Definition Of The Six Trigonometric Functions Notes

LESSON 2 DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE Topics in this lesson: 1. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A CIRCLE OF RADIUS r 2. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE 3. THE SPECIAL ANGLES IN TRIGONOMETRY 4. TEN THINGS EASILY OBTAINED FROM UNIT CIRCLE TRIGONOMETRY 5. THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE SPECIAL ANGLES IN THE FIRST QUADRANT BY ROTATING COUNTERCLOCKWISE 6. ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL ANGLES OF 6 ( 30 ) , 4 ( 45 ) , AND ( 60 ) 3 7. THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE SPECIAL ANGLES 1. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A CIRCLE OF RADIUS r y r Pr ( ) s ( x, y) -r r -r r x x2 y2 r2 Definition Let Pr ( ) ( x , y ) be the point of intersection of the terminal side of 2 y2 r 2 . Then we define the with the circle whose equation is x the angle following six trigonometric functions of the angle cos x r sec r , provided that x x 0 sin y r csc r , provided that y y 0 tan y , provided that x x cot x y y , provided that 0 0 NOTE: By definition, the secant function is the reciprocal of the cosine function. The cosecant function is the reciprocal of the sine function. The cotangent function is the reciprocal of the tangent function. Back to Topics List 2. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE Since you can use any size circle to define the six trigonometric functions, the best circle to use would be the Unit Circle, whose radius r is 1. Using the Unit Circle, we get the following special definition. x , y ) be the point of intersection of the terminal side of Definition Let P ( ) the angle with the Unit Circle. Since r 1 for the Unit Circle, then by the definition above, we get the following definition for the six trigonometric functions of the angle using the Unit Circle cos x sec 1 , provided that x x 0 sin y tan y , provided that x x 0 csc 1 y y , provided that 0 cot x y y , provided that 0 y 1 P( ) s ( x, y ) -1 1 -1 1 x2 x y2 1 (The Unit Circle) NOTE: The definition of the six trigonometric functions of the angle in terms of the Unit Circle says that the cosine of the angle is the x-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle. This definition also says that the sine of the angle is the y-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle. The tangent of the angle is the y-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle divided by the x-coordinate of the point of intersection. The secant function is still the reciprocal of the cosine function, the cosecant function is still the reciprocal of the sine function, and the cotangent function is still the reciprocal of the tangent function. Examples Find the exact value of the six trigonometric functions for the following angles. 1. 0 P(0 ) 2. 1, 0 ) cos 0 x 1 sec 0 1 cos 0 1 1 sin 0 y 0 csc 0 1 sin 0 1 = undefined 0 tan 0 y x cot 0 1 tan 0 1 0 = undefined 0 1 0 2 NOTE: This is the 9 0 angle in units of degrees. P 2 ( 0 , 1) 1 cos sin tan 2 1 = undefined 0 y x 2 1 2 cos csc 1 y 2 sec 0 x cot 1 0 undefined 2 1 2 sin 1 1 2 1 2 tan x y 2 3. NOTE: This is the 18 0 angle in units of degrees. P( ) cos x sin y 1, 0 ) 1 0 sec 1 cos csc 1 sin 1 1 1 1 1 0 = undefined 0 1 0 4. 0 y x tan 1 1 tan cot 0 270 NOTE: This is the 3 angle in units of radians. 2 P ( 270 ) cos 270 5. 1 = undefined 0 0 , 1) sec 270 0 1 = undefined 0 sin 270 1 csc 270 1 tan 270 1 = undefined 0 cot 270 0 1 2 NOTE: This is the 36 0 angle in units of degrees. 0 P(2 ) 6. 1, 0 ) cos 2 1 sec 2 1 sin 2 0 csc 2 1 = undefined 0 tan 2 0 1 cot 2 1 = undefined 0 0 90 NOTE: This is the 2 angle in units of radians. P ( 90 ) 0 , 1) cos ( 90 ) 7. sec ( 90 ) 0 sin ( 90 ) 1 csc ( 90 ) tan ( 90 ) 1 = undefined 0 cot ( 90 ) 1 = undefined 0 1 0 0 1 180 NOTE: This is the P ( 180 ) angle in units of radians. 1, 0 ) cos ( 180 ) sin ( 180 ) tan ( 180 ) 1 sec ( 180 ) 0 0 1 0 1 csc ( 180 ) 1 = undefined 0 cot ( 180 ) 1 = undefined 0 3 2 8. NOTE: This is the 27 0 angle in units of degrees. P 9. 3 2 ( 0 , 1) cos 3 2 0 sec 3 2 1 0 = undefined sin 3 2 1 csc 3 2 1 tan 3 2 cot 3 2 0 1 1 0 = undefined 360 NOTE: This is the 2 angle in units of radians. 