4 Chapter Trigonometric Functions
Chris Reed
4.1 - Introduction
The trigonometric functions have domains that are sets of real numbers, but the applications of trigonometry (hence forth called trig by me), often involve triangles and the angles of their vertices. We will begin by exploring the connection between angles and the set of real numbers.
4.1 - Vocabulary
An angle consists of two rays, or half-lines, that originate at a common point called the vertex, which we will denote O. One of the rays is called the initial side of the angle, and the other ray is called the terminal side.
O Angles are commonly denoted using lowercase Greek letters, such as .
4.2 Measuring Angles
Angles can be defined in a number of different ways. An angle formed by one complete rotation, by definition, consists of 360 equal parts called degrees. Degrees are denoted using the symbol . An angle of 1 is 1/360 of a revolution.
Radian Measure
A more natural way of measuring angles, (one that is not based on the rather artificial 360 degrees), is called radian measure. Radian measure is the measure used most often in calculus. The measure of an angle in radians is determined by the length of the arc of a unit circle that is cut by the rays of the angle.
Radian Measure
The radian measure of an angle with vertex at the center of a unit circle is the length of the arc made by the angles.
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Conversion Between Degrees and Radians
Conversion: Since 180 = radians, 1 = / 180 0.01745 radians 1 radian = 180 / 57.296 Keep these conversions in mind!
Example 1
Express 135 and 420 in radians and 5 / 6 in degrees.
Express 135 and 420 in radians and 5 / 6 in degrees.
To convert from degrees to radians, multiply the number of degrees by /180, the number of radians in one degree. To convert from radians to degrees, multiply the number of radians by 180/, the number of degrees in one radian. Go ahead and do this for the 135 example...
Express and 420 in radians and 5 / 6 in degrees.
135 = 135 ( /180) = radians 420 = 420 ( /180) = 7 / 3 radians 5 / 6 = (5 / 6) (180 / ) = 150 Notice, we do not worry about the decimal equivalents...they are very rarely required.
Degree and Radian Equivalents for Standard Angles
Degrees Radians 0 0 30 /6 45 /4 60 /3 90 /2
Radian Measure and Terminal Points
For any real number t, the angle generated by rotating counterclockwise from the positive xaxis to the point P(t) on the unit circle has (0,1) /2 radian measure t.
(-1,0) - (0,-1) - /2 (1,0)
Length of a Circular Arc
In a circle of radius r, the length s of an arc with angle radians is:
s = r .
Example 2
A. Find the length of an arc of a circle with radius 5 and angle 30. B. The arc of a circle of radius 3 associated with an angle has length 5. What is the radian measure of ?
A. Find the length of an arc of a circle with radius 5 and angle 30.
For A. The angle first has to be converted into radians in order to use our length formula. So, what is 30 in radians? 30 = /6 radians. (You could get this using the conversion factors, or by looking at the chart that you s...