a function is periodic if it has a repeating pattern as the values of $x$ move from left to right

for a circular arc of length $s$ on a circle of radius 1, the radian measure of the associated angle $\theta$ is $s$

trigonometric identity

an equation involving trigonometric functions that is true for all angles $\theta$ for which the functions in the equation are defined

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Calculus Volume 1 · 1.3 Trigonometric Functions

Trigonometric Functions: practice problems

33 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 113
For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$
$240 °$
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Exercises 115
For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$
$- 60 °$
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Exercises 117
For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$
$330 °$
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Exercises 119
For the following exercises, convert each angle in radians to degrees.
$\frac{7 π}{6} rad$
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Exercises 121
For the following exercises, convert each angle in radians to degrees.
$- 3 π rad$
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Exercises 123
Evaluate the following functional values.
$cos (\frac{4 π}{3})$
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Exercises 125
Evaluate the following functional values.
$sin (- \frac{3 π}{4})$
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Exercises 127
Evaluate the following functional values.
$sin (\frac{π}{12})$
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Exercises 129
For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places.
$a = 4, c = 7$
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Exercises 131
For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places.
$a = 85.3, b = 125.5$
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Exercises 133
For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places.
$a = 84, b = 13$
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Exercises 135
For the following exercises, $P$ is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle $θ$ with a terminal side that passes through point $P .$ Rationalize denominators.
$P (\frac{7}{25}, y), y > 0$
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Exercises 137
For the following exercises, $P$ is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle $θ$ with a terminal side that passes through point $P .$ Rationalize denominators.
$P (x, \frac{7}{3}), x < 0$
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Exercises 139
For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.
$tan^{2} x + sin x csc x$
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Exercises 141
For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.
$\frac{t a n ^{2} x}{s e c ^{2} x}$
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Exercises 143
For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.
$(1 + tan θ)^{2} - 2 tan θ$
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Exercises 145
For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.
$\frac{c o s t}{s i n t} + \frac{s i n t}{1 + c o s t}$
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Exercises 155
For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$
$2 sin θ - 1 = 0$
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Exercises 157
For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$
$2 tan^{2} θ = 2$
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Exercises 159
For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$
$3 cot θ + 1 = 0$
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Exercises 161
For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$
$2 cos θ sin θ = sin θ$
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Exercises 163
For the following exercises, each graph is of the form $y = A sin B x$ or $y = A cos B x,$ where $B > 0.$ Write the equation of the graph.
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Exercises 165
For the following exercises, each graph is of the form $y = A sin B x$ or $y = A cos B x,$ where $B > 0.$ Write the equation of the graph.
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Exercises 167
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
$y = sin (x - \frac{π}{4})$
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Exercises 169
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
$y = \frac{- 1}{2} sin (\frac{1}{4} x)$
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Exercises 171
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
$y = - 3 sin (π x + 2)$
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Exercises 173
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
[T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of $120 °,$ how many inches does it move? Approximate to the nearest whole inch.
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Exercises 175
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
[T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, $ω,$ and is given by $ω = θ / t,$ where $θ$ is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.
a. $θ = \frac{7 π}{4} rad, t = 10$ sec b. sec c. min d. min
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Exercises 177
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
[T] The area of an isosceles triangle with equal sides of length x is
$\frac{1}{2} x^{2} sin θ,$
where $θ$ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle $θ = 5 π / 12$ rad.
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Exercises 179
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
[T] An alternating current for outlets in a home has voltage given by the function
$V (t) = 150 cos 368 t,$
where V is the voltage in volts at time t in seconds.
Find the period of the function and interpret its meaning.
Determine the number of periods that occur when 1 sec has passed.
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Exercises 181
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
[T] Suppose that $T = 50 + 10 sin [\frac{π}{12} (t - 8)]$ is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week.
Determine the amplitude and period.
Find the temperature 7 hours after midnight.
At what time does
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Review Exercises 321
Simplify the following trigonometric expressions.
$cos^{2} x - sin^{2} x$
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Review Exercises 323
Solve the following trigonometric equations on the interval $x = [- 2 π, 2 π]$ exactly.
$sec^{2} x - 2 sec x + 1 = 0$
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Key terms · 4

periodic function: a function is periodic if it has a repeating pattern as the values of $x$ move from left to right
radians: for a circular arc of length $s$ on a circle of radius 1, the radian measure of the associated angle $\theta$ is $s$
trigonometric functions: functions of an angle defined as ratios of the lengths of the sides of a right triangle
trigonometric identity: an equation involving trigonometric functions that is true for all angles $\theta$ for which the functions in the equation are defined

Learn 1.3 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 1.3 in the textbook · This section on openstax.org

Trigonometric Functions: practice problems

33 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 113

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$

$240 °$

Exercises 115

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$

$- 60 °$

Exercises 117

For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of $π .$

$330 °$

Exercises 119

For the following exercises, convert each angle in radians to degrees.

$\frac{7 π}{6} rad$

Exercises 121

For the following exercises, convert each angle in radians to degrees.

$- 3 π rad$

Exercises 123

Evaluate the following functional values.

$cos (\frac{4 π}{3})$

Exercises 125

Evaluate the following functional values.

$sin (- \frac{3 π}{4})$

Exercises 127

Evaluate the following functional values.

$sin (\frac{π}{12})$

Exercises 129

For the following exercises, consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, simplify to a fraction or round to three decimal places.

An image of a triangle. The three corners of the triangle are labeled “A”, “B”, and “C”. Between the corner A and corner C is the side b. Between corner C and corner B is the side a. Between corner B and corner A is the side c. The angle of corner C is marked with a right triangle symbol. The angle of corner A is marked with an angle symbol.

$a = 4, c = 7$

Exercises 131

$a = 85.3, b = 125.5$

Exercises 133

$a = 84, b = 13$

Exercises 135

For the following exercises, $P$ is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle $θ$ with a terminal side that passes through point $P .$ Rationalize denominators.

$P (\frac{7}{25}, y), y > 0$

Exercises 137

$P (x, \frac{7}{3}), x < 0$

Exercises 139

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

$tan^{2} x + sin x csc x$

Exercises 141

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

$\frac{t a n ^{2} x}{s e c ^{2} x}$

Exercises 143

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

$(1 + tan θ)^{2} - 2 tan θ$

Exercises 145

For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

$\frac{c o s t}{s i n t} + \frac{s i n t}{1 + c o s t}$

Exercises 155

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$2 sin θ - 1 = 0$

Exercises 157

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$2 tan^{2} θ = 2$

Exercises 159

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$3 cot θ + 1 = 0$

Exercises 161

For the following exercises, solve the trigonometric equations on the interval $0 \leq θ < 2 π .$

$2 cos θ sin θ = sin θ$

Exercises 163

For the following exercises, each graph is of the form $y = A sin B x$ or $y = A cos B x,$ where $B > 0.$ Write the equation of the graph.

An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -5 to 5. The graph is of a curved wave function that starts at the point (-4, 0) and decreases until the point (-2, 4). After this point the function begins increasing until it hits the point (2, 4). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-4, 0), (0, 0), and (4, 0). The y intercept is at the origin.

Exercises 165

For the following exercises, each graph is of the form $y = A sin B x$ or $y = A cos B x,$ where $B > 0.$ Write the equation of the graph.