# Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) ) sin 11π 1 ) ( - 1) - ( - 1) ( - 1) - ( - 1) ) sin 1 ) ( - 1) - ( + 1) ( + 1) - ( - 1) ) sin ) - ( + 1) ( + 1) - ( - 1) ( - 1) ) tan ) ) sin cos - cos sin ) ) sin cos 1 + cos sin 1 ) ) cos π 1 cos π + sin π 1 sin π ) ) cos π 1 cos π - sin π 1 sin π )

2 ) tan 0 - tan (-0 ) 1 + tan 0 tan (-0 ) ) Find the exact value under the given conditions. 11) sin α = 0, 0 < α < π 1 ; cos β = 1, 0 < β < π Find cos (α + β). 11) 1 1 1) tan α = 1, π < α < π ; cos β = -, π 1 < β < π Find sin (α + β). 0 1) 1) sin α =, π < α < π; cos β =, 0 < β < π Find cos (α - β). 1) ) sin α = -, π < α < π ; tan β = - 1 1, π < β < π Find cos (α + β). 1) ) cos α = 1, 0 < α < π ; sin β = - 1, - π < β < 0 Find tan(α + β). 1) ) cos α = - 1, π < α < π; sin β = 1 1, π < α < π Find tan(α - β). 1) Solve the problem. 1) If sin θ = 1, θ in quadrant II, find the exact value of cos θ + π 1)

3 1) If cos θ = 1, θ in quadrant IV, find the exact value of tan θ + π 1) Use the figures to evaluate the function if f(x) = sin x, g(x) = cos x, and h(x) = tan x. x + y = x + y = 1 (x, ) 1, y 1) f(α + β) 1) ) g(α + β) 0) ) h(α - β) 1) ) f(α - β) ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Establish the identity. ) sin x + π = cos x ) ) cos x + π = cos x - 1 sin x ) ) tan x - π = tan x tan x )

4 ) tan π + x = -cot x ) ) cos π - θ = -sin θ ) ) csc π + u = sec u ) ) cos(α + β) = cot β - tan α ) cos α sin β 0) cos(x - y) - cos(x + y) = sin x sin y 0) 1) cos(x - y) 1 + tan x tan y = cos(x + y) 1 - tan x tan y 1) ) cot(π - θ) = - cot θ ) Solve the problem. ) If tan α = x + 1 and tan β = x - 1, show that cot(α + β) = - x x ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. ) sin cos sin-1 ) 1 0 ) cos tan -1 - sin-1 ) 1 ) tan tan -1 + sin-1 1 ) ) cos sin tan-1 1 )

5 ) cos tan cos-1 ) 1 1 Use the information given about the angle θ, 0 θ π, to find the exact value of the indicated trigonometric function. ) sin θ = 1 1, 0 < θ < π Find cos(θ). ) ) tan θ =, π < θ < π Find sin(θ). 0) - - 1) csc θ = -, tan θ > 0 Find cos(θ). 1) ) sin θ =, tan θ < 0 Find sin(θ). ) - - ) tan θ =, π < θ < π Find cos(θ). ) - - ) sin θ = -, π < θ < π Find tan(θ). ) - - ) cos θ = - 1, π < θ < π Find tan(θ). ) ) cos θ = -, π < θ < π Find sin θ. ) - -

6 Use the figures to evaluate the function given that f(x) = sin x, g(x) = cos x, and h(x) = tan x. x + y = x + y = 1 (a, ) - 1, b ) f(α) ) ) f(β) ) - - Find the exact value of the expression. ) sin cos -1 - ) ) sin sin -1 0) ) cos sin ) ) tan cos -1 - ) ) sec tan -1 )

7 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. ) The path of a projectile fired at an inclination θ (in degrees) to the horizontal with an initial speed v0 is a parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the formula ) R = v 0 g sin(θ) where g is the acceleration due to gravity. The maximum height H of the projectile is v 0 H = (1 - cos(θ)) g Find the range R and the maximum height H in terms of g if the projectile is fired with an initial speed of 00 meters per second at an angle of 1 and then at an angle of.. Do not use a calculator, but simplify the answers. Establish the identity. ) tan u (1 + cos(u)) = 1 - cos(u) ) ) cot(θ)= csc θ - cot θ ) ) cos(x) = cos x - sin x cos x ) ) cot u = csc u + cot u csc u - cot u ) ) cos(u) = cos (u) - 1 ) 0) sin (x) = 1 (sin(x))(1 - cos(x)) 0) 1) sin(θ) = sin θ - cos θ sin θ - cos θ 1) ) cos(θ) = cos θ - sin θ cos θ + sin θ ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the information given about the angle θ, 0 θ π, to find the exact value of the indicated trigonometric function. ) sin θ = 1, 0 < θ < π Find sin θ. )

