Unit Circle Trigonometry Text 2

Definitions of the Six Trigonometric Functions: Special Case of the Unit Circle Let θ be an angle drawn in standard position, and let P ( x, y ) represent the point where the terminal side of the angle intersects the unit circle x 2 + y 2 = 1 . The six trigonometric functions are defined as follows: sin (θ ) = y csc (θ ) = 1 1 = sin (θ ) y ( y ≠ 0) cos (θ ) = x sec (θ ) = 1 1 = cos (θ ) x ( x ≠ 0) cot (θ ) = 1 x cos (θ ) = = tan (θ ) y sin (θ ) tan (θ ) = y sin (θ ) = x cos (θ ) ( x ≠ 0) ( y ≠ 0) Note that these definitions apply ONLY to the unit circle! Note that the value of a trigonometric function for any given angle remains constant regardless of the radius of the circle. For example, let us suppose that we want to find sin ( 60D ) in two different ways, using the following two diagrams. (Only the first quadrant is shown, since that is all that is needed for the problem.) ( ) Find sin 60D . Diagram 1: y ( 12 P 6, 6 3 Solution: Since the circle shown has radius 12, we must use the general definition for the sine function, y sin (θ ) = . Since the point given is r 6, 6 3 , the y value is 6 3 . We know from ) ( 60o 12 x ) the diagram that r = 12 . Therefore, sin ( 60D ) = Unit Circle Trigonometry y 6 3 3 = = . r 12 2 Definitions of the Six Trigonometric Functions ( ) Find sin 60D . Diagram 2: y ( 1 1 2 , 3 2 Solution: Since the circle shown has radius 1, we can use the shortened definition of the sine function sin (θ ) = y . Since the point given is 12 , 23 , the ) ( 3 2 y value is 60o . ( ) Therefore, sin 60D = y = x 1 ) 3 2 . We will use the unit circle (and thus the shortened definitions) to evaluate the trigonometric functions of special angles. (We will see some examples later in the unit where the general definitions of the trigonometric functions can be useful.) Examples Find the exact values of the following: ( ) csc (180 ) tan ( 45 ) 1. cos 90D 4. 7. ( ) sec ( 60 ) sin ( 270 ) 2. sin 45D D 5. D 8. ( ) cot ( 0 ) sec ( 30 ) 3. tan 30D D D 6. D D 9. Solutions: First, recall our unit circle with the coordinates which we have filled in so far: y ( 0, 1) 90o ( 1 2 , 3 2 60o 1 45 ) ( o 2 2 30o ( −1, 0 ) 180 , 2 2 ( ) 3 2 , 12 ) (1, 0 ) o 1 -1 0o 360o x -1 270o ( 0, Unit Circle Trigonometry − 1) Definitions of the Six Trigonometric Functions ( ) 1. cos 90D The terminal side of a 90o angle intersects the unit circle at the point ( 0, 1) . ( ) Using the definition cos (θ ) = x , we conclude that cos 90D = 0 . ( ) 2. sin 45D ( 2 2 , ( 3 2 , 12 . = 12 ⋅ 2 3 = The terminal side of a 45o angle intersects the unit circle at the point ( ) Using the definition sin (θ ) = y , we conclude that sin 45D = 2 2 2 2 ). . ( ) 3. tan 30D The terminal side of a 30o angle intersects the unit circle at the point ( ) Using the definition tan (θ ) = xy , we conclude that tan 30D = ( ) Rationalizing the denominator, tan 30D = ( 4. csc 180D 1 3 ⋅ 3 3 = 3 3 1 2 3 2 ) 1 3 . . ) The terminal side of a 180o angle intersects the unit circle at the point ( −1, 0 ) . Using the definition csc (θ ) = undefined, since 1 0 1 sin (θ ) ( ) = 1y , we conclude that csc 180D is is undefined. ( ) 5. sec 60D The terminal side of a 60o angle intersects the unit circle at the point ( ) Using the definition sec (θ ) = 1x , we conclude that sec 60D = 1 1 2 ( 1 2 , 3 2 ). = 1⋅ 12 = 2 . ( ) 6. cot 0D The terminal side of the 0o angle intersects the unit circle at the point (1, 0 ) . ( ) Using the definition cot (θ ) = xy , we conclude that cot 0D is undefined, since 1 0 is undefined. Unit Circle Trigonometry Definitions of the Six Trigonometric Functions ( ) 7. tan 45D The terminal side of a 45o angle intersects the unit circle at the point Using the definition tan (θ ) = xy , we conclude that tan ( 45D ) = 2 2 2 2 = ( 2 2 2 2 ⋅ , 2 2 2 2 ). =1. 