Trigonometric Functions

Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions.

Radian Measure

To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle

θ,

be the length of the corresponding arc on the unit circle ([link]). We say the angle corresponding to the arc of length 1 has radian measure 1.

The Six Basic Trigonometric Functions

Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.

Common Angles Expressed in Degrees and Radians
Degrees	Radians	Degrees	Radians
0	0	120	$2 π / 3$
30	$π / 6$	135	$3 π / 4$
45	$π / 4$	150	$5 π / 6$
60	$π / 3$	180	$π$
90	$π / 2$

To define the trigonometric functions, first consider the unit circle centered at the origin and a point

P = (x, y)

An angle in this position is said to be in standard position ([link]). We can then define the values of the six trigonometric functions for

θ

The values of the other trigonometric functions can be expressed in terms of

x, y,

[link] shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of

sin θ

Values of $sin θ$ and $cos θ$ at Major Angles $θ$ in the First Quadrant
$θ$	$s i n θ$	$c o s θ$
$0$	$0$	$1$
$\frac{π}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$\frac{π}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
$\frac{π}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
$\frac{π}{2}$	$1$	$0$

As mentioned earlier, the ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at either of the acute angles of the triangle. Let

θ

be the length of the hypotenuse. By inscribing the triangle into a circle of radius

H,

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for all angles

θ

for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.

Solving Trigonometric Equations

For each of the following equations, use a trigonometric identity to find all solutions.

$1 + cos (2 θ) = cos θ$
$sin (2 θ) = tan θ$

Using the double-angle formula for $cos (2 θ),$
we see that
$θ$
is a solution of

$1 + cos (2 θ) = cos θ$

if and only if

$1 + 2 {cos}^{2} θ - 1 = cos θ,$

which is true if and only if

$2 {cos}^{2} θ - cos θ = 0 .$

To solve this equation, it is important to note that we need to factor the left-hand side and not divide both sides of the equation by
$cos θ .$
The problem with dividing by
$cos θ$
is that it is possible that
$cos θ$
is zero. In fact, if we did divide both sides of the equation by
$cos θ,$
we would miss some of the solutions of the original equation. Factoring the left-hand side of the equation, we see that
$θ$
is a solution of this equation if and only if

$cos θ (2 cos θ - 1) = 0 .$

Since
$cos θ = 0$
when

$θ = \frac{π}{2}, \frac{π}{2} \pm π, \frac{π}{2} \pm 2 π,…,$

and
$cos θ = 1 / 2$
when

$θ = \frac{π}{3}, \frac{π}{3} \pm 2 π,… or θ = - \frac{π}{3}, - \frac{π}{3} \pm 2 π,…,$

we conclude that the set of solutions to this equation is

$θ = \frac{π}{2} + n π, θ = \frac{π}{3} + 2 n π, and θ = - \frac{π}{3} + 2 n π, n = 0, \pm 1, \pm 2, … .$
Using the double-angle formula for $sin (2 θ)$
and the reciprocal identity for
$tan (θ),$
the equation can be written as

$2 sin θ cos θ = \frac{sin θ}{cos θ} .$

To solve this equation, we multiply both sides by
$cos θ$
to eliminate the denominator, and say that if
$θ$
satisfies this equation, then
$θ$
satisfies the equation

$2 sin θ {cos}^{2} θ - sin θ = 0 .$

However, we need to be a little careful here. Even if
$θ$
satisfies this new equation, it may not satisfy the original equation because, to satisfy the original equation, we would need to be able to divide both sides of the equation by
$cos θ .$
However, if
$cos θ = 0,$
we cannot divide both sides of the equation by
$cos θ .$
Therefore, it is possible that we may arrive at extraneous solutions. So, at the end, it is important to check for extraneous solutions. Returning to the equation, it is important that we factor
$sin θ$
out of both terms on the left-hand side instead of dividing both sides of the equation by
$sin θ .$
Factoring the left-hand side of the equation, we can rewrite this equation as

$sin θ (2 {cos}^{2} θ - 1) = 0 .$

Therefore, the solutions are given by the angles
$θ$
such that
$sin θ = 0$
or
${cos}^{2} θ = 1 / 2 .$
The solutions of the first equation are
$θ = 0, \pm π, \pm 2 π,….$
The solutions of the second equation are
$θ = π / 4, (π / 4) \pm (π / 2), (π / 4) \pm π,….$
After checking for extraneous solutions, the set of solutions to the equation is

$θ = n π and θ = \frac{π}{4} + \frac{n π}{2}, n = 0, \pm 1, \pm 2, … .$

Graphs and Periods of the Trigonometric Functions

We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let

P = (x, y)

are the same. Consequently, the trigonometric functions are periodic functions. The period of a function

f

Just as with algebraic functions, we can apply transformations to trigonometric functions. In particular, consider the following function:

changes the period. This transformed sine function will have a period

2 π / \| B \| .

