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  • Convert angle measures between degrees and radians.
  • Recognize the triangular and circular definitions of the basic trigonometric functions.
  • Write the basic trigonometric identities.
  • Identify the graphs and periods of the trigonometric functions.
  • Describe the shift of a sine or cosine graph from the equation of the function.

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 , let s 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.

An image of a circle. At the exact center of the circle there is a point. From this point, there is one line segment that extends horizontally to the right a point on the edge of the circle and another line segment that extends diagonally upwards and to the right to another point on the edge of the circle. These line segments have a length of 1 unit. The curved segment on the edge of the circle that connects the two points at the end of the line segments is labeled “s”. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta = s radians”.
The radian measure of an angle θ is the arc length s of the associated arc on the unit circle.

Since an angle of 360 ° corresponds to the circumference of a circle, or an arc of length 2 π , we conclude that an angle with a degree measure of 360 ° has a radian measure of 2 π . Similarly, we see that 180 ° is equivalent to π radians. [link] shows the relationship between common degree and radian values.

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

Converting between radians and degrees

  1. Express 225 ° using radians.
  2. Express 5 π / 3 rad using degrees.

Use the fact that 180 ° is equivalent to π radians as a conversion factor: 1 = π rad 180 ° = 180 ° π rad .

  1. 225 ° = 225 ° · π 180 ° = 5 π 4 rad
  2. 5 π 3 rad = 5 π 3 · 180 ° π = 300 °
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Express 210 ° using radians. Express 11 π / 6 rad using degrees.

7 π / 6 ; 330°

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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.

To define the trigonometric functions, first consider the unit circle centered at the origin and a point P = ( x , y ) on the unit circle. Let θ be an angle with an initial side that lies along the positive x -axis and with a terminal side that is the line segment O P . 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 in terms of the coordinates x and y .

An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment that extends diagonally upwards and to the right to another point on the edge of the circle. This point is labeled “P = (x, y)”. These line segments have a length of 1 unit. From the point “P”, there is a dotted vertical line that extends downwards until it hits the x axis and thus the horizontal line segment. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta”.
The angle θ is in standard position. The values of the trigonometric functions for θ are defined in terms of the coordinates x and y .

Definition

Let P = ( x , y ) be a point on the unit circle centered at the origin O . Let θ be an angle with an initial side along the positive x -axis and a terminal side given by the line segment O P . The trigonometric functions    are then defined as

sin θ = y csc θ = 1 y cos θ = x sec θ = 1 x tan θ = y x cot θ = x y

If x = 0 , sec θ and tan θ are undefined. If y = 0 , then cot θ and csc θ are undefined.

Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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