8. sin ( 270D ) The terminal side of a 270o angle intersects the unit circle at the point ( 0, − 1) . ( ) Using the definition sin (θ ) = y , we conclude that sin 270D = −1 . ( ) 9. sec 30D The terminal side of a 30o angle intersects the unit circle at the point Using the definition sec (θ ) = ( ) sec 30D = sec ( 30D ) = 1 = 1⋅ 3 2 2 3 ⋅ 3 3 = 2 3 = 2 3 3 2 3 1 cos (θ ) ( 3 2 ) , 12 . = 1x , we conclude that . Rationalizing the denominator, . Example Find tan 90D and cot 90D . ( ) ( ) Solution: The terminal side of a 90o angle intersects the unit circle at the point ( 0, 1) . ( ) To find tan 90D , we use the definition tan (θ ) = is undefined, since 1 0 and conclude that tan 90D x y and conclude that is undefined. ( ) To find cot 90D , we use the definition cot (θ ) = ( ) ( ) y x cot 90D = 10 = 0 . ( ) Important Note: If we want to find cot 90D and we instead use the definition cot (θ ) = 1 undefined 1 tan (θ ) ( ) , knowing that tan 90D is undefined, we cannot compute . In this case, we need to instead use the direct formula cot (θ ) = x y as we have done above. Unit Circle Trigonometry Definitions of the Six Trigonometric Functions Coordinates of All Special Angles on the Unit Circle We will now label the coordinates of all of the special angles in the unit circle so that we can apply the trigonometric functions to angles in any quadrant. First, recall our unit circle with the coordinates we have filled in so far. y ( 0, 1) 90o ( 1 2 , 3 2 60o 1 45 ) ( o 30o ( −1, 0 ) 180 , 2 2 ( ) 3 2 , 12 2 2 ) (1, 0 ) 0o 360o o 1 -1 x -1 270o ( 0, − 1) We will now draw all of the angles (between 0o and 360o) that have a 30o reference angle. Definition of Reference Angle When an angle is drawn in standard position, its reference angle is the positive acute angle measured from x-axis to the angle’s terminal side. Angles with Reference angles of 30o: Points A, B, C, and D each represent the point where the terminal side of an angle intersects the unit circle. For each of these points, we wish to determine the angle measure (in standard position from 0o to 360o) as well as the coordinates of that particular angle. y 1 B A -1 30o 30o 30o 30o C 1 x D -1 Unit Circle Trigonometry Coordinates of All Special Angles on the Unit Circle Point A is located along the terminal side of a 30o angle. This point should already be familiar to us; the coordinates of point A are 23 , 12 . ( ) Point B is located along the terminal side of a 150o angle, since 180D − 30D = 150D . The coordinates of point B are almost identical to those of point A, except that the x-value is negative in the second quadrant, so the coordinates of point B are − 23 , 12 . ( ) Point C is located along the terminal side of a 210o angle, since 180D + 30D = 210D . The coordinates of point C are almost identical to those of point A, except that the x-value and y-value are negative in the third quadrant, so the coordinates of point C are − 23 , − 12 . ( ) Point D is located along