Browse Trigonometry problems in the Studdy Learning Bank!

Problem 101

Find the measure of an angle if its supplement is $24$ degrees less than $5$ times the angle.

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Problem 102

Prove that the exact value of the cosecant of 105 degrees is the square root of 6 minus the square root of 2.

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Problem 103

Find the derivative of $\sec x \cos x$ .

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Problem 104

Find the value of $x$ given the equation $x + 98^\circ + 48^\circ = 180^\circ$ .

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Problem 105

Evaluate the sine of 1.2 using a calculator.

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Problem 106

Find the cosecant of $\frac{5 \pi}{3}$ .

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Problem 107

Find the limit of $\frac{\sin 2x}{x^2 \cot 4x}$ as $x$ approaches 0.

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Problem 108

Find the exact values of $s$ in the interval $[0, 2\pi)$ that satisfy $\cos^2 s = \frac{3}{4}$ .

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Problem 109

Use the periodic property of $y=\cos x$ to find the equivalent expression for $\cos(-3\pi/4)$ .

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Problem 110

Evaluate the expression $\sin x \csc (-x)$ .

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Problem 111

Find the exact values of $\sin \theta$ and $\csc \theta$ given $\tan \theta = 8/15$ and $\cos \theta < 0$ .

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Problem 112

Find $\tan (\theta)$ for a triangle with $h=60$ and $\ell=12$ , where $\ell$ is the opposite and $h$ is the hypotenuse.

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Problem 113

Find $\cos A$ , $\tan A$ , and $\sin A$ for a right triangle with side lengths 8, 15, and 17.

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Problem 114

Find the derivative of $y = \cos^{-1}(\sin(x))$ .

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Problem 115

Calculate the percent grade and angle of elevation for a highway with a vertical rise of $140$ feet over $2000$ feet of horizontal distance.

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Problem 116

Find the angle whose cosine is $-\sqrt{2}/2$ .

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Problem 117

Find the length of the kite string if the kite's height is $12 \mathrm{m}$ and the string makes a $30^{\circ}$ angle with the ground.

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Problem 118

Find the measure of $\angle X$ to the nearest degree if $\sin X = \frac{4}{9}$ .

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Problem 119

Find the true statement about the cotangent function $y=\cot x$ .
A. Zeros are $\frac{n \pi}{2}$ where $n$ is an integer. B. Domain excludes $x$ where $\cos x=0$ . C. Principal cycle is on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ . D. Infinitely many vertical asymptotes at $x=n \pi$ where $n$ is an integer.

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Problem 120

Find the inverse sine of 0.

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Problem 121

Solve for $x$ to nearest hundredth, where $0 \leq x < 2\pi$ , given $\cos x = -1/\sqrt{3}$ , $\sin x = 0.832$ , $\sec x = 2.8$ , and $\cot x = -2.341$ .

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Problem 122

Find the value of $x$ if $m \angle P = (7x - 22)^\circ$ and $\angle P$ is a right angle.

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Problem 123

Find the value of the tangent function for 33°42'. Round the result to four decimal places. $\tan 33^{\circ} 42^{\prime} \approx \square$

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Problem 124

Simplify $\sin(x - 3\pi/4)$

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Problem 125

Find the coterminal angle $\theta$ from $0 \leq \theta < 2\pi$ , quadrant, and reference angle for the rotation of $\frac{14\pi}{3}$ radians.

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Problem 126

Determine number of triangles and solve for angles/sides given $a=8$ , $c=6$ , and $\angle C=14^\circ$ . Number of solutions $=$ , $\angle A_1 \approx$ , $\angle B_1 \approx$ , $b_1 \approx$ , $\angle A_2 \approx$ , $\angle B_2 \approx$ , $b_2 \approx$ .

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Problem 127

Find the exact value of $6 \cos \frac{\pi}{3} - 3 \tan \frac{\pi}{6}$ without using a calculator.

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Problem 128

Find the value of $f(x) = 5 \sin^{-1}(\sin(x)) + 3 \cos^{-1}(\sin(4x))$ at $x = \pi/3$ without a calculator.

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Problem 129

Convert degrees and minutes to decimal degrees, then subtract.
$90^{\circ} - (48.517^{\circ}) = \boxed{41.483^{\circ}}$

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Problem 130

Rewrite $\cot \theta$ in terms of $\sin \theta$ only.