0 P ( 360 ) 1, 0 ) cos ( 360 ) 1 sec ( 360 ) 1 sin ( 360 ) 0 csc ( 360 ) 1 = undefined 0 tan ( 360 ) 0 1 cot ( 360 ) 1 = undefined 0 0 Back to Topics List 3. THE SPECIAL ANGLES IN TRIGONOMETRY The Special Angles (in radians) in trigonometry are 0, 6 , 4 , 3 , 2 , 2 , 3 3 , 4 7 , 6 5 , 4 5 , 6 4 , 3 , 3 , 2 5 , 3 7 , 4 11 , 6 2 The Special Angles (in degrees) in trigonometry are 0 30 45 210 , 60 225 , 90 120 240 , 135 270 , 150 300 , 315 , FROM 330 , UNIT 360 4. TEN THINGS EASILY TRIGONOMETRY 1. The x-coordinate of any point on the Unit Circle satisfies the condition 1 x 1 . Since we get the cosine of any angle from the x-coordinate of the point of intersection of the terminal side of the angle with the Unit 1 cos 1 for all . Thus, the range of the cosine Circle, then function is the closed interval [ 1, 1] . 2. The y-coordinate of any point on the Unit Circle satisfies the condition 1 y 1 . Since we get the sine of any angle from the y-coordinate of the point of intersection of the terminal side of the angle with the Unit 1 sin 1 Circle, then for all . Thus, the range of the sine function is also the closed interval [ 1, 1] . 3. OBTAINED 180 CIRCLE y , where x is the x-coordinate and y is the yx coordinate of the point of intersection of the terminal side of the angle with By definition, tan the Unit Circle. Since cos sin tan for all . cos x and sin y by definition, then 4. Similarly, since sec 1 by definition, then sec x 1 cos for all . 5. Similarly, since csc 1 by definition, then csc y 1 sin for all . 6. Similarly, since cot x cot y by definition, then cos sin for all . 7. 1 . This equation says that if y The equation of the Unit Circle is x you take the x-coordinate and the y-coordinate of a point on the Unit Circle, square them and then add, you will get the value of 1. Since we get the cosine of any angle from the x-coordinate and the sine of the angle from the y-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle, then we obtain the equation 2 ( cos ) 2 2 ( sin ) 2 1, which we write as cos 2 sin 2 1 for all . This equation is the first of the three Pythagorean Identities. 8. 2 Taking the equation cos equation by cos cos2 sin2 2 cos 2 sin 2 1 and dividing both sides of the we obtain the second Pythagorean Identity: 1 cos2 cos2 cos2 sin 2 cos2 1 cos2 sec2 9. 2 sin cos 1 1 cos tan 2 tan 2 1 2 Similarly, taking the equation cos csc2 cot 2 2 sin 2 1 and dividing both sides you will obtain the third Pythagorean Identity: 1 for all . The sign of the cosine, sine, and tangent functions in the Four Quadrants: ( , ( cos y ) sin ( , ) ( cos sin ) ) ( ( y x tan ( tan y x ( ( ( ( y x tan ( , ) ( cos sin ) 5. sec2 1 for all . of the equation by sin 10. 2 ( x ( , ) ( cos sin ) ( tan y x ( ( ( THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE SPECIAL ANGLES IN THE FIRST QUADRANT BY ROTATING COUNTERCLOCKWISE THE POINT OF INTERSECTION OF THE ANGLE THE UNIT CIRCLE 6 ( 30 ) WITH y 1 P 1 1 3 1 , 2 2 6 x 1 This picture shows us that the x-coordinate is greater than the y-coordinate of the ( 30 ) with the point P 6 , which is the point of intersection of the angle 6 Unit Circle. Thus, cos sin 3 2 6 6 1 2 sec csc 2 3 6 6 2 tan 6 1 2 3 1 3 cot 2 3 6 THE POINT OF INTERSECTION OF THE ANGLE 4 UNIT CIRCLE ( 45 ) WITH THE y 1 P 1 2 , 2 4 1 2 2 x 1 This picture shows us that the x-coordinate is same as the y-coordinate of the ( 45 ) with the , which is the point of intersection of the angle 4 4 Unit Circle. Thus, point P cos sin tan 2 4 sec 2 2 4 2 4 2 2 2 csc 1 cot 2 2 4 2 2 2 4 4 2 2 1 THE POINT OF INTERSECTION OF THE ANGLE UNIT CIRCLE 3 ( 60 ) WITH THE y 1 P 1 3 1 , 2 2 3 1 1 x This picture shows us that the x-coordinate is less than the y-coordinate of the ( 60 ) with the , which is the point of intersection of the angle 3 3 Unit Circle. Thus, point P cos sin 3 1 2 sec 3 2 3 csc 3 2 2 3 3 3 tan 6. 2 1 2 3 3 cot 1 3 3 ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL ANGLES OF 6 ( 30 ) , ( 45 ) , AND ( 60 ) 4 3 You will need to remember the three numbers 1 , angles 6 ( 30 ) , 4 3 , and 1 3 . Arrange the ( 45 ) , and ( 60 ) from smallest to largest. Then 3 arrange the three numbers from smallest to largest: 6 ( 30 ) | 1 3 4 ( 45 ) | 3 ( 60 ) | 1 In other words, the tangent of the smallest angle of number of 1 3 , the tangent of the largest angle of number of 3 , and the tangent of the middle angle of number of 1 . 7. Tangent 3 ( 30 ) is the smallest 6 3 ( 60 ) is the largest 4 ( 45 ) is the middle THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE SPECIAL ANGLES To find the trigonometric functions for the special angles in the II, III, and IV quadrants by rotating counterclockwise and for all the special angles in the IV, III, II, and I quadrants by rotating clockwise, we make use of the reference angle of these angles.