the terminal side of a 330o angle, since 360D − 30D = 330D . The coordinates of point D are almost identical to those of point A, except that the y-value is negative in the fourth quadrant, so the coordinates of point D are 3 1 2 ,− 2 . ( ) Below is a diagram with all of the 30o reference angles between 0o and 360o, along with their respective coordinates: y 1 (− 3 2 ) , 12 150D -1 (− 3 2 30D 30o 30o 30o 30o ) , − 12 210D ( 3 2 ) x 1 330D , 12 ( 3 2 , − 12 ) -1 We have gone into great detail describing the justification for the angles and coordinates related to 30o reference angles. Since the explanations for the 45o and 60o reference angles are so similar, we will skip the details and simply give the final diagrams with the angles and coordinates labeled. Unit Circle Trigonometry Coordinates of All Special Angles on the Unit Circle Angles with Reference angles of 45o: Below is a diagram with all of the 45o reference angles between 0o and 360o, along with their respective coordinates: y (− 2 2 2 2 , ) 135 2 2 ,− 2 2 45D 45o 45o -1 (− 1 D ) 225 45o 45o D ( , 2 2 ) x 1 ( 315D -1 2 2 2 2 ,− 2 2 ) Angles with Reference angles of 60o: Below is a diagram with all of the 60o reference angles between 0o and 360o, along with their respective coordinates: y (− 1 2 , 3 2 ) 120 -1 (− 1 2 ,− 3 2 ) 240 1 D 60D 60o 60o 60o 60o D -1 ( 1 2 , 3 2 x 1 300D ) ( 1 2 ,− 3 2 ) We will now put all of the previous information together in one diagram which includes the coordinates of all special angles on the unit circle. Unit Circle Trigonometry Coordinates of All Special Angles on the Unit Circle Coordinates of All Special Angles on the Unit Circle y 1 2 (− (− 3 2 ( − , ) 120 , ) 135 2 2 3 2 90D ( 0, 1) D 1 60D ( 1 2 , ) 45D D 2 2 3 2 ) ( 2 2 1 ) , − 12 210D (− 2 2 ,− 2 2 (− ) 3 2 , 12 ) 0D (1, 0 ) -1 3 2 ( 30D , 12 150D ( −1, 0 ) 180D (− 2 2 , 330D ) 225 D 1 2 ,− 3 2 315D ) 240 D -1 270 ( 0, − 1) D 300D ( 1 2 ,− ( ) x 360o 2 2 ( , − 12 3 2 ,− 2 2 ) ) 3 2 Evaluating Trigonometric Functions of Any Special Angle Now that we know the coordinates of all the special angles on the unit circle, we can use these coordinates to find trigonometric functions of any special angle (i.e. any multiple of 30o, 45o, or 60o). First, let us review the concept of coterminal angles. Definition of Coterminal Angles Coterminal Angles are angles drawn in standard position that share a terminal side. For any angle θ , an angle coterminal with θ can be obtained by using the formula θ + k ⋅ 360D , where k is any integer. ( ) Unit Circle Trigonometry Evaluating Trigonometric Functions of Any Special Angle We already have discussed the fact that a 0o angle is coterminal with a 360o angle. Below are more examples of coterminal angles. Example Find four angles that are coterminal with a 30o angle drawn in standard position. Solution: To find angles coterminal with 30o, we add to it any multiple of 360o, since adding 360o means adding exactly one revolution, which keeps the terminal side of the angle in the same position. There are an infinite number of coterminal angles for any given angle, so the following solutions are not unique. ( ) + 2 ⋅ ( 360 ) = 750 + ( −1) ⋅ ( 360 ) = −330 + ( −2 ) ⋅ ( 360 ) = −690 30D + 1⋅ 360D = 390D 30D 30D 30D D D D D D D Therefore, four angles coterminal with 30o are 390o, 750o, -330o, and -690o. Since coterminal angles share the same terminal