A shifted sine curve arises naturally when graphing the number of hours of daylight in a given location as a function of the day of the year. For example, suppose a city reports that June 21 is the longest day of the year with

15.7

Key Concepts

Key Equations

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

240 °

\frac{4 π}{3} rad

15 °

−60 °

\frac{− π}{3}

−225 °

330 °

\frac{11 π}{6} rad

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

\frac{π}{2} rad

\frac{7 π}{6} rad

210 °

\frac{11 π}{2} rad

−3 π rad

−540 °

\frac{5 π}{12} rad

Evaluate the following functional values.

cos (\frac{4 π}{3})

−0.5

tan (\frac{19 π}{4})

sin (- \frac{3 π}{4})

- \frac{\sqrt{2}}{2}

sec (\frac{π}{6})

sin (\frac{π}{12})

\frac{\sqrt{3} - 1}{2 \sqrt{2}}

cos (\frac{5 π}{12})

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, round to one decimal place.

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

a. $b = 5.7$

b. $sin A = \frac{4}{7}, cos A = \frac{5.7}{7}, tan A = \frac{4}{5.7}, csc A = \frac{7}{4}, sec A = \frac{7}{5.7}, cot A = \frac{5.7}{4}$

a = 21, c = 29

a = 85.3, b = 125.5

a. $c = 151.7$

b. $sin A = 0.5623, cos A = 0.8273, tan A = 0.6797, csc A = 1.778, sec A = 1.209, cot A = 1.471$

b = 40, c = 41

a = 84, b = 13

a. $c = 85$

b. $sin A = \frac{84}{85}, cos A = \frac{13}{85}, tan A = \frac{84}{13}, csc A = \frac{85}{84}, sec A = \frac{85}{13}, cot A = \frac{13}{84}$

b = 28, c = 35

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

a. $y = \frac{24}{25}$

b. $sin θ = \frac{24}{25}, cos θ = \frac{7}{25}, tan θ = \frac{24}{7}, csc θ = \frac{25}{24}, sec θ = \frac{25}{7}, cot θ = \frac{7}{24}$

P (\frac{−15}{17}, y), y < 0

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

a. $x = \frac{− \sqrt{2}}{3}$

b. $sin θ = \frac{\sqrt{7}}{3}, cos θ = \frac{− \sqrt{2}}{3}, tan θ = \frac{− \sqrt{14}}{2}, csc θ = \frac{3 \sqrt{7}}{7}, sec θ = \frac{−3 \sqrt{2}}{2}, cot θ = \frac{− \sqrt{14}}{7}$

P (x, \frac{− \sqrt{15}}{4}), x > 0

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

{sec}^{2} x

sec x sin x cot x

\frac{{tan}^{2} x}{{sec}^{2} x}

{sin}^{2} x

sec x - cos x

{(1 + tan θ)}^{2} - 2 tan θ

{sec}^{2} θ

sin x (csc x - sin x)

\frac{cos t}{sin t} + \frac{sin t}{1 + cos t}

\frac{1}{sin t} = csc t

\frac{1 + {tan}^{2} α}{1 + {cot}^{2} α}

For the following exercises, verify that each equation is an identity.

\frac{tan θ cot θ}{csc θ} = sin θ

\frac{{sec}^{2} θ}{tan θ} = sec θ csc θ

\frac{sin t}{csc t} + \frac{cos t}{sec t} = 1

\frac{sin x}{cos x + 1} + \frac{cos x - 1}{sin x} = 0

cot γ + tan γ = sec γ csc γ

{sin}^{2} β + {tan}^{2} β + {cos}^{2} β = {sec}^{2} β

\frac{1}{1 - sin α} + \frac{1}{1 + sin α} = 2 {sec}^{2} α

\frac{tan θ - cot θ}{sin θ cos θ} = {sec}^{2} θ - {csc}^{2} θ

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

2 sin θ - 1 = 0

{\frac{π}{6}, \frac{5 π}{6}}

1 + cos θ = \frac{1}{2}

2 {tan}^{2} θ = 2

{\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}}

4 {sin}^{2} θ - 2 = 0

\sqrt{3} cot θ + 1 = 0

{\frac{2 π}{3}, \frac{5 π}{3}}

3 sec θ - 2 \sqrt{3} = 0

2 cos θ sin θ = sin θ

{0, π, \frac{π}{3}, \frac{5 π}{3}}

{csc}^{2} θ + 2 csc θ + 1 = 0

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.](/calculus-book/resources/CNX_Calc_Figure_01_03_202.jpg)

y = 4 sin (\frac{π}{4} x)

