Pre-Calculus II: Trigonometry Exam 1 Preparation Solutions. Math&142 November 8, 2016

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1 Pre-Calculus II: Trigonometry Exam 1 Preparation Solutions Math&1 November 8, Convert the angle in degrees to radian. Express the answer as a multiple of π. a 87 π rad 180 = 87π 180 rad b 16 π rad 180 = 11π 1 rad. Convert the following from radian to degrees. a π b π rad 180 π rad = 7 rad 180 π rad = 00. If s denotes the length of the arc of a circle of radius r subtended by a central angle, find the missing quantity. r = 1 feet, s = feet, then = s = r = s r = = 1 rad 1/. The area of a sector of a circle with central angle rad is 16 m. Find the radius of the circle. A = 1 r where = rad and A = 16 m ; 16 = 1 r r = m. Find the exact value of the expression. Do not use a calculator. a cos60 + tan0 = = = b cos π 1 =

2 6. In the problem, t is a real number and P = x, y is the point on the unit circle that corresponds to t. Find the exact value of the indicated trigonometric function of t. a 7, 7 Find csct and cott csct = 1 sint = 1 /7 = 7 cott = cost sint = /7 /7 = 7. Use the even-odd properties to find the exact value of the expression. Do not use a calculator. a sin π b cos π 6 a sin π = sin π cos π 6 = = 1 8. Find the exact value of the indicated trigonometric function of. a Given sin = 1, tan > 0, Find sec and cot sec = hyp adj = 1 1 cot = adj opp = 1 = 1 b Given sin =, cos = 1, Find tan tan = sin cos = / = 1/ 9. A building 90 feet tall casts a 80 foot long shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building to the nearest degree? Assume the person s eyes are level with the top of the building. tan =

3 10. Without graphing the function y = sin x + π, determine its amplitude, period, and phase shift. Re-write the function: y = sin x + π = sin [x π] amplitude is = period is π = π phase shift is ϕ ω = π = π 11. Wildlife management personnel use predator-prey equations to model the populations of certain predators and their prey in the wild. Suppose the population M of a predator after t months is given by π M = sin 6 t while the population N of its primary prey is given by π N = 1, cos t Find the period of each of these functions. Period for predators: Period for prey: T = π π/6 = 1 T = π π/ = 8 1. For what numbers x, 0 x π, does sinx = 0? x = 0, x = π, x = π 1. Make a sketch of each of the functions below: a y = sin π x b y = sin 1 x c y = cos π x d y = cos 1 x 1. What is the y-intercept of y = cscx? Since y = cscx = 1 sinx 1. For what numbers x, π x π, does secx = 1? when x = 0 yields an undefined value, there is NO y-intercept. secx = 1 when cosx = 1 this happens when x = π, x = 0, x = π 16. For what numbers x, π x π, does the graph of y = cotx have vertical asymptotes? y = cotx = cosx sinx 17. Find the phase shift of the function, y = sin 1 x π Vertical Asymptotes where sinx = 0 where x = π, π, 0, π, π y = sin 1 x π = sin 1 x π phase shift π units right.

4 18. Graph the function. Show at least one period. where a y = cosx π Take y = cosx shift π units right, flip over x-axis and multiply by. b y = sinx Here the period: T = π = π 1 cycle in π units. Length of ONE sub-interval: 1 π = π 8 New Equilibrium: at y = and has an amplitude of units above/below equilibrium. c y = sinx + π We can rewrite this as: y = sin[x π] Amplitude: Period: π = π Phase Shift: π

5 1 d y = sec x Sketch y = cos Period: 1 x then flip the humps of the graph. π = π One cycle in π units 1/ Length of ONE sub-interval: 1 π = π 19. Evaluate the following exactly: a sin 1 = MEANS sin = = π b cos 1 = MEANS cos = = π c tan 1 1 = MEANS tan = 1 = π

6 d cos [sin 1 ] Let = sin 1 then sin = sin = = opp hyp Quad IV, Form Right Triangle, 1 So, cos = 1 [ ] e sec tan 1 Let = tan 1 then tan = tan = = opp adj 1 Quad I, Form Right Triangle, So, sec = 1 cos = 1 1 / 1 = 0. Sketch y = tanx and y = tan 1 x. DONE IN CLASS, LOOK AT YOUR NOTES OR TEXT- BOOK. 1. Solve each equation on the interval 0 < π a cos + 1 = 0 cos = 1 = π, π b cosx cosx sinx = 0 cosx[1 { cosx = 0 x = π sinx] = 0, π sinx = 1 x = π, π c sinx cosx = 0 sinx = cosx x = π, π. Solve each equation. Find a GENERAL formula for all solutions: { = π a cos = 1 = + nπ = π 6 + nπ = π + nπ = n Z. π 6 + nπ

7 b sin = sin { = = = π + nπ = π + 6nπ n Z + nπ = π + 6nπ = π c cos + sin = 1 write cos in terms of sine [1 sin ] + sin 1 0 sin + sin 1 = 0 sin + sin + 1 = 0 sin sin 1 = 0 sin + 1sin { 1 = 0 = 7π sin = nπ = 11π 6 + nπ, n Z sin = 1 = π + nπ for any integer n. d tan + π = 1 We know tan = 1 where = π + πn, n Z Must have + π = π + πn = π π + πn = π 1 + πn = π 6 + π n, n Z