Section 1 - Radford University
Supplemental Section: Trigonometry
Practice HW # 1-5 at end of these notes
Angle Measurement
Angles can be measured in degrees.
Counterclockwise Angles Clockwise Angles
Important Degree Relationships
1 revolution = [pic]
[pic] revolution = [pic]
[pic] revolution = [pic]
[pic] revolution = [pic]
Angles can also be measured in radians.
Consider the unit circle
[pic]
Important Radian Relationships
1 revolution = [pic] radians
[pic] revolution = [pic] radians
[pic] revolution = [pic] radians
[pic] revolution = [pic] radians
Example 1: Draw, in standard position, the angle [pic]
Solution:
█
Example 2: Draw, in standard position, the angle [pic].
Solution:
█
Angle Measurement Conversions
Formula for converting from degree to radian measure
[pic]
Formula for converting from degree to radian measure
[pic]
Example 3: Convert [pic] from degrees to radians.
Solution:
█
Example 4: Convert [pic] from radians to degrees.
Solution: Using the formula
[pic]
we obtain the result
[pic]
█
The Sine and Cosine Functions
We can define the cosine and sine of an angle [pic] in radians, denoted as [pic] and [pic], as the x and y coordinates respectively of a point P on the unit circle that is determined by the angle [pic].
Consider the unit circle
[pic]
Sine and Cosine of Basic Angle Values
|[pic] Degrees |[pic] Radians |[pic] |[pic] |
|0 |0[pic] | | |
|30 |[pic] | | |
|45 |[pic] | | |
|60 |[pic] | | |
|90 |[pic] | | |
|180 |[pic][pic] | | |
|270 |[pic] | | |
|360 |[pic] | | |
Note: If we know the cosine and sine values for an angle [pic] in the first quadrant, we can determine the sine and cosine of related angles in other quadrants.
Sign Diagram for Cosine and Sine Values
Example 5: Compute the exact values of the sine and cosine for the angle [pic]
Solution:
█
Example 6: Compute the exact values of the sine and cosine for the angle [pic].
Solution:
█
Other Trigonometric Functions
Tangent: [pic] Secant: [pic]
Cosecant: [pic] Cotangent: [pic]
Example 7: Find the exact values of the six trigonometric functions for the angle [pic].
Solution:
█
Note: [pic], [pic]
Basic Trigonometric Identities
Pythagorean Identities Double Angle Formula
1. [pic] [pic]
2. [pic]
3. [pic]
Example 8: Find the values of x in the interval [pic] that satisfy the equation [pic].
Solution: We solve the equation using the following steps:
[pic]
In the interval [pic], [pic] when [pic]. In the interval [pic], [pic] when [pic]. Thus the five solutions are
[pic]
█
Graphs of Sine and Cosine
Period – distance on the x axis required for a trigonometric function to repeat its output values.
The period of [pic] and [pic] is[pic].
Example 9: Graph [pic]
Solution: The graph can be plotted using the following Maple command:
> plot(sin(x), x = -4*Pi..4*Pi, y = -2..2, color = red, thickness = 2);
[pic]
█
Example 10: Graph [pic]
Solution: The graph can be plotted using the following Maple command:
> plot(cos(x), x = -4*Pi..4*Pi, y = -2..2, color = red, thickness = 2);
[pic]
█
Graph of the Tangent Function
Period of the tangent function is [pic] radians. The vertical asymptotes of the tangent function [pic] are the values of x where [pic], that is the odd multiples of [pic], [pic].
Example 11: Graph[pic].
Solution: The graph is given by the following:
[pic]
█
Practice Problems
1. Convert from degrees to radians.
a. [pic] c. [pic]
b. [pic] d. [pic]
2. Convert from radians to degrees.
a. [pic] c. [pic]
b. [pic] d. [pic]
3. Draw, in standard position, the angle whose measure is given.
a. [pic] c. [pic] rad
b. [pic] rad d. [pic] rad
4. Find the exact trigonometric ratios for the angle whose radian measure is given.
a. [pic] c. [pic]
b. [pic] d. [pic]
5. Find all values of x in the interval [pic] that satisfy the equation.
a. [pic] b. [pic]
Selected Answers
1. a. [pic], b. [pic]
2. a. [pic], b. [pic]
4. a. [pic],[pic], [pic], [pic], [pic], [pic]
b. [pic],[pic], [pic], [pic], [pic], [pic]
5. [pic][pic]
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y
x
y
x
y
x
y
x
y
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