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Evaluating Trig Functions I. Definition and Properties of the Unit Circle a. Definition: A Unit Circle is the circle with a radius of one ( r 1 ), centered at the origin (0,0) . b. Equation: x 2 y 2 1 c. Arc Length Since arc length can be found using the formula: s r (where s = arc length, r = radius, = central angle in radians) For the unit circle, since r = 1, s (1) Therefore s The arc length of a sector of a unit circle equals the radian measure of angle . d. Circumference: C 2r 2 (1) 2 The arc length (circumference) of 2 is also the radian measure of the angle corresponding to 360 . 2 radians = 360 degrees radians = 180 degrees e. Relating Coordinate Values to Trig Functions For any point P( x, y) on the unit circle, x cos and y sin where is any central angle with: 1) initial side = positive x axis 2) terminal side = radius through pt. P In the first quadrant this can be verified: opp y y sin y hyp r 1 cos adj x x x hyp r 1 www.rit.edu/asc Page 1 of 10 f. The x and y axes can be labeled using the radian measure of the angle which corresponds to the points where the unit circle intersects the axes. (0,1) (1,0) (-1,0) 0, 2 (0,-1) We can also see this graphically: y sin x ( ,1) 2 (2 ,0) ( ,0) (0,0) (3 ,1) 2 y cos x (2 ,1) (0,1) ( ,0) 2 (3 ,0) 2 ( ,1) www.rit.edu/asc Page 2 of 10 y tan x (0,0) ( ,0) (2 ,0) Finding the sine and cosine values of quadrantal angles is now easy. 3 For example, to find sin use the point (0,-1) which corresponds 2 3 3 to a central angle of . Since sin is the y coordinate of the 2 2 3 3 point, sin 1 . Similarly, cos 0 (the x coordinate of the 2 2 point). g. Examples: i. Find the arc length of a sector in a unit circle with a central angle of 120 . Solution: In a unit circle, arc length = central angle measured in radians, s . Since radians equals 180 , multiply 120 by the conversion radians ratio of . 180 2 Thus the arc length and the measure of are 120 = 180 3 2 both radians. 3 www.rit.edu/asc Page 3 of 10 ii. Find cos 2 and sin 2 . Solution: cos P). 2 2 implies the angle is a right angle ( 90 ), so P (0,1) . Hence, 0 (the x coordinate of P) and sin 2 1 (the y coordinate of iii. Find sin and cos Solution: If , P (1,0) . So sin 0 (y value) and cos 1 (x coordinate of P). II. More Properties of the Unit Circle a. If 4 (which is equivalent to 45 ), then for the point P on the unit circle, x y 2 . 2 P( 2 If 2 2 . ( 45 ), then P , 4 2 2 1 x 2 , 2 ) 2 y Explanation: x 2 y 2 1 (equation of unit circle) x 2 x 2 1 ( x y since it is an isosceles 45 45 90 triangle) 2x 2 1 1 1 2 x y 2 2 2 www.rit.edu/asc Page 4 of 10 b. If 6 (which is equivalent to 30 ), then for the point P on the 1 3 and y . 2 2 unit circle, x Using properties of 30 60 90 triangles with hypotenuse of length 1 (since r 1 ): y If 3 1 ( 30 ), then P , . 2 2 6 c. If 3 1 3 1 P , 2 2 1 2 3 2 x (which is equivalent to 60 ), then for the point P on the unit circle, x 1 3 and y . 