?We can determine the half-angle formula for \(\tan \left(\frac{x}{2}\right)=\pm

Explain how to determine the reduction identities from the double-angle identity \(cos(2x) = cos^{2} x ? sin^{2} x\). Text Transcription: cos(2x) = cos^2 x ? sin^2 x

We can determine the half-angle formula for \(\tan \left(\frac{x}{2}\right)=\pm \frac{\sqrt{1-\cos x}}{\sqrt{1+\cos x}}\) by dividing the formula for \(\sin \left(\frac{x}{2}\right)\) by \(\cos \left(\frac{x}{2}\right)\). Explain how to determine two formulas for \(\tan \left(\frac{x}{2}\right)\) that do not involve any square roots. Text Transcription: tan (x/2) = ± sqrt 1 - cos x / sqrt 1 + cos x sin (x/2) cos (x/2) tan (x/2)

For the half-angle formula given in the previous exercise for \(\tan \left(\frac{x}{2}\right)\), explain why dividing by 0 is not a concern. (Hint: examine the values of cos x necessary for the denominator to be 0.) Text Transcription: tan (x/2)

For the following exercises, find the exact values of a) sin(2x), b) cos(2x), and c) tan(2x) without solving for x. If \(sin x = \frac {1}{8}\), and x is in quadrant I. Text Transcription: sin x = 1/8

For the following exercises, find the exact values of a) sin(2x), b) cos(2x), and c) tan(2x) without solving for x. If \(cos x = \frac {2}{3}\), and x is in quadrant I. Text Transcription: cos x = 2/3

For the following exercises, find the exact values of a) sin(2x), b) cos(2x), and c) tan(2x) without solving for x. If \(cos x = ? \frac {1}{2}\), and x is in quadrant III. Text Transcription: cos x = - 1/2

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold. \(\cos (2 \theta)=\frac{3}{5}\) and 90° ? ? ? 180° Text Transcription: cos(2?) = 3 / 5

For the following exercises, find the values of the six trigonometric functions if the conditions provided hold. \(\cos (2 \theta)=\frac{1}{\sqrt{2}}\) and 180° ? ? ? 270° Text Transcription: cos(2?) = 1 / sqrt 2

For the following exercises, simplify to one trigonometric expression. \(2 \sin \left(\frac{\pi}{4}\right) 2 \cos \left(\frac{\pi}{4}\right)\) Text Transcription: 2 sin (?/4) 2 cos (?/4)

For the following exercises, simplify to one trigonometric expression. \(4 \sin \left(\frac{\pi}{8}\right) \cos \left(\frac{\pi}{8}\right)\) Text Transcription: 4 sin (?/8) cos (?/8)

For the following exercises, find the exact value using half-angle formulas. \(\sin \left(\frac{\pi}{8}\right)\) Text Transcription: sin (?/8)

For the following exercises, find the exact value using half-angle formulas. \(\cos \left(-\frac{11\pi}{12}\right)\) Text Transcription: cos (-11?/12)

For the following exercises, find the exact value using half-angle formulas. \(\sin \left(\frac{11\pi}{12}\right)\) Text Transcription: sin (11?/12)

For the following exercises, find the exact value using half-angle formulas. \(\cos \left(\frac{7\pi}{8}\right)\) Text Transcription: cos (7?/8)

For the following exercises, find the exact value using half-angle formulas. \(\tan \left(\frac{5\pi}{12}\right)\) Text Transcription: tan (5?/12)

For the following exercises, find the exact value using half-angle formulas. \(\tan \left(-\frac{3\pi}{12}\right)\) Text Transcription: tan (- 3?/12)

For the following exercises, find the exact value using half-angle formulas. \(\tan \left(-\frac{3\pi}{8}\right)\) Text Transcription: tan (- 3?/8)