8 ) sin θ = 1, tan θ > 0 Find cos θ. ) ) tan θ = 1, π < θ < π Find sin θ. ) ) tan θ = 1, π < θ < π Find cos θ. ) ) cot θ = -, sec θ > 0 Find sin θ. ) ) tan θ =, cos θ < 0 Find sin θ. ) ) cos θ = -, π < θ < π Find cos θ. ) ) cos θ = -, sin θ > 0 Find cos θ. 0) ) cos(θ) = 1, 0 < θ < π Find cos θ. 1) - - ) cos(θ) = 1, 0 < θ < π Find sin θ. ) - -

9 Use the Half-angle Formulas to find the exact value of the trigonometric function. ) sin ) ) cos 1 ) ) sin ) ) cos ) ) cos - π ) ) sin π ) Find the exact value of the expression. ) sin 1 cos-1 ) ) cos 1 sin-1 0) 1 1

10 Use the figures to evaluate the function given that f(x) = sin x, g(x) = cos x, and h(x) = tan x. x + y = x + y = 1 (a, ) - 1, b 1) f α 1) ) f β ) - - ) h β ) - - Express the product as a sum containing only sines or cosines. ) sin(θ) cos(θ) ) 1 [sin(1θ) + sin(θ)] 1 [cos(1θ) - cos(θ)] 1 [sin(1θ) + cos(θ)] sin cos(θ ) ) cos(θ) cos(θ) ) cos (θ ) 1 [ cos θ + cos(θ)] 1 [cos(θ) - cos θ] 1 [cos(θ) - sin θ] ) sin(θ) cos(θ) ) 1 [cos(θ) - cos θ] sin cos(0θ ) 1 [cos(θ) + sin θ] 1 [sin(θ) - sin θ]

11 ) cos θ cos θ ) 1 [cos(θ) - sin(θ)] 1 [cos(θ) + cos(θ)] 1 cos (θ) 1 [cos(θ) - sin(θ)] ) sin θ cos θ ) 1 [cos(θ) - sin(θ)] 1 sin cos(θ) 1 [cos(θ) + sin(θ)] 1 [sin(θ) - sin(θ)] Complete the identity. ) sin(θ) sin(θ) cos(θ) cos(θ) =? cos (0θ) sin (0θ) cos (1θ) + cos (θ) cos (θ) - cos (1θ) ) Express the sum or difference as a product of sines and/or cosines. 0) sin(θ) + sin(θ) sin(1θ) cos(θ) sin(θ) sin(θ) sin(θ) sin(θ) cos(θ) 0) 1) cos(θ) - cos(θ) - cos(θ) sin(θ) cos(θ) - sin(θ) sin(θ) cos(θ) cos(θ) 1) ) sin(θ) - sin(θ) sin(θ) cos(θ) cos(θ) sin(θ) cos(θ) sin(θ) cos(θ) ) ) cos θ + cos θ ) cos(θ) sin(θ) sin θ sin(θ) sin θ cos(θ) cos θ ) sin 11θ + sin θ ) sin θ sin θ cos(θ) sin θ sin θ cos θ sin(θ) ) sin(θ) - sin(θ) sin(θ) cos θ sin(θ) cos θ sin θ cos(θ) sin θ cos(θ) ) 11

12 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Establish the identity. sin(θ) + sin(θ) ) cos(θ) + cos(θ) = tan(θ) ) ) sin(θ) + sin(θ) sin(θ) - sin(θ) = - tan(θ) tan(θ) ) ) cos(θ) - cos(θ) cos(θ)+ cos(θ) = - tan(θ) tan(θ) ) ) sin θ[sin θ + sin(θ)] = cos(θ)[cos(θ) - cos(θ)] ) 0) sin α - sin β sin α + sin β = tan α - β cot α + β 0) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Complete the identity. 1) 1 - cos(θ) + cos(θ) - cos(θ) =? sin θ sin(θ) sin(θ) cos θ cos(θ) sin(θ) cos θ cos(θ) cos(θ) sin θ cos(θ) sin(θ) 1) Solve the equation on the interval 0 θ < π. ) cos θ + = π, 11π π, π π, π π, π ) ) sin θ = 1 π, π, π, 11π π, π ) π, π, π, π π, π ) sin θ - = 0 π, π, π, π π, π, π, 11π ) π, π π, π 1

