)+\sin ^{2} x=1\) Text Transcription: sin^2(pi/2 - x) + sin^2 x = 1

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\sec ^{2} y-\cot ^{2}\left(\frac{\pi}{2}-y\right)=1\) Text Transcription: sec^2 y - cot^2(pi/2 - y) = 1

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\sin x \csc x\left(\frac{\pi}{2}-x\right)=\tan x\) Text Transcription: sin x csc x(pi/2 - x) = tan x

Verifying a Trigonometric Identity In Exercises 41-48, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. \(\sec ^{2}\left(\frac{\pi}{2}-x\right)-1=\cot ^{2} x\) Text Transcription: sec^2(pi/2 - x) - 1 = cot^2 x

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(2 \sec ^{2} x-2 \sec ^{2} x \sin ^{2} x-\sin ^{2} x-\cos ^{2} x=1\) Text Transcription: 2 sec^2 x - 2 sec^2 x sin^2 x - sin^2 x - cos^2 x = 1

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\csc x(\csc x-\sin x)+\frac{\sin x-\cos x}{\sin x}+\cot x=\csc ^{2} x\) Text Transcription: csc x(csc x - sin x) + sin x - cos x/sin x + cot x = csc^2 x

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\frac{\cot x \tan x}{\cos x}=\sec x\) Text Transcription: cot x tan x / cos x = sec x

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\frac{1+\csc \theta}{\sec \theta}-\cot \theta=\cos \theta\) Text Transcription: 1 + csc theta/sec theta - cot theta = cos theta

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\sec \theta \cot \theta=\csc \theta\) Text Transcription: sec theta cot theta = csc theta

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\sin \theta \csc \theta-\sin ^{2} \theta=\cos ^{2} \theta\) Text Transcription: sin theta csc theta - sin^2 theta = cos^2 theta

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta\) Text Transcription: 1 + sin theta/cos theta + cos theta/1 + sin theta = 2 sec theta

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\frac{\tan \theta}{1+\sec \theta}+\frac{1+\sec \theta}{\tan \theta}=2 \csc \theta\) Text Transcription: tan theta/1 + sec theta + 1 + sec theta/tan theta = 2 csc theta

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\frac{\sin \beta}{1-\cos \beta}=\frac{1+\cos \beta}{\sin \beta}\) Text Transcription: sin beta/1 - cos beta = 1 + cos beta/sin beta

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\frac{\cot \alpha}{\csc \alpha-1}=\frac{\csc \alpha+1}{\cot \alpha}\) Text Transcription: cot alpha/csc alpha - 1 = csc alpha + 1/cot alpha

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{1+\sin \alpha}{|\cos \alpha|}\) Text Transcription: sqrt 1 + sin alpha/1 - sin alpha = 1 + sin alpha/|cos alpha|

Verifying a Trigonometric Identity In Exercises 49-60, verify the identity algebraically. Use a graphing utility to check your result graphically. \(\sqrt{\frac{1-\cos \beta}{1+\cos \beta}}=\frac{1-\cos \beta}{|\sin \beta|}\) Text Transcription: sqrt 1 - cos beta/1 + cos beta = 1 - cos beta/|sin beta|

Graphing a Trigonometric Function In Exercises 61-64, use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. \(y=\frac{1}{\cot x+1}+\frac{1}{\tan x+1}\) Text Transcription: y = 1/cot x+1 + 1/tan x+1

Graphing a Trigonometric Function In Exercises 61-64, use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. \(y=\frac{\cos x}{1-\tan x}+\frac{\sin x \cos x}{\sin x-\cos x}\) Text Transcription: y = cos x/1-tan x + sin x cos x/sin x - cos x

Graphing a Trigonometric Function In Exercises 61-64, use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. \(y=\frac{1}{\sin x}-\frac{\cos ^{2} x}{\sin x}\) Text Transcription: y = 1/sin x - cos^2 x/sin x

Graphing a Trigonometric Function In Exercises 61-64, use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. \(y=\sin t+\frac{\cot ^{2} t}{\csc t}\) Text Transcription: y = sin t + cot^2 t/csc t

Verifying an Identity Involving Logarithms In Exercises 65 and 66, use the properties of logarithms and trigonometric identities to verify the identity. \(\ln |\cot \theta|=\ln |\cos \theta|-\ln |\sin \theta|\) Text Transcription: ln|cot theta| = ln|cos theta| - ln|sin theta|