![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, -2) and increases until the point (-3, 2). After this point the function decreases until it hits the point (-2, -2). After this point the function increases until it hits the point (-1, 2). After this point the function decreases until it hits the point (0, -2). After this point the function increases until it hits the point (1, 2). After this point the function decreases until it hits the point (2, -2). After this point the function increases until it hits the point (3, 2). After this point the function begins decreasing again. The x intercepts of the function on this graph are at (-3.5, 0), (-2.5, 0), (-1.5, 0), (-0.5, 0), (0.5, 0), (1.5, 0), (2.5, 0), and (3.5, 0). The y intercept is at the (0, -2).](/calculus-book/resources/CNX_Calc_Figure_01_03_203.jpg)

![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. There are many periods and only a few will be explained. The function begins decreasing at the point (-1, 1) and decreases until the point (-0.5, -1). After this point the function increases until it hits the point (0, 1). After this point the function decreases until it hits the point (0.5, -1). After this point the function increases until it hits the point (1, 1). After this point the function decreases again. The x intercepts of the function on this graph are at (-0.75, 0), (-0.25, 0), (0.25, 0), and (0.75, 0). The y intercept is at (0, 1).](/calculus-book/resources/CNX_Calc_Figure_01_03_204.jpg)

y = cos (2 π x)

![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. There are many periods and only a few will be explained. The function begins decreasing at the point (-1.25, 0.75) and decreases until the point (-0.75, -0.75). After this point the function increases until it hits the point (0.25, 0.75). After this point the function decreases until it hits the point (0.25, -0.75). After this point the function increases until it hits the point (0.75, 0.75). After this point the function decreases again. The x intercepts of the function on this graph are at (-1, 0), (-0.5, 0), (0, 0), and (0.5, 0). The y intercept is at the origin.](/calculus-book/resources/CNX_Calc_Figure_01_03_205.jpg)

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})

a. 1 b. $2 π$

c. $\frac{π}{4}$

units to the right

y = 3 cos (2 x + 3)

y = \frac{−1}{2} sin (\frac{1}{4} x)

a. $\frac{1}{2}$

b. $8 π$

c. No phase shift

y = 2 cos (x - \frac{π}{3})

y = −3 sin (π x + 2)

a. 3 b. $2$

c. $\frac{2}{π}$

units to the left

y = 4 cos (2 x - \frac{π}{2})

[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.

Approximately 42 in.

[T] Find the length of the arc intercepted by central angle $θ$

in a circle of radius r. Round to the nearest hundredth.

a. $r = 12.8$

cm, $θ = \frac{5 π}{6}$

rad b. $r = 4.378$

cm, $θ = \frac{7 π}{6}$

rad c. $r = 0.964$

cm, $θ = 50 °$

d. $r = 8.55$

cm, $θ = 325 °$

[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. $θ = \frac{3 π}{5} rad, t = 8$

sec c. $θ = \frac{2 π}{9} rad, t = 1$

min d. $θ = 23.76 rad, t = 14$

min

a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min

[T] A total of 250,000 m² of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.

Find the radius of the circular land area.
If the land area is to form a $45 °$
sector of a circle instead of a whole circle, find the length of the curved side.

[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.

\approx 30.9 {in}^{2}

[T] A particle travels in a circular path at a constant angular speed $ω .$

The angular speed is modeled by the function $ω = 9 \| cos (π t - π / 12) \| .$

Determine the angular speed at $t = 9$

sec.

[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.

a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods

[T] The number of hours of daylight in a northeast city is modeled by the function

N (t) = 12 + 3 sin [\frac{2 π}{365} (t - 79)],

where t is the number of days after January 1.

Find the amplitude and period.
Determine the number of hours of daylight on the longest day of the year.
Determine the number of hours of daylight on the shortest day of the year.
Determine the number of hours of daylight 90 days after January 1.
Sketch the graph of the function for one period starting on January 1.

[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 $T = 60 ° ?$
Sketch the graph of $T$
over
$0 \leq t \leq 24 .$

a. Amplitude = $10; period = 24$

b. $47.4 ° F$

c. 14 hours later, or 2 p.m. d.* * *

[T] The function $H (t) = 8 sin (\frac{π}{6} t)$

models the height H (in feet) of the tide t hours after midnight. Assume that $t = 0$

is midnight.

Find the amplitude and period.
Graph the function over one period.
What is the height of the tide at 4:30 a.m.?