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Problem 131

Find the calculator approximation of $\sec(8.0032)$ in radian mode.

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Problem 132

Determine the principal cycle, period, vertical asymptotes, center, and halfway points of $y=\tan(\frac{1}{6}x-\pi)$ . Sketch the graph.
The interval of the principal cycle is $[-\pi,\pi]$ . The period is $6\pi$ . The equation of the left vertical asymptote is $x=\frac{7\pi}{3}$ and the right is $x=\frac{11\pi}{3}$ . The coordinates of the center point are $\left(3\pi,0\right)$ . The coordinates of the left-most halfway point are $\left(\frac{\pi}{3},0\right)$ . The coordinates of the right-most halfway point are $\left(\frac{5\pi}{3},0\right)$ .

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Problem 133

Find the values of $x$ between $0$ and $4\pi$ where $\cos x = \frac{1}{2}$ . List the answers in ascending order.

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Problem 134

Find $\sin(A/2)$ given $\sin(A) = 4/5$ and $0^\circ < A < 360^\circ$ in the first quadrant.

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Problem 135

Find the exact value of $y = \operatorname{arcsec}(2)$ without using a calculator.

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Problem 136

Solve the trigonometric equation $\sec^2 \theta + 2 \sec \theta = 0$ for $\theta$ .

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Problem 137

Solve for the acute angle $\theta$ where $3 \cos \theta = \frac{1}{9}$ .

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Problem 138

Find the cotangent of 14 degrees.

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Problem 139

Graph the cosine function with period $2\pi$ .

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Problem 140

Evaluate the indefinite integral $\int(3 - 12 \sec^2 4x) dx$ .

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Problem 141

Find the trigonometric ratios and angle $\theta$ when $\cos \theta = \frac{-4}{\sqrt{20}}$ and $0^{\circ} \leq \theta \leq 180^{\circ}$ .

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Problem 142

Find the value of $x$ given that $\tan 40 = \frac{x}{100}$ .

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Problem 143

Find the inverse cosine of -0.38 and select the correct answer.
$\cos ^{-1}(-0.38)$
A. $\cos ^{-1}(-0.38)=2.0944$ radians B. The value does not exist.

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Problem 144

Simplify the equation $\tan^2(x) - \sin^2(x) = \tan^2(x)\sin^2(x)$ .

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Problem 145

Find the distance, to the nearest tenth of a mile, from each of two fire-lookout stations 29 miles apart to a fire, given the bearings from the stations are $N 35^{\circ} E$ and $N 30^{\circ} W$ .

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Problem 146

Find the angle of elevation (in degrees) of a right triangle with opposite side 18m and adjacent side 26m.

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Problem 147

Match sine graph features to values: $y=-1$ is Amplitude, $3$ is Frequency, $2\pi$ is Period, $1$ is Midline.

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Problem 148

Find the inverse cosine of 0.873 and round the result to the nearest thousandth in radians.

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Problem 149

Find the equation of a shifted cosine curve with amplitude 3, period $3 \pi$ , and axis at $y=2$ . Evaluate the function at $x=\frac{\pi}{2}, \frac{3 \pi}{4}, \frac{11 \pi}{6}$ .

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Problem 150

Find all exact radian solutions to $\sin (2 x) \cos (4 x)+\cos (2 x) \sin (4 x)=1$ .

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Problem 151

Find an equation for a trigonometric function that starts at (0,0), goes to ( $\pi/2$ ,1), down to ( $\pi$ ,0), down to ( $3\pi/2$ ,-1), and up to (2 $\pi$ ,0).

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Problem 152

Find the ratio equivalent to $\frac{1}{\cos(x)}$ .

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Problem 153

Find the angle of depression $x$ for a hiker descending a mountain at a rate of $0.01$ miles in elevation per mile of hiking.

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Problem 154

Find the exact values of $\sin 2\theta$ , $b\cdot\cos 2\theta$ , and $\tan 2\theta$ given $\cos\theta=\frac{20}{29}$ and $\theta$ lies in quadrant IV.

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Problem 155

Find the reference angle, in radians, for the angle $\theta = \frac{16\pi}{15}$ . Write the exact answer without rounding.