For the following exercises, find the exact values of a) \(\), b) \(\), and c) \(\) without solving for x, when 0 ? x ? 360°. If \(tan x = ? \frac {4}{3}\), and x is in quadrant IV. Text Transcription: sin (x/2) cos (x/2) tan (x/2) tan x = - 4/3

For the following exercises, find the exact values of a) \(\), b) \(\), and c) \(\) without solving for x, when 0 ? x ? 360°. If \(sin x = ? \frac {12}{3}\), and x is in quadrant III. Text Transcription: sin (x/2) cos (x/2) tan (x/2) sin x = - 12/13

For the following exercises, find the exact values of a) \(\), b) \(\), and c) \(\) without solving for x, when 0 ? x ? 360°. If csc x = 7, and x is in quadrant II. Text Transcription: sin (x/2) cos (x/2) tan (x/2)

For the following exercises, find the exact values of a) \(\), b) \(\), and c) \(\) without solving for x, when 0 ? x ? 360°. If sec x = ?4, and x is in quadrant II. Text Transcription: sin (x/2) cos (x/2) tan (x/2)

For the following exercises, use Figure 5 to find the requested half and double angles. Find sin(2?), cos(2?), and tan(2?).

For the following exercises, use Figure 5 to find the requested half and double angles. Find sin(2?), cos(2?), and tan(2?).

For the following exercises, use Figure 5 to find the requested half and double angles. Find sin \(\left(\frac{\theta}{2}\right)\) , cos \(\left(\frac{\theta}{2}\right)\), and tan \(\left(\frac{\theta}{2}\right)\). Text Transcription: (?/2)

For the following exercises, use Figure 5 to find the requested half and double angles. Find sin \(\left(\frac{\alpha}{2}\right)\), cos \(\left(\frac{\alpha}{2}\right)\), and tan \(\left(\frac{\alpha}{2}\right)\) Text Transcription: (?/2)

For the following exercises, simplify each expression. Do not evaluate. \(cos^{2}(28°) ? sin^{2}(28°)\) Text Transcription: cos^2(28°) ? sin^2(28°)

For the following exercises, simplify each expression. Do not evaluate. \(2 cos^{2}(37°) ? 1\) Text Transcription: 2 cos^2(37°) ? 1

For the following exercises, simplify each expression. Do not evaluate. \(1 ? 2 sin^{2}(17°)\) Text Transcription: 1 ? 2 sin^2(17°)

For the following exercises, simplify each expression. Do not evaluate. \(cos^{2}(9x) ? sin^{2}(9x)\) Text Transcription: cos^2(9x) ? sin^2(9x)

For the following exercises, prove the identity given. \((sin t ? cos t)^{2} = 1 ? sin(2t)\) Text Transcription: (sin t ? cos t)^2 = 1 ? sin(2t)

For the following exercises, prove the identity given. \(\frac{\sin (2 \theta)}{1+\cos (2 \theta)} \tan ^{2} \theta=\tan ^{3} \theta\) Text Transcription: sin(2?) / 1 + cos(2?) tan^2 ? = tan^3 ?

For the following exercises, rewrite the expression with an exponent no higher than 1. \(cos^{2}(5x)\) Text Transcription: cos^2(5x)

For the following exercises, rewrite the expression with an exponent no higher than 1. \(cos^{2}(6x)\) Text Transcription: cos^2(6x)

For the following exercises, rewrite the expression with an exponent no higher than 1. \(sin^{4}(8x)\) Text Transcription: sin^4(8x)

For the following exercises, rewrite the expression with an exponent no higher than 1. \(sin^{4}(3x)\) Text Transcription: sin^4(3x)

For the following exercises, rewrite the expression with an exponent no higher than 1. \(cos^{2} x sin^{4} x\) Text Transcription: cos^2 x sin^4 x

For the following exercises, rewrite the expression with an exponent no higher than 1. \(cos^{4} x sin^{2} x\) Text Transcription: cos^4 x sin^2 x

