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LESSON 5 DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS DETERMINED BY A POINT AND A LINE IN THE xy-PLANE Topics in this lesson: 1. DEFINITION AND EXAMPLES OF THE SIX TRIGONOMETRIC FUNCTIONS DETERMINED BY A POINT IN THE xy-PLANE 2. EXAMPLES OF THE SIX TRIGONOMETRIC FUNCTIONS DETERMINED BY A LINE IN THE xy-PLANE 1. DEFINITION AND EXAMPLES OF THE SIX TRIGONOMETRIC FUNCTIONS DETERMINED BY A POINT IN THE xy-PLANE y r Pr ( ) ( x, y ) r -r r x -r Definition Let Pr ( ) ( x , y ) be a point on the terminal side of the angle which is not the origin ( 0 , 0 ) . Then we define the following six trigonometric functions of the angle cos x r sec r , provided that x x 0 sin y r csc r y y , provided that 0 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 y , provided that x x tan where r x2 0 cot x , provided that y y 0 y2 NOTE: This is exactly the same definition that was given in Section 1 of Lesson 2 . y 2 is the distance from the point Pr ( ) ( x , y ) to the origin ( 0 , 0 ) , 2 y2 r 2 . The definition which is the radius of the circle whose equation is x above is saying that in order to define any one of the six trigonometric functions, you only need to know the coordinates of any point on the terminal side of the angle , which is not the origin ( 0 , 0 ) , and you do not need the circle. However, hopefully, you would agree that the Unit Circle has been helpful for us and that you would continue to make use of it. I know that I use it all the time to help me. r x2 Notice that for the examples below, we will not have to make use of a reference angle for the problem. The sign of the answer will follow from the definition of the trigonometric function. Examples Find the exact value of the six trigonometric functions of the angle the given point is on the terminal side of . 1. if ( 8, 6 ) This point is in the second quadrant and lies on a circle of radius r 64 36 100 10 . Animation of the point and the angle. cos x r 8 10 sin y r 6 10 tan y x 6 8 4 5 3 5 csc 3 4 5 4 sec cot Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 5 3 4 3 is approximately 143.1 NOTE: In Lesson 9, we will find that the angle or any angle coterminal to this angle. 2. ( 3, 9) This point is in the fourth quadrant and lies on a circle of radius r 3 81 84 2 21 . Animation of the point and the angle. cos cos sin 3 84 84 1 2 sec 7 y r 2 y x 1 28 9 9 3 3 3 2 9 21 2 ( 21) 7 3 21 2(7) 7 14 3 21 14 2 21 9 3 3 cot NOTE: In Lesson 9, we will find that the angle or any angle coterminal to this angle. 3. 7 2(7) 1 2 7 9 2 21 csc 21 1 28 28 9 84 9 sin tan 3 x r 1 3 3 3 9 is approximately 280.9 ( 24 , 0 ) This point is on the negative x-axis and lies on a circle of radius r ( 24 ) 2 0 242 24 . Animation of the point and the angle. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 cos x r sin y r tan y x 24 24 0 24 1 sec 0 0 24 0 NOTE: We already know that the angle to this angle. 4. ( 12 , 1 csc = undefined cot = undefined is 180 or any angle coterminal 18 ) This point is in the third quadrant and lies on a circle of radius ( 12 ) 2 r 62 4 angle. 629 ( 18 ) 2 62 ( 4 122 9) 12 cos x r sin y r 6 13 tan y x 18 12 6 13 . 2 6 13 13 18 3 13 3 2 182 (6 2) 2 ( 6 3)2 Animation of the point and the 13 sec 2 13 csc cot NOTE: In Lesson 9, we will find that the angle or any angle coterminal to this angle. Back to Topics List Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 3 2 3 is approximately 236.3 2. EXAMPLES OF THE SIX TRIGONOMETRIC DETERMINED BY A LINE IN THE xy-PLANE FUNCTIONS Example The terminal side of the angle is in the fourth quadrant and lies on the 3 y x . Find the exact value of the six trigonometric functions of the angle line 7 . Pick a point on the given line that lies in the fourth quadrant. Since x-coordinates are positive in the fourth quadrant, you will need to choose a positive number for 3 (7) 3 . Thus, the point the x-coordinate of the point. If x = 7, then y 7 3 x in the fourth quadrant. Since the terminal side ( 7 , 3 ) lies on the line y 7 3 x , then the point ( 7 , 3 ) lies on the terminal side of of lies on the line y 7 49 9 58 . Animation of the . This point lies on a circle of radius r line, point, and the angle. cos 7 58 3 sin tan 58 3 7 58 sec 7 58 csc 3 7 3 cot NOTE: In Lesson 9, we will find that the angle angle coterminal to this angle. Question: What was the slope of the given line y is approximately 336.8 3 x ? Answer: 7 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 3 7 or any What was the tangent of the angle 3 7 ? Answer: Example The terminal side of the angle is in the third quadrant and lies on the line 10 x 8 y 0 . Find the exact value of the six trigonometric functions of the angle . First, take the equation for the line and solve for y. 10 x 8y 0 10 x 8y y 10 x 8 y 5 x 4 Pick a point on this line that lies in the third quadrant. Since x-coordinates are negative in the third quadrant, you will need to choose a negative number for the x5 y ( 4) 5 . Thus, the point x 4 coordinate of the point. If , then 4 5 x in the third quadrant. Since the terminal side of ( 4 , 5 ) lies on the line y 4 5 x , then the point ( 4 , 5 ) lies on the terminal side of lies on the line y 4 16 25 41 . Animation of the . This point lies on a circle of radius r line, point, and the angle. 4 cos 41 5 sin tan 41 5 4 41 sec 4 41 csc 5 4 5 cot NOTE: In Lesson 9, we will find that the angle angle coterminal to this angle. is approximately 231.3 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 or any 5 5 x ? Answer: 4 4 5 What was the tangent of the angle ? Answer: 4 Question: What was the slope of the line y Example The terminal side of the angle is in the second quadrant and lies on the 4 x . Find the exact value of the six trigonometric functions of the angle line y . Pick a point on the given line that lies in the second quadrant. Since x-coordinates are negative in the second quadrant, you will need to choose a negative number for 4 ( 1) 4 . Thus, the point 1 , then y the x-coordinate of the point. If x ( 1, 4 ) lies on the line y 4 x in the second quadrant. Since the terminal side 4 x , then the point ( 1, 4 ) lies on the terminal side of of lies on the line y . This point lies on a circle of radius r line, point, and the angle. 1 cos sin tan 17 4 17 4 1 16 17 . Animation of the sec 17 17 csc 4 1 4 cot NOTE: In Lesson 9, we will find that the angle angle coterminal to this angle. is approximately 104.0 4 x ? Answer: Question: What was the slope of the given line y What was the tangent of the angle ? Answer: 4 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 4 or any Given the graph of the line y m x and the angle will be proved in the next lesson. below, then tan y y mx x Back to Topics List Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 m . This