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Problem 156

Simplify the expression $17 \sin^4 x$ using power-reducing trigonometric identities.

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Problem 157

Find the exact value of $y = \sec^{-1}(\sqrt{2})$ without a calculator.

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Problem 158

Find the value of $\tan(\arcsin(1/2))$ .

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Problem 159

Find the values of $x$ in the interval $[0, 2\pi]$ that satisfy the equation $2\csc^2 x - 8 = 0$ .

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Problem 160

Find the sine of 292 degrees as a function of a positive acute angle.

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Problem 161

Find the value of $\cot(2A)$ given that $\cos A = -\frac{3}{\sqrt{10}}$ and $A$ is in the second quadrant.

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Problem 162

Given $\cos 2x = \frac{4}{5}$ with $\frac{\pi}{2} < x < \pi$ , find $\cos x = \pm \sqrt{\frac{3}{5}}$ .

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Problem 163

Sketch a cycle of $y=3+3 \sec (2 x)$ . Find its period, asymptotes, and range. Determine the vertical asymptotes: $x=\frac{\pi}{4}+k \frac{\pi}{2}$ , for any integer $k$ .

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Problem 164

Evaluate the inverse cotangent of the square root of 3.

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Problem 165

Determine if $x = \frac{7 \pi}{4}$ is a solution to the equation $\sec(x) = \sqrt{2}$ .

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Problem 166

Classify each expression as a trigonometric equation or identity: a) $\tan 2x=\frac{2\sin x\cos x}{1-2\sin^2 x}$ b) $\sec^2 x-\tan^2 x=\cos x$ c) $\csc^2 x-\cot^2 x=\sin^2 x+\cos^2 x$ d) $\tan^2 x=1$

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Problem 167

Find the reference angle of the angle $-60^{\circ}$ .

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Problem 168

Find the value of $\cot 80^{\circ}$ rounded to the nearest thousandth.

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Problem 169

Simplify the expression $\frac{8}{\cot^2(x) + 1}$ using fundamental identities. Verify the result numerically using a graphing utility.

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Problem 170

Find the two possible lengths of the third side of a non-right triangle with $A=40^\circ$ , $a=47$ ft, and $b=57$ ft. Round to the nearest foot.

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Problem 171

Find quadrant of $\theta$ when $\sin \theta > 0$ and $\sec \theta > 0$ .

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Problem 172

Find the solutions to $\tan(x + \pi/3) = \sqrt{3}/3$ over $[0, 2\pi]$ .

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Problem 173

Find the value of constant $K$ in the formula $\frac{d \omega}{d \theta} = K \cos (5 \theta) \sin (5 \theta) / \sin (2 \omega)$ derived from $7 \cos (2 \omega) = 4 + 4 [\cos (5 \theta)]^{2}$ using implicit differentiation. Round the answer to three decimal places.

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Problem 174

Complete the cosine of 30°, 45°, and 60° using the given options. a) $\cos 30^{\circ}=\frac{\sqrt{3}}{2}$ b) $\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ c) $\cos 60^{\circ}=\frac{1}{2}$

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Problem 175

Solve the equation $2 \cos^2(x) + \cos(x) - 1 = 0$ and express the solution(s) as a comma-separated list in exact form, using symbolic fractions and $k$ for any integer.

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Problem 176

Solve for $p$ when $\frac{9}{p}=\cos 49^{\circ}$ . Give your answer to 2 decimal places.

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Problem 177

Find the equivalent measure of $\pi/9$ . Options: A) 30 B) 20 C) 25 D) 10

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Problem 178

Convert polar equation $r=3 \cos (\theta)$ to Cartesian equation.

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Problem 179

Solve the equation $\sec \frac{5 \theta}{4} = 2$ on the interval $0 \leq \theta < 2 \pi$ .

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Problem 180

Find the maximum value of $r=8 \sin \theta$ as an ordered pair $(x, y)$ .

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Problem 181

Find the phase shift of the function $f(x) = 4 \cos(x + \pi)$ . Simplify the result.

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Problem 182

Select the correct equation for the direct variation between an object's height $h$ and the length of its shadow $l$ , where $k$ is the constant of variation. A. $l=kh$ B. $h=k$ C. $l=hk$ D. $l=h+k$

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Problem 183

Determine the angle of elevation of the sun given a 12-meter tall tree and a 9-meter shadow.