13 ) tan θ = ) π, π π π π, π ) sec θ = - ) π, π, π, 11π π, π π, π π, π, 11π ) cot θ = - ) π, π π, π, 1π π, π, 1π, π π, π ) cot θ - 1 = 0 π, π π, π π, 11π π, π ) ) csc θ - 1 = ) π π π π 1) cos(θ) = 1) π π, 11π π 1, 11π 1, 1π 1, π 1 π 111) cos θ - π = 111) π, π π, π, 11π π, π, π, and 1π π, π 1

14 11) cot θ - π = 1 11) π π, π, 11π 1π, and π, π π, π, π, and 1π Solve the equation. Give a general formula for all the solutions. 11) cos θ = 1 {θ θ = kπ} θ θ = π + kπ 11) {θ θ = π + kπ} θ θ = π + kπ 11) sin θ = 1 θ θ = π + kπ θ θ = π + kπ 11) {θ θ = π + kπ} {θ θ = kπ} 11) sin θ = 11) θ θ = π + kπ, θ = π + kπ θ θ = π + kπ, θ = π + kπ θ θ = π + kπ, θ = π + kπ θ θ = π + kπ, θ = π + kπ 11) csc θ = 11) θ θ = π + kπ {θ θ = π + kπ} 1 θ θ = π + kπ θ θ = π + kπ Use a calculator to solve the equation on the interval 0 θ < π. Round the answer to two decimal places. 11) sin θ = 0. 0., ,. 0.,. 0.,.1 11) 11) tan θ =. 1.1,. 1.1, ,.0 1.1,.1 11) 11) csc θ = ,. 0.1, ) ) cot θ = -.,.1.,.1.,..,. ) 1

15 Solve the equation on the interval [0, π). 11) Suppose f(x) = cos θ -1. Solve f(x) = 0. π 0 π π 11) 1) Suppose f(x) = cos θ + 1. Solve f(x) = 0. 1) π, π π π, π π, π Solve the problem. 1) What are the x-intercepts of the graph of f(x) = sin(x) + on the interval [0, π]? π, π, π, 11π, 1π, 1π π 1, 11π 1, 1π 1, π 1 π, π, π, 11π, 1π, 1π π, π 1) 1) Given f(x) = tan x, for what values of x is f(x) > - on the interval - π, π? 1) - π, π - π, π 0, π - π, π Solve the problem using Snellʹs Law: sin θ 1 sin θ = v 1 v. 1) A light beam in air travels at. meters per second. If its angle of incidence to a second medium is and its angle of refraction in the second medium is, what is its speed in the second medium (to two decimal places)? 1.0 mps.01 mps. mps 1. mps 1) The index of refraction of light passing from air into a second medium is 1.. If the angle of incidence is, what is the angle of refraction (to two decimal places)? ) 1) 1) A light beam in air travels at. meters per second. If its angle of incidence to a second medium is and its angle of refraction in the second medium is, what is its speed in the second medium (to two decimal places)?.0 mps. mps. mps.0 mps 1) Solve the problem. 1) A weight suspended from a spring is vibrating vertically with up being the positive direction. The function f(t) = sin πt - π represents the distance in centimeters of the weight from its rest position as a function of time t, where t is measured in seconds. Find the smallest positive value of t for which the displacement of the weight above its rest position is cm. Round answer to three decimal places, if necessary. 0. sec 0. sec 1. sec. sec 1) 1

16 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) 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 1) M = sin π t while the population N of its primary prey is given by N = 1, cos π t Find the values of t, 0 t < 1, for which the predator population is. Find the values of t, 0 t < 1, for which the prey population is,. ) You are flying a kite and want to know its angle of elevation. The string on the kite is meters long and the kite is level with the top of a building that you know is meters high. Use an inverse trigonometric function to find the angle of elevation of the kite. Round to two decimal places. ) 1