For the following exercises, rewrite the expression with an exponent no higher than 1. \(tan^{2} x sin^{2} x\) Text Transcription: tan^2 x sin^2 x

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(tan^{4} x\) Text Transcription: tan^4 x

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(sin^{2}(2x)\) Text Transcription: sin^2(2x)

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(sin^{2} x cos^{2} x\) Text Transcription: sin^2 x cos^2 x

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(tan^{2} x sin x\) Text Transcription: tan^2 x sin x

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(tan^{4} x cos^{2} x\) Text Transcription: tan^4 x cos^2 x

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(cos^{2} x sin(2x)\) Text Transcription: cos^2 x sin(2x)

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(cos^{2} (2x)sin x\) Text Transcription: cos^2 (2x)sin x

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. \(\tan ^{2}\left(\frac{x}{2}\right) \sin x\) Text Transcription: tan^2 (x/2) sin x

For the following exercises, prove the identities. \(\sin (2 x)=\frac{2 \tan x}{1+\tan ^{2} x}\) Text Transcription: sin (2x) = 2 tan x / 1 + 2 tan^2 x

For the following exercises, prove the identities. \(\cos (2 \alpha)=\frac{1-\tan ^{2} \alpha}{1+\tan ^{2} \alpha}\) Text Transcription: cos(2a) = 1 - tan^2 a / 1 + tan^2 a

For the following exercises, prove the identities. \(\tan (2 x)=\frac{2 \sin x \cos x}{2 \cos ^{2} x-1}\) Text Transcription: tan(2x) = 2 sin x cos x / 2 cos^2 x - 1

For the following exercises, prove the identities. \((sin^{2} x ? 1)^}{2} = cos(2x) + sin^{4} x\) Text Transcription: (sin^2 x ? 1)^2 = cos(2x) + sin^4 x

For the following exercises, prove the identities. \(sin(3x) = 3 sin x cos^{2} x ? sin^{3} x\) Text Transcription: sin(3x) = 3 sin x cos^2 x ? sin^3 x

For the following exercises, prove the identities. \(cos(3x) = cos^{3} x ? 3 sin^{2} x cos x\) Text Transcription: cos(3x) = cos^3 x ? 3 sin^2 x cos x

For the following exercises, prove the identities. \(\frac{1+\cos (2 t)}{\sin (2 t)-\cos t}=\frac{2 \cos t}{2 \sin t-1}\) Text Transcription: 1 + cos(2t) / sin(2t) - cos t = 2 cos t / 2 sin t - 1

For the following exercises, prove the identities. \(cos(16x) = (cos^{2} (4x) ? sin^{2} (4x) ? sin(8x))(cos^{2} (4x) ? sin^{2} (4x) + sin(8x))\) Text Transcription: cos(16x) = (cos^2 (4x) ? sin^2 (4x) ? sin(8x))(cos^2 (4x) ? sin^2 (4x) + sin(8x))

Explain how to determine the double-angle formula for tan(2x) using the double-angle formulas for cos(2x) and sin(2x).

For the following exercises, find the exact values of a) sin(2x), b) cos(2x), and c) tan(2x) without solving for x. If tan x = ?8, and x is in quadrant IV.

For the following exercises, simplify each expression. Do not evaluate. 4 sin(8x) cos(8x)

For the following exercises, simplify each expression. Do not evaluate. 6 sin(5x) cos(5x)

For the following exercises, prove the identity given. sin(2x) = ?2 sin(?x) cos(?x)

For the following exercises, prove the identity given. cot x ? tan x = 2 cot(2x)

For the following exercises, algebraically find an equivalent function, only in terms of sin x and/or cos x, and then check the answer by graphing both equations. sin(4x)

For the following exercises, algebraically find an equivalent function, only in terms of sin x and/or cos x, and then check the answer by graphing both equations. cos(4x)

For the following exercises, prove the identities. sin(16x) = 16 sin x cos x cos(2x)cos(4x)cos(8x)