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Problem 184

Find the simplified expression equivalent to $5 \tan^2 x - 5 \sec^2 x$ .

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Problem 185

Solve the equation $2 \cos^2 \theta - 5 \cos \theta - 3 = 0$ using trigonometric substitution.

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Problem 186

Solve the trigonometric equation $3 \sin x = 2 \cos^{2} x$ for $x$ .

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Problem 187

Use double angle identities to simplify $\cos^2 \frac{\pi}{14} - \sin^2 \frac{\pi}{14}$ .

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Problem 188

Find the value of acute angle $x$ in degrees, given $\tan(x) = 11.7$ . Round your answer to the nearest hundredth.

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Problem 189

Solve the trigonometric equation $2 \sin x + \sin 2x = 0$ for $x$ .

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Problem 190

Simplify $9 \sec^{2} x - 9$ to a single trigonometric function with no denominator.

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Problem 191

Find $\theta$ in degrees given $\tan(\theta) = -6.2$ and $-90^{\circ} < \theta < 90^{\circ}$ . Round answer to nearest hundredth.

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Problem 192

Find the length of the side opposite a $64^\circ$ angle in a right triangle with adjacent side of 11.

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Problem 193

Solve for the coefficients A, B, C, and D in the given trigonometric identity involving $\cos^2 x$ and $\sin^4 x$ .

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Problem 194

Sine law creates ambiguous case if information is in ASA, ASS, SAS, or SSS form. If angles in Q have same reference angle, their sine ratio is negative, and they add up to 180°. If $\sin \theta = \frac{7}{12}$ , $\theta$ could be $53.13^\circ$ or $126.87^\circ$ .

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Problem 195

Find $\cos A$ given $a=8, c=10$ .

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Problem 196

Find the positive value of $\cos \frac{1}{2} x$ when $\cos x = \frac{4}{5}$ , in simplest radical form with a rational denominator.

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Problem 197

Find the inverse tangent of -1.

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Problem 198

Find the value of $B$ if $\sec B = 1$ .

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Problem 199

Find the exact radian measure between the hour and minute hands on a grandfather clock showing 7 o'clock.

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Problem 200

A girl swims across a river flowing at $3 \mathrm{~m} / \mathrm{s}$ . She swims at $9 \mathrm{~m} / \mathrm{s}$ relative to water. What angle $\theta$ (in degrees) should she swim to reach the opposite point?

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Trigonometry

Problem 101

Find the measure of an angle if its supplement is 242424 degrees less than 555 times the angle.

Problem 102

Prove that the exact value of the cosecant of 105 degrees is the square root of 6 minus the square root of 2.

Problem 103

Find the derivative of sec⁡xcos⁡x\sec x \cos xsecxcosx.

Problem 104

Find the value of xxx given the equation x+98∘+48∘=180∘x + 98^\circ + 48^\circ = 180^\circx+98∘+48∘=180∘.

Problem 105

Evaluate the sine of 1.2 using a calculator.

Problem 106

Find the cosecant of 5π3\frac{5 \pi}{3}35π​.

Problem 107

Find the limit of sin⁡2xx2cot⁡4x\frac{\sin 2x}{x^2 \cot 4x}x2cot4xsin2x​ as xxx approaches 0.

Problem 108

Find the exact values of sss in the interval [0,2π)[0, 2\pi)[0,2π) that satisfy cos⁡2s=34\cos^2 s = \frac{3}{4}cos2s=43​.

Problem 109

Use the periodic property of y=cos⁡xy=\cos xy=cosx to find the equivalent expression for cos⁡(−3π/4)\cos(-3\pi/4)cos(−3π/4).

Problem 110

Evaluate the expression sin⁡xcsc⁡(−x)\sin x \csc (-x)sinxcsc(−x).

Problem 111

Find the exact values of sin⁡θ\sin \thetasinθ and csc⁡θ\csc \thetacscθ given tan⁡θ=8/15\tan \theta = 8/15tanθ=8/15 and cos⁡θ<0\cos \theta < 0cosθ<0.