side, they intersect the same point on the unit circle. Therefore, since 30o intersects the unit circle at the point 23 , 12 , the angles 390o, 750o, -330o, and -690o ( also intersect the unit circle at the point ( 3 2 , 12 ) ) . This allows us to find trigonometric functions of special angles other than those that we have drawn on our unit circle between 0o and 360o. Next, recall the definitions of the trigonometric functions as they apply to the unit circle: sin (θ ) = y csc (θ ) = 1 sin (θ ) = 1 y ( y ≠ 0) cos (θ ) = x sec (θ ) = 1 cos(θ ) = 1 x ( x ≠ 0) cot (θ ) = 1 tan (θ ) = xy = tan (θ ) = xy = sin (θ ) cos (θ ) ( x ≠ 0) cos (θ ) sin (θ ) ( y ≠ 0) We can now use these definitions to find trigonometric functions of any special angle. Unit Circle Trigonometry Evaluating Trigonometric Functions of Any Special Angle Examples Find the exact values of the following: ( ) sec ( 30 ) sin ( −240 ) 1. sin 240D 4. 7. ( ) tan (135 ) tan ( −150 ) 2. cos 315D D 5. D 8. D D ( ) csc ( 630 ) sec ( 810 ) 3. cot 210D 6. 9. D D Solutions: ( 1. sin 240D ) The terminal side of a 240o angle intersects the unit circle at the point − 12 , − 23 . Using the definition sin (θ ) = y , we conclude that ( ) ( ) ( ) sin 240D = − 2. cos 315D 3 2 . The terminal side of a 315o angle intersects the unit circle at the point 2 2 2 , − 2 . Using the definition cos (θ ) = x , we conclude that ( ) ( ) cos 315D = ( 3. cot 210D 2 2 . ) The terminal side of a 210o angle intersects the unit circle at the point − 23 , − 12 . Using the definition cot (θ ) = xy , we conclude that ( ) ( ) cot 210D = − 3 2 −1 2 = − 3 2 ⋅ −12 = 3 . . ( ) 4. sec 30D The terminal side of a 30o angle intersects the unit circle at the point Using the definition sec (θ ) = ( ) sec ( 30 ) = sec 30D = D 1 3 2 2 3 = 1⋅ ⋅ Unit Circle Trigonometry 3 3 2 3 = = 2 3 3 2 3 1 cos(θ ) ( 3 2 ) , 12 . = 1x , we conclude that . Rationalizing the denominator, . Evaluating Trigonometric Functions of Any Special Angle ( 5. tan 135D ) The terminal side of a 135o angle intersects the unit circle at the point − 22 , 22 . Using the definition tan (θ ) = xy , we conclude that ( ) ( ) tan 135D = 2 2 − 2 2 = 2 2 ⋅ −22 = −1 . 6. csc ( 630D ) A 630o angle is coterminal with a 270o angle (since 630D − 360D = 270D ) and its terminal side intersects the unit circle at the point ( 0, − 1) . Using the definition csc (θ ) = 1 sin (θ ) ( ) = 1y , we conclude that csc 630D = 1 −1 = −1 . 7. sin ( −240D ) A -240o angle is coterminal with a 120o angle (since −240D + 360D = 120D ) and its terminal side intersects the unit circle at the point − 12 , 23 . Using the ( ) ( definition sin (θ ) = y , we conclude that sin −240D = ) 3 2 . 8. tan ( −150D ) A -150o angle is coterminal with a 210o angle (since −150D + 360D = 210D ) and its terminal side intersects the unit circle at the point − 23 , − 12 . Using the ( ( ) definition tan (θ ) = xy , we conclude that tan −150D = ) − 12 = − 23 Rationalizing the denominator, we obtain tan ( −150D ) = 1 3 9. sec ( 810D ) ⋅ 3 3 ( −1 2 ⋅ −23 = = 1 3 . 3 3 ) A 810o angle is coterminal with a 90o angle (since 810D − 2 ⋅ 360D = 90D ) and its terminal side intersects the unit circle at the point ( 0, 1) . Using the definition sec (θ ) = 1 0 1 cos (θ ) ( ) = 1x , we conclude that sec 810D is undefined, since is undefined. Unit Circle Trigonometry Evaluating Trigonometric Functions of Any Special Angle