2 2 Using properties of 30 60 90 triangles with hypotenuse of length 1 (since r 1 ): y 1 3 P , 2 2 1 3 2 1 2 If x 1 3 . ( 60 ), then P , 2 2 3 NOTE: The larger side, 3 , is always opposite the larger angle, , 3 2 and the smaller side, 1 , is opposite the smaller angle, . 2 6 www.rit.edu/asc Page 5 of 10 d. Examples: Find : (i) sin (ii) csc 6 (iii) tan 3 3 3 ( , ) 2 2 ( 4 2 2 , ) 2 2 1 (y coordinate when ) 6 2 6 1 1 2 2 3 (ii) csc 3 3 3 3 sin 3 2 2 sin 4 (iii) tan 2 1 4 cos 2 4 2 (i) sin 6 3 1 ( , ) 2 2 x III. 4 Solution: y1 More on Evaluating Trig Functions Using the Unit Circle It is important to recognize the radian measure of the standard angles , , and located in quadrants II, III, and IV. 6 4 3 related to Graph y x y x y x Reference Angle QI QII QIII QIV Point P 6 6 5 6 7 6 11 6 3 1 2 , 2 4 4 3 4 5 4 7 4 2 2 2 , 2 3 3 2 3 4 3 5 3 1 3 , 2 2 The signs of the x and y values of point P can be determined by knowing the quadrant the angle terminates in. It’s as simple as remembering: www.rit.edu/asc Page 6 of 10 All Students Take Calculus Quadrant I Quadrant II Sine & cosecant are positive are positive Quadrant III Quadrant IV Cosine & secant Tangent & cotangent are positive Examples: 1. All trig functions Evaluate sin are positive 5 . 4 Solution: 5 has a reference angle of . 4 4 5 , it is in quadrant III. Since 4 4 P ( sin 2 2 , ) 2 2 5 2 4 2 P NOTE: Our final answer will be negative because only tan and cot are positive in the third quadrant. (since x and y are negative in QIII) (the y coordinate of P) www.rit.edu/asc Page 7 of 10 2. Evaluate cos 5 . 6 P Solution: 5 has a reference angle of . 6 6 Since P ( cos 3. 5 , it is in quadrant II. 6 6 3 1 , ) 2 2 NOTE: Our final answer will be negative because only sin and csc are positive in the second quadrant. (since x negative in QII) 5 3 6 2 (the x coordinate of P) Evaluate tan 11 . 3 Solution: 11 has a reference angle of . 3 3 Since 11 2 , it is in quadrant IV. 3 3 P NOTE: Our final answer will be negative because only cos and sec are positive in the fourth quadrant. 1 3 P ( , ) 2 2 3 11 3 2 3 tan 1 3 cos 3 2 sin IV. Other Facts Derived From the Unit Circle www.rit.edu/asc Page 8 of 10 1. Fundamental Identity: For any point P on the unit circle, P ( x, y) (cos , sin ) . Substituting x cos and y sin into the equation of the circle: x2 y 2 1 (cos )2 (sin )2 1 sin 2 cos 2 1 2. Other identities: The x coordinates of points for and are the same, so: cos( ) cos( ) Therefore cos( ) is an even function. The y coordinates of points for and are the opposite, so: sin( ) sin( ) Therefore sin( ) is an odd function. ( x, y) ( x, y) Practice Exercises: Use the unit circle to answer the following problems. 1. Evaluate: a. cos b. sin 3 2 2. Find the six trigonometric functions values at the following values of : 5 a. 3 c. tan(3 ) d. csc b. 3 4 c. 7 6 d. 2 www.rit.edu/asc 11 6 Page 9 of 10 Solutions: Unit Circle Trig 1. a) -1 b) -1 c) 0 d) 1 2. a. Angle Quadrant Point P 5 3 7 6 c. d. 11 6 QII QIII QI 1 3 , 2 2 2 2 2 , 2 3 1 2 , 2 3 1 2 ,2 3 2 cos 1 2 tan 3 2 3 sec cot 3 4 QIV sin csc b. 2 2 1 3 2 2 1 1 3 3 3 2 2 2 2 3 3 2 2 2 2 1 www.rit.edu/asc 1 2 1 2 3 2 3 2 3 3 1 3 2 2 3 3 3 3 2 2 3 3 2 3 2 3 3 3 Page 10 of 10