Problem 112

Find tan⁡(θ)\tan (\theta)tan(θ) for a triangle with h=60h=60h=60 and ℓ=12\ell=12ℓ=12, where ℓ\ellℓ is the opposite and hhh is the hypotenuse.

Problem 113

Find cos⁡A\cos AcosA, tan⁡A\tan AtanA, and sin⁡A\sin AsinA for a right triangle with side lengths 8, 15, and 17.

Problem 114

Find the derivative of y=cos⁡−1(sin⁡(x))y = \cos^{-1}(\sin(x))y=cos−1(sin(x)).

Problem 115

Calculate the percent grade and angle of elevation for a highway with a vertical rise of 140140140 feet over 200020002000 feet of horizontal distance.

Problem 116

Find the angle whose cosine is −2/2-\sqrt{2}/2−2​/2.

Problem 117

Find the length of the kite string if the kite's height is 12m12 \mathrm{m}12m and the string makes a 30∘30^{\circ}30∘ angle with the ground.

Problem 118

Find the measure of ∠X\angle X∠X to the nearest degree if sin⁡X=49\sin X = \frac{4}{9}sinX=94​.

Problem 119

Problem 120

Find the inverse sine of 0.

Problem 121

Solve for xxx to nearest hundredth, where 0≤x<2π0 \leq x < 2\pi0≤x<2π, given cos⁡x=−1/3\cos x = -1/\sqrt{3}cosx=−1/3​, sin⁡x=0.832\sin x = 0.832sinx=0.832, sec⁡x=2.8\sec x = 2.8secx=2.8, and cot⁡x=−2.341\cot x = -2.341cotx=−2.341.

Problem 122

Find the value of xxx if m∠P=(7x−22)∘m \angle P = (7x - 22)^\circm∠P=(7x−22)∘ and ∠P\angle P∠P is a right angle.

Problem 123

Find the value of the tangent function for 33°42'. Round the result to four decimal places. tan⁡33∘42′≈□\tan 33^{\circ} 42^{\prime} \approx \squaretan33∘42′≈□

Problem 124

Simplify sin⁡(x−3π/4)\sin(x - 3\pi/4)sin(x−3π/4)

Problem 125

Find the coterminal angle θ\thetaθ from 0≤θ<2π0 \leq \theta < 2\pi0≤θ<2π, quadrant, and reference angle for the rotation of 14π3\frac{14\pi}{3}314π​ radians.

Problem 126

Problem 127

Find the exact value of 6cos⁡π3−3tan⁡π66 \cos \frac{\pi}{3} - 3 \tan \frac{\pi}{6}6cos3π​−3tan6π​ without using a calculator.

Problem 128

Find the value of f(x)=5sin⁡−1(sin⁡(x))+3cos⁡−1(sin⁡(4x))f(x) = 5 \sin^{-1}(\sin(x)) + 3 \cos^{-1}(\sin(4x))f(x)=5sin−1(sin(x))+3cos−1(sin(4x)) at x=π/3x = \pi/3x=π/3 without a calculator.

Problem 129

Convert degrees and minutes to decimal degrees, then subtract.90∘−(48.517∘)=41.483∘90^{\circ} - (48.517^{\circ}) = \boxed{41.483^{\circ}}90∘−(48.517∘)=41.483∘​

Problem 130

Rewrite cot⁡θ\cot \thetacotθ in terms of sin⁡θ\sin \thetasinθ only.

Problem 131

Find the calculator approximation of sec⁡(8.0032)\sec(8.0032)sec(8.0032) in radian mode.

Problem 132

Problem 133

Find the values of xxx between 000 and 4π4\pi4π where cos⁡x=12\cos x = \frac{1}{2}cosx=21​. List the answers in ascending order.

Problem 134

Find sin⁡(A/2)\sin(A/2)sin(A/2) given sin⁡(A)=4/5\sin(A) = 4/5sin(A)=4/5 and 0∘<A<360∘0^\circ < A < 360^\circ0∘<A<360∘ in the first quadrant.

Problem 135

Find the exact value of y=arcsec⁡(2)y = \operatorname{arcsec}(2)y=arcsec(2) without using a calculator.

Problem 136

Solve the trigonometric equation sec⁡2θ+2sec⁡θ=0\sec^2 \theta + 2 \sec \theta = 0sec2θ+2secθ=0 for θ\thetaθ.

