Math 165 - Chapter 6 - some problems to practice Name___________________________________ Write the equation of a sine function with the given characteristics. 1) Amplitude: 4 Period: 6Δ Δ Phase Shift: 6 Graph the function. Show at least one period. Δ 2) y = 3cos(5x + ) 2 1 3) Given that f(x) = 2tanx, g(x) = 3cosx, k(x) = -sinx and h(x) = 2x, find each of the following 13Δ a) f( ) 4 b) (goh)(pi/8) c) (hok)(11pi/6) d) (g - f)(4pi/3) e) Find the average rate of change of g(x) over the interval [pi/2, pi] 2 Find the exact value of the requested trigonometric function of Ό. 2 4) cos Ό = and tan Ό < 0, find sin Ό. 5 _________________________________________________________________________________________ Find an equation for the graph. 5) a) Find an equation for the graph with no phase shift b) Find an equation for the graph with phase shift = pi/2 c) Find an equation for the graph with phase shift = 3pi/4 __________________________________________________________________________________________ Use the given values of the sine and cosine to find the function value. 5 -2 6 6) sin Ό = - , cos Ό = . Find cotΌ. 7 7 3 Name the quadrant in which the angle Ό lies. 7) sin Ό > 0 and cos Ό < 0 ___________________________________________________________________________________ Evaluate. Δ ŘΔ 8) csc - + sec - 3 3 ___________________________________________________________________________________ 9) Which function increases more rapidly over the interval [0, pi/4]? f(x) = sinx or g(x) = tan x? __________________________________________________________________________________________ 10) If sin x = 2/3, what is a) sin(x+pi) b) sin (pi - x) c) sin (2pi - x) _______________________________________________________________________________________________ 11) Which trigonometric functions are odd functions? 4 12) What is the range of the cosecant function? _____________________________________________________________________________________ 13) If cos x = 0.4, what is a) cos (-x) b) cos (x + 2pi) b) sin (pi - x) __________________________________________________________________________________________ 14) Evaluate 2 tan(pi) - 3 sin (-3pi/2) - 4 cos (-5pi) _____________________________________________________________________________________________ Find the exact value of the expression. Do not use a calculator. 19Δ 15) sec 4 ____________________________________________________________________________________________ Solve the problem. 16) What is the range of the tangent function? 5 17) Find the exact value of sin 135° - sin 270°. Do not use a calculator. ____________________________________________________________________________________________ Find the exact value of the expression. 18) If angle A is 30°, find the exact value of 4 sin A - 2 csc2 A + tan (90° - A). _____________________________________________________________________________________________ Solve the problem. 19) Find the exact value of cos 495° + tan -120°. Do not use a calculator. _____________________________________________________________________________________________ 20) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the formula 2a t = g sinΌ cosΌ where a is the length (in feet) of the base and g Y 32 feet per second per second is the acceleration of gravity. How long does it take a block to slide down an inclined plane with base a = 8 when Ό = 45°? Round the final answer to the nearest tenth of a second. 6 A point on the terminal side of angle Ό is given. Find the exact value of the given trigonometric function. 21) (-15, 36); Find sin Ό. ___________________________________________________________________________________________ Find the amplitude, phase shift and period of the function. Δ 22) y = -2 sin (4x - ) 2 ____________________________________________________________________________ 23) Give the period of each of the six trigonometric functions and skech a rough sketch for each. 7 Answer Key Testname: REVCHAP6-SUMMER 2018 1 1 1) y = 4 sin x - Δ 3 18 2) (-3-4 3) 3) (a) 2, (b) (3 2) / 2; (c) 1; (d) ; (e) -6/pi 2 21 4) - 5 5) a) y = 3 sin (2x); (b) y = -3sin(2x - pi); (c) y=-3cos(2x-3pi/2) 2 6 6) 5 7) Quadrant II 2 3 8) -2 - 3 2 2 9) y = tan x. The ARC for tangent is 4/pi, which is bigger than the ARC for sine that is pi 10) (a) -2/3; (b) 2/3; (c) -2/3 11) sine, cosecant, tangent, cotangent 12) all real numbers greater than or equal to 1 or less than or equal to -1 13) (a) 0.4; (b) 0.4; (c) 0.84 14) 1 15) - 2 16) all real numbers 2 + 2 17) 2 18) 3 - 6 - 2 + 2 3 19) 2 20) 1 sec 12 21) 13 22) 2, pi/2, pi/8 23) you do this one 8.