The University of Toledo - Department of Mathematics and ...
Pre-Class Problems 6 for Monday, February 18
These are the type of problems that you will be working on in class. These problems are from Lesson 5.
Solution to Problems on the Pre-Exam.
You can go to the solution for each problem by clicking on the problem letter.
Objective of the following problem: To find the exact value of the six trigonometric functions for an angle if given the coordinates of a point on the terminal side of the angle. This will require the use of Radius r Trigonometry.
1. Find the exact value of the six trigonometric functions of the angle [pic] if the given point is on the terminal side of [pic].
a. [pic] b. [pic] c. [pic]
d. [pic] e. [pic] f. [pic]
g. [pic]
Objective of the following problems: To find the exact value of the six trigonometric functions for an angle whose terminal side lies on the graph of a line and the equation of the line is given. This will require the use of Radius r Trigonometry.
2. Find the exact value of the six trigonometric functions for the following angles.
a. The terminal side of the angle [pic] is in the fourth quadrant and lies on the line [pic].
b. The terminal side of the angle [pic] is in the first quadrant and lies on the line [pic].
c. The terminal side of the angle [pic] is in the third quadrant and lies on the line [pic].
d. The terminal side of the angle [pic] is in the second quadrant and lies on the line [pic].
e. The terminal side of the angle [pic] is in the third quadrant and lies on the line [pic].
f. The terminal side of the angle [pic] is in the fourth quadrant and lies on the line [pic].
Additional problems available in the textbook: Page 505 … 15 – 20, 33 - 36. Page 498 … Example 1.
Solutions:
1a. [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 1.
1b. [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
NOTE: Here is an easier way to simplify [pic]:
[pic]
[pic]
Back to Problem 1.
1c. [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 1.
1d. [pic]
[pic]
[pic] [pic]
[pic] [pic]undefined
[pic] [pic]undefined
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is 9.
Back to Problem 1.
1e. [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is 3.
Back to Problem 1.
1f. [pic]
[pic]
[pic] [pic] undefined
[pic] [pic]
[pic]undefined [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 1.
1g. [pic]
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 1.
2a. The terminal side of the angle [pic] is in the fourth quadrant and lies on the line [pic].
NOTE: We will need to find a point that is on the line [pic] in the fourth quadrant. Since x-coordinates are positive in the fourth quadrant, then we will need to pick a positive number for x. We will find the y-coordinate of the point using the equation [pic]. Picking the number 4 for x, we have the following.
x y
4 [pic] [pic]
NOTE: The point [pic] is in the fourth quadrant and lies on the line [pic]. Since the terminal side of the angle [pic] is in the fourth quadrant and lies on this line, then the point [pic] also lies on the terminal side of [pic].
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is 5.
Back to Problem 2.
2b. The terminal side of the angle [pic] is in the first quadrant and lies on the line [pic].
NOTE: We will need to find a point that is on the line [pic] in the first quadrant. Since x-coordinates are positive in the first quadrant, then we will need to pick a positive number for x. We will find the y-coordinate of the point using the equation [pic]. This is the equation that we obtain when we solve the given equation [pic] for y.
[pic]
Picking the number 2 for x, we have the following.
x y
2 5 [pic]
NOTE: The point [pic] is in the first quadrant and lies on the line [pic]. Since the terminal side of the angle [pic] is in the first quadrant and lies on this line, then the point [pic] also lies on the terminal side of [pic].
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 2.
2c. The terminal side of the angle [pic] is in the third quadrant and lies on the line [pic].
NOTE: We will need to find a point that is on the line [pic] in the third quadrant. Since x-coordinates are negative in the third quadrant, then we will need to pick a negative number for x. We will find the y-coordinate of the point using the equation [pic]. Picking the number [pic] for x, we have the following.
x y
[pic] [pic] [pic]
NOTE: The point [pic] is in the third quadrant and lies on the line [pic]. Since the terminal side of the angle [pic] is in the third quadrant and lies on this line, then the point [pic] also lies on the terminal side of [pic].
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is 4.
Back to Problem 2.
2d. The terminal side of the angle [pic] is in the second quadrant and lies on the line [pic].
NOTE: We will need to find a point that is on the line [pic] in the second quadrant. Since x-coordinates are negative in the second quadrant, then we will need to pick a negative number for x. We will find the y-coordinate of the point using the equation [pic]. This is the equation that we obtain when we solve the given equation [pic] for y.
[pic]
Picking the number [pic] for x, we have the following.
x y
[pic] 4 [pic]
NOTE: The point [pic] is in the second quadrant and lies on the line [pic]. Since the terminal side of the angle [pic] is in the second quadrant and lies on this line, then the point [pic] also lies on the terminal side of [pic].
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 2.
2e. The terminal side of the angle [pic] is in the third quadrant and lies on the line [pic].
NOTE: We will need to find a point that is on the line [pic] in the third quadrant. Since x-coordinates are negative in the third quadrant, then we will need to pick a negative number for x. We will find the y-coordinate of the point using the equation [pic]. This is the equation that we obtain when we solve the given equation [pic] for y.
[pic]
Picking the number [pic] for x, we have the following.
x y
[pic] [pic] [pic]
NOTE: The point [pic] is in the third quadrant and lies on the line [pic]. Since the terminal side of the angle [pic] is in the third quadrant and lies on this line, then the point [pic] also lies on the terminal side of [pic].
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 2.
2f. The terminal side of the angle [pic] is in the fourth quadrant and lies on the line [pic].
NOTE: We will need to find a point that is on the line [pic] in the fourth quadrant. Since x-coordinates are positive in the fourth quadrant, then we will need to pick a positive number for x. We will find the y-coordinate of the point using the equation [pic]. Picking the number 6 for x, we have the following.
x y
6 [pic] [pic]
NOTE: The point [pic] is in the fourth quadrant and lies on the line [pic]. Since the terminal side of the angle [pic] is in the fourth quadrant and lies on this line, then the point [pic] also lies on the terminal side of [pic].
[pic]
[pic] [pic]
[pic] [pic]
[pic] [pic]
NOTE: In addition to the point [pic] lying on the terminal side of the angle [pic], it also lies on the graph of the circle [pic]. This is the circle whose center is at the origin and whose radius is [pic].
Back to Problem 2.
Solution to Problems on the Pre-Exam: Back to Page 1.
9. If the terminal side of [pic] passes through the point [pic], then find the exact value of [pic] and [pic].
[pic]
[pic]
[pic]
[pic] [pic]
10. If the terminal side of [pic] is in the III quadrant and lies on the line [pic], then find the exact value of [pic] and [pic].
[pic]
x y
[pic] [pic] [pic]
[pic]
[pic]
[pic]
[pic] [pic]
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