Problem 137

Solve for the acute angle θ\thetaθ where 3cos⁡θ=193 \cos \theta = \frac{1}{9}3cosθ=91​.

Problem 138

Find the cotangent of 14 degrees.

Problem 139

Graph the cosine function with period 2π2\pi2π.

Problem 140

Evaluate the indefinite integral ∫(3−12sec⁡24x)dx\int(3 - 12 \sec^2 4x) dx∫(3−12sec24x)dx.

Problem 141

Find the trigonometric ratios and angle θ\thetaθ when cos⁡θ=−420\cos \theta = \frac{-4}{\sqrt{20}}cosθ=20​−4​ and 0∘≤θ≤180∘0^{\circ} \leq \theta \leq 180^{\circ}0∘≤θ≤180∘.

Find the measure of an angle if its supplement is $24$ degrees less than $5$ times the angle.

Find the derivative of $\sec x \cos x$ .

Find the value of $x$ given the equation $x + 98^\circ + 48^\circ = 180^\circ$ .

Find the cosecant of $\frac{5 \pi}{3}$ .

Find the limit of $\frac{\sin 2x}{x^2 \cot 4x}$ as $x$ approaches 0.

Find the exact values of $s$ in the interval $[0, 2\pi)$ that satisfy $\cos^2 s = \frac{3}{4}$ .

Use the periodic property of $y=\cos x$ to find the equivalent expression for $\cos(-3\pi/4)$ .

Evaluate the expression $\sin x \csc (-x)$ .

Find the exact values of $\sin \theta$ and $\csc \theta$ given $\tan \theta = 8/15$ and $\cos \theta < 0$ .

Find $\tan (\theta)$ for a triangle with $h=60$ and $\ell=12$ , where $\ell$ is the opposite and $h$ is the hypotenuse.

Find $\cos A$ , $\tan A$ , and $\sin A$ for a right triangle with side lengths 8, 15, and 17.

Find the derivative of $y = \cos^{-1}(\sin(x))$ .

Calculate the percent grade and angle of elevation for a highway with a vertical rise of $140$ feet over $2000$ feet of horizontal distance.

Find the angle whose cosine is $-\sqrt{2}/2$ .

Find the length of the kite string if the kite's height is $12 \mathrm{m}$ and the string makes a $30^{\circ}$ angle with the ground.

Find the measure of $\angle X$ to the nearest degree if $\sin X = \frac{4}{9}$ .

Solve for $x$ to nearest hundredth, where $0 \leq x < 2\pi$ , given $\cos x = -1/\sqrt{3}$ , $\sin x = 0.832$ , $\sec x = 2.8$ , and $\cot x = -2.341$ .

Find the value of $x$ if $m \angle P = (7x - 22)^\circ$ and $\angle P$ is a right angle.

Find the value of the tangent function for 33°42'. Round the result to four decimal places. $\tan 33^{\circ} 42^{\prime} \approx \square$

Simplify $\sin(x - 3\pi/4)$

Find the coterminal angle $\theta$ from $0 \leq \theta < 2\pi$ , quadrant, and reference angle for the rotation of $\frac{14\pi}{3}$ radians.

Find the exact value of $6 \cos \frac{\pi}{3} - 3 \tan \frac{\pi}{6}$ without using a calculator.

Find the value of $f(x) = 5 \sin^{-1}(\sin(x)) + 3 \cos^{-1}(\sin(4x))$ at $x = \pi/3$ without a calculator.

Convert degrees and minutes to decimal degrees, then subtract.
$90^{\circ} - (48.517^{\circ}) = \boxed{41.483^{\circ}}$

Rewrite $\cot \theta$ in terms of $\sin \theta$ only.

Find the calculator approximation of $\sec(8.0032)$ in radian mode.

Find the values of $x$ between $0$ and $4\pi$ where $\cos x = \frac{1}{2}$ . List the answers in ascending order.

Find $\sin(A/2)$ given $\sin(A) = 4/5$ and $0^\circ < A < 360^\circ$ in the first quadrant.

Find the exact value of $y = \operatorname{arcsec}(2)$ without using a calculator.

Solve the trigonometric equation $\sec^2 \theta + 2 \sec \theta = 0$ for $\theta$ .