Verifying an Identity Involving Logarithms In Exercises 65 and 66, use the properties of logarithms and trigonometric identities to verify the identity. \(\ln |\sec \theta|=-\ln |\cos \theta|\) Text Transcription: ln|sec theta| = - ln|cos theta|

Using Cofunction Identities In Exercises 67-70, use the cofunction identities to evaluate the expression without using a calculator. \(\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ}\) Text Transcription: sin^2 25 degrees + sin^2 65 degrees

Using Cofunction Identities In Exercises 67-70, use the cofunction identities to evaluate the expression without using a calculator. \(\cos ^{2} 18^{\circ}+\cos ^{2} 72^{\circ}\) Text Transcription: cos^2 18 degrees + cos^2 72 degrees

Using Cofunction Identities In Exercises 67-70, use the cofunction identities to evaluate the expression without using a calculator. \(\cos ^{2} 20^{\circ}+\cos ^{2} 52^{\circ}+\cos ^{2} 38^{\circ}+\cos ^{2} 70^{\circ}\) Text Transcription: cos^2 20 degrees + cos^2 52 degrees + cos^2 38 degrees + cos^2 70 degrees

Using Cofunction Identities In Exercises 67-70, use the cofunction identities to evaluate the expression without using a calculator. \(\sin ^{2} 18^{\circ}+\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}+\sin ^{2} 72^{\circ}\) Text Transcription: sin^2 18 degrees + sin^2 40 degrees + sin^2 50 degrees + sin^2 72 degrees

Examples from Calculus In Exercises 71-74, powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. \(\cot ^{6} x=\cot ^{4} x \csc ^{2} x-\cot ^{4} x\) Text Transcription: cot^6 x = cot^4 x csc^2 x - cot^4 x

Examples from Calculus In Exercises 71-74, powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. \(\sec ^{4} x \tan ^{2} x=\left(\tan ^{2} x+\tan ^{4} x\right) \sec ^{2} x\) Text Transcription: sec^4 x tan^2 x = (tan^2 x + tan^4 x)sec^2 x

Examples from Calculus In Exercises 71-74, powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. \(\cos ^{3} x \sin ^{2} x=\left(\sin ^{2} x-\sin ^{4} x\right) \cos x\) Text Transcription: cos^3 x sin^2 x = (sin^2 x - sin^4 x)cos x

Examples from Calculus In Exercises 71-74, powers of trigonometric functions are rewritten to be useful in calculus. Verify the identity. \(\sin ^{4} x+\cos ^{4} x=1-2 \cos ^{2} x+2 \cos ^{4} x\) Text Transcription: sin^4 x + cos^4 x = 1 - 2 cos^2 x + 2 cos^4 x

Verifying a Trigonometric Identity In Exercises 75-78, verify the identity. \(\tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^{2}}}\) Text Transcription: tan(sin^-1 x) = x / sqrt 1 - x^2

Verifying a Trigonometric Identity In Exercises 75-78, verify the identity. \(\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}\) Text Transcription: cos(sin^-1 x) = sqrt 1 - x^2

Verifying a Trigonometric Identity In Exercises 75-78, verify the identity. \(\tan \left(\sin ^{-1} \frac{x-1}{4}\right)=\frac{x-1}{\sqrt{16-(x-1)^{2}}}\) Text Transcription: tan(sin^-1 x-1/4) = x-1/sqrt 16-(x-1)^2

Verifying a Trigonometric Identity In Exercises 75-78, verify the identity. \(\tan \left(\cos ^{-1} \frac{x+1}{2}\right)=\frac{\sqrt{4-(x+1)^{2}}}{x+1}\) Text Transcription: tan(cos^-1 x+1/2) = sqrt 4-(x+1)^2/x+1

Why you should learn it (p. 497) The length s of a shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is \(\theta\) (see figure) can be modeled by \(s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}\) (a) Verify that the expression for s is equivalent to h \(\cot \theta\). (b) Use a graphing utility to complete the table. Let h = 5 feet. (c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is \(90^{\circ}\)? Text Transcription: theta s = h sin(90 degrees - theta) / sin theta cot theta 90 degrees

Rate of Change The rate of change of the function f(x) = sin x + csc x is given by cos x ? csc x cot x. Show that the expression for the rate of change can also be given by \(-\cos x \cot ^{2} x\). Text Transcription: - cos x cot^2 x