17 Answer Key Testname: UNTITLED1 1) D ) C ) A ) C ) D ) A ) D ) A ) D ) B 11) D 1) C 1) D 1) C 1) B 1) D 1) C 1) C 1) A 0) C 1) C ) D ) sin x + π = sin x cos π + sin π cos x = (sin x)(0) + (1)(cos x) = cos x. ) cos x + π = cos x cos π - sin x sin π = cos x - 1 sin x. ) tan x - π = tan x - tan π/ 1 + (tan x)(tan π/) = tan x tan x. ) tan π sin ((π/) + x) sin (π/) cos x + sin x cos (π/) + x = = cos ((π/) + x) cos (π/) cos x - sin (π/) sin x = 1 cos x + sin x 0 0 cos x - 1 sin x π ) cos - θ = cos π cos θ + sin π sin θ = 0 cos θ - 1 sin θ = - sin θ ) csc π + u = 1 sin (π/) cos u + cos (π/) sin u = 1 = sec u. 1 cos u + 0 sin u ) cos(α + β) cos α cos β - sin α sin β cos α cos β = = cos α sin β cos α sin β cos α sin β = -cot x. sin α sin β - cos α sin β = cos β sin β - sin α = cot β - tan α cos α 0) cos (x - y) - cos (x + y) = cos x cos y + sin x sin y - ( cos x cos y - sin x sin y) = sin x sin y. cos (x - y) cos x cos y + sin x sin y 1/(cos x cos y) cos x cos y + sin x sin y 1) = = cos (x + y) cos x cos y - sin x sin y 1/(cos x cos y) cos x cos y - sin x sin y = 1 + tan x tan y 1 - tan x tan y. ) cot(π - θ) = ) cot(α + β) = ) D ) B cos(π - θ) sin(π - θ) cos π cos θ + sin π sin θ (-1) cos θ + 0 sin θ = = sin π cos θ - cos π sin θ 0 cos θ - (-1) sin θ = - cos θ sin θ = - cot θ tan α tan β 1 - (x + 1)(x - 1) = = = 1 - (x - 1) = - x tan(α + β) tan α + tan β (x + 1) + (x - 1) x x 1

18 Answer Key Testname: UNTITLED1 ) B ) A ) B ) A 0) C 1) C ) D ) C ) B ) D ) B ) C ) D ) C 0) D 1) D ) D ) D ) θ = 1 : R = 0,000, H = g θ =. : R = 000( - ) ; g 0,000( ) 000( - ), H = g g ) tan u (1 + cos(u)) = 1 - cos(u) (1 + cos(u)) = 1 - cos(u) 1 + cos(u) ) cot(θ) = cos(θ) sin(θ) = 1 - sin θ sin θ cos θ = 1 sin θ - cos θ sin θ = csc θ - cot θ ) cos(x) = cos(x + x) = cos(x) cos x - sin(x) sin x = (cos x - sin x) cos x - sin x cos x sin x = cos x - sin x cos x - sin x cos x = cos x - sin x cos x. ) cot u = 1 tan u = 1 + cos u csc u + cot u = 1 - cos u csc u - cot u ) cos(u) = cos[(u)] = cos (u) - 1 0) sin (x) = (sin (x))(sin(x)) = 1 - cos(x) (sin(x)) = 1 (sin(x))(1 - cos(x)). 1) sin θ - cos θ sin θ - cos θ = sin θ + sin θ cos θ + cos θ = 1 + sin θ cos θ = sin(θ) ) cos(θ) = cos[(θ)] = cos (θ) - sin (θ) = (cos θ - sin θ) - ( sin θ cos θ) = cos θ - sin θ cos θ + sin θ - sin θ cos θ = cos θ - sin θ cos θ + sin θ ) A ) A ) C ) B ) A 1

19 Answer Key Testname: UNTITLED1 ) D ) B 0) A 1) B ) B ) A ) B ) A ) C ) B ) C ) A 0) A 1) B ) D ) D ) A ) B ) D ) B ) D ) D 0) D 1) C ) D ) D ) C ) D sin (θ) + sin (θ) sin (θ) cos (θ) sin (θ) ) = = cos (θ) + cos (θ) cos (θ) cos (θ) cos (θ) ) ) sin (θ) + sin (θ) sin (θ) - sin (θ) = sin (θ) cos (θ) sin (θ) cos (θ) = sin (θ) cos (θ) = tan (θ) cos (θ) tan (θ) = sin (θ) tan (θ) cos (θ) - cos (θ) - sin (θ) sin (θ) sin (θ) = = - cos (θ) + cos (θ) cos (θ) cos (θ) cos (θ) sin(θ) = - tan (θ) tan (θ) cos(θ) ) sin θ[sin θ + sin(θ)] = sin θ[ sin(θ) cos(θ)] = cos(θ)[sin θ sin(θ)] = cos(θ) 1 (cos(θ) - cos(θ)) = cos( 0) θ)[cos(θ) - cos(θ)] sin α - β cos α + β sin α - sin β sin α + sin β = sin α + β cos α - β = sin α - β cos α - β cos α + β sin α + β = tan α - β cot α + β 1) D ) C ) A ) A ) B ) D ) B 1

20 Answer Key Testname: UNTITLED1 ) B ) B 1) C 111) B 11) C 11) A 11) B 11) B 11) B 11) D 11) C 11) C ) D 11) B 1) A 1) C 1) D 1) C 1) D 1) D 1) A 1) M =, t = ; N =,, t =, ) 0. 0

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