Solve for the acute angle $\theta$ where $3 \cos \theta = \frac{1}{9}$ .

Graph the cosine function with period $2\pi$ .

Evaluate the indefinite integral $\int(3 - 12 \sec^2 4x) dx$ .

Find the trigonometric ratios and angle $\theta$ when $\cos \theta = \frac{-4}{\sqrt{20}}$ and $0^{\circ} \leq \theta \leq 180^{\circ}$ .

Find the value of $x$ given that $\tan 40 = \frac{x}{100}$ .

Find the inverse cosine of -0.38 and select the correct answer.
$\cos ^{-1}(-0.38)$
A. $\cos ^{-1}(-0.38)=2.0944$ radians B. The value does not exist.

Simplify the equation $\tan^2(x) - \sin^2(x) = \tan^2(x)\sin^2(x)$ .

Find the distance, to the nearest tenth of a mile, from each of two fire-lookout stations 29 miles apart to a fire, given the bearings from the stations are $N 35^{\circ} E$ and $N 30^{\circ} W$ .

Match sine graph features to values: $y=-1$ is Amplitude, $3$ is Frequency, $2\pi$ is Period, $1$ is Midline.

Find the equation of a shifted cosine curve with amplitude 3, period $3 \pi$ , and axis at $y=2$ . Evaluate the function at $x=\frac{\pi}{2}, \frac{3 \pi}{4}, \frac{11 \pi}{6}$ .

Find all exact radian solutions to $\sin (2 x) \cos (4 x)+\cos (2 x) \sin (4 x)=1$ .

Find an equation for a trigonometric function that starts at (0,0), goes to ( $\pi/2$ ,1), down to ( $\pi$ ,0), down to ( $3\pi/2$ ,-1), and up to (2 $\pi$ ,0).

Find the ratio equivalent to $\frac{1}{\cos(x)}$ .

Find the angle of depression $x$ for a hiker descending a mountain at a rate of $0.01$ miles in elevation per mile of hiking.

Find the exact values of $\sin 2\theta$ , $b\cdot\cos 2\theta$ , and $\tan 2\theta$ given $\cos\theta=\frac{20}{29}$ and $\theta$ lies in quadrant IV.

Find the reference angle, in radians, for the angle $\theta = \frac{16\pi}{15}$ . Write the exact answer without rounding.

Simplify the expression $17 \sin^4 x$ using power-reducing trigonometric identities.

Find the exact value of $y = \sec^{-1}(\sqrt{2})$ without a calculator.

Find the value of $\tan(\arcsin(1/2))$ .

Find the values of $x$ in the interval $[0, 2\pi]$ that satisfy the equation $2\csc^2 x - 8 = 0$ .

Find the value of $\cot(2A)$ given that $\cos A = -\frac{3}{\sqrt{10}}$ and $A$ is in the second quadrant.

Given $\cos 2x = \frac{4}{5}$ with $\frac{\pi}{2} < x < \pi$ , find $\cos x = \pm \sqrt{\frac{3}{5}}$ .

Sketch a cycle of $y=3+3 \sec (2 x)$ . Find its period, asymptotes, and range. Determine the vertical asymptotes: $x=\frac{\pi}{4}+k \frac{\pi}{2}$ , for any integer $k$ .

Determine if $x = \frac{7 \pi}{4}$ is a solution to the equation $\sec(x) = \sqrt{2}$ .

Find the reference angle of the angle $-60^{\circ}$ .

Find the value of $\cot 80^{\circ}$ rounded to the nearest thousandth.

Simplify the expression $\frac{8}{\cot^2(x) + 1}$ using fundamental identities. Verify the result numerically using a graphing utility.

Find the two possible lengths of the third side of a non-right triangle with $A=40^\circ$ , $a=47$ ft, and $b=57$ ft. Round to the nearest foot.

Find quadrant of $\theta$ when $\sin \theta > 0$ and $\sec \theta > 0$ .

Find the solutions to $\tan(x + \pi/3) = \sqrt{3}/3$ over $[0, 2\pi]$ .

Complete the cosine of 30°, 45°, and 60° using the given options. a) $\cos 30^{\circ}=\frac{\sqrt{3}}{2}$ b) $\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ c) $\cos 60^{\circ}=\frac{1}{2}$