True or False? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. The equation \(\sin ^{2} \theta+\cos ^{2} \theta=1+\tan ^{2} \theta\) is an identity, because \(\sin ^{2}(0)+\cos ^{2}(0)=1\) and \(1+\tan ^{2}(0)=1\) Text Transcription: sin^2 theta + cos^2 theta = 1 + tan^2 theta sin^2(0) + cos^2(0) = 1 1 + tan^2(0) = 1

True or False? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer. \(\sin \left(x^{2}\right)=\sin ^{2}(x)\) Text Transcription: sin(x^2) = sin^2(x)

Verifying a Trigonometric Identity In Exercises 83 and 84, (a) verify the identity and (b) determine whether the identity is true for the given value of x. Explain. \(\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x},\quad\ \ x=0\) Text Transcription: sin x/1+cos x = 1-cos x/sin x, x = 0

Verifying a Trigonometric Identity In Exercises 83 and 84, (a) verify the identity and (b) determine whether the identity is true for the given value of x. Explain. \(\frac{\sec x}{\tan x}=\frac{\tan x}{\sec x-\cos x},\quad\ \ x=\pi\) Text Transcription: sec x/tan x = tan x/sec x - cos x, x = pi

Using Trigonometric Substitution In Exercises 85-88, use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta\), where \(0<\theta<\pi / 2\). Assume a > 0. \(\sqrt{a^2-u^2},\quad\ \ u=a\sin\theta\) Text Transcription: theta 0 < theta < pi/2 sqrt a^2 - u^2, u = a sin theta

Using Trigonometric Substitution In Exercises 85-88, use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta\), where \(0<\theta<\pi / 2\). Assume a > 0. \(\sqrt{a^2-u^2},\quad\ \ u=a\cos\theta\) Text Transcription: theta 0 < theta < pi/2 sqrt a^2 - u^2, u = a cos theta

Using Trigonometric Substitution In Exercises 85-88, use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta\), where \(0<\theta<\pi / 2\). Assume a > 0. \(\sqrt{a^2+u^2},\quad\ \ u=a\tan\theta\) Text Transcription: theta 0 < theta < pi/2 sqrt a^2 + u^2, u = a tan theta

Using Trigonometric Substitution In Exercises 85-88, use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta\), where \(0<\theta<\pi / 2\). Assume a > 0. \(\sqrt{u^2-a^2},\quad\ \ u=a\sec\theta\) Text Transcription: theta 0 < theta < pi/2 sqrt u^2 - a^2, u = a sec theta

Think About It In Exercises 89-92, explain why the equation is not an identity and find one value of the variable for which the equation is not true. \(\sqrt{\sec ^{2} x-1}=\tan x\) Text Transcription: sqrt sec^2 x-1 = tan x

Think About It In Exercises 89-92, explain why the equation is not an identity and find one value of the variable for which the equation is not true. \(\sin \theta=\sqrt{1-\cos ^{2} \theta}\) Text Transcription: sin theta = sqrt 1 - cos^2 theta

Think About It In Exercises 89-92, explain why the equation is not an identity and find one value of the variable for which the equation is not true. \(1-\cos \theta=\sin \theta\) Text Transcription: 1 - cos theta = sin theta

Think About It In Exercises 89-92, explain why the equation is not an identity and find one value of the variable for which the equation is not true. 1 + tan x = sec x

Verifying a Trigonometric Identity Verify that for all integers n, \(\sin \left[\frac{(12 n+1) \pi}{6}\right]=\frac{1}{2}\). Text Transcription: sin[(12n+1)pi/6] = 1/2

How Do You See It? Explain how to use the figure to derive the identity \(\frac{\sec ^{2} \theta-1}{\sec ^{2} \theta}=\sin ^{2} \theta\) given in Example 1. Text Transcription: sec^2 theta - 1 / sec^2 theta = sin^2 theta

Graphing an Exponential Function In Exercises 95-98, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. \(f(x)=2^{x}+3\) Text Transcription: f(x) = 2^x + 3

Graphing an Exponential Function In Exercises 95-98, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. \(f(x)=-2^{x-3}\) Text Transcription: f(x) = - 2^x-3

Graphing an Exponential Function In Exercises 95-98, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. \(f(x)=2^{-x}-1\) Text Transcription: f(x) = 2^-x - 1

Graphing an Exponential Function In Exercises 95-98, use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. \(f(x)=2^{x+1}+5\) Text Transcription: f(x) = 2^x+1 + 5