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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.130tan2 x + 69tan x
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.2tan2 x + 7tan x + 6 =
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.20sin2 x − 27sin x +
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.2cos2 x − cos x + 15
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.100tan2 x + 20tan
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.8cos2 x − 2cos x −
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros. 6sin2 x − 5sin
For the following exercises, find exact solutions on the interval [0,2π). Look for opportunities to use trigonometric identities.cos3 t = cos t
For the following exercises, find exact solutions on the interval [0,2π). Look for opportunities to use trigonometric identities.tan x = 3sin x
For the following exercises, prove the identity. tan (7-1)= 1 tan t 1 + tan t
For the following exercises, find exact solutions on the interval [0,2π). Look for opportunities to use trigonometric identities.12sin2 t + cos t − 6 = 0
For the following exercises, prove the identities.cos(16x) = (cos2 (4x) − sin2 (4x) − sin(8x))(cos2 (4x) − sin2 (4x) + sin(8x))
For the following exercises, prove the identity.(cos(2x) − cos(4x))2 + (sin(4x) + sin(2x))2 = 4 sin2 (3x)
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.6cos2 x + 7sin x − 8 = 0
For the following exercises, prove the identities.sin(16x) = 16 sin x cos x cos(2x)cos(4x)cos(8x)
For the following exercises, prove the identity.cos x − cos(3x) = 4 sin2 x cos x
For the following exercises, find exact solutions on the interval [0,2π). Look for opportunities to use trigonometric identities.8cos2 θ = 3 − 2cos θ
For the following exercises, prove the identities. 1 + cos(2t) sin(2t) - cost 2 cos t 2 sin t - 1
For the following exercises, prove the identity. sin (10x) sin(2x) cos (10x)+ cos(2x) tan(4x)
For the following exercises, model the described behavior and find requested values.The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to
For the following exercises, find exact solutions on the interval [0,2π). Look for opportunities to use trigonometric identities.8sin2 x + 6sin x + 1 = 0
For the following exercises, prove the identity. cos(2y) - cos(4y) sin(2y) + sin(4y) tan y
For the following exercises, prove the identities.cos(3x) = cos3 x − 3 sin2 x cos x
For the following exercises, model the described behavior and find requested values. An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.4cos2 x − 4 = 15cos x
For the following exercises, prove the identity. cos(6y)+cos(8y) sin(6y) - sin(4y) = cot y cos(7y) sec(5y)
For the following exercises, prove the identities.sin(3x) = 3 sin x cos2 x − sin3 x
For the following exercises, find the amplitude, frequency, and period of the given equations.y = −2sin(16xπ)
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.−3sin t = 15cos t sin t
For the following exercises, prove the identities.(sin2 x − 1)2 = cos(2x) + sin4 x
For the following exercises, prove the identity. cos(3x) + cos x cos(3x) - cos x -cot (2x)cot x
For the following exercises, find the amplitude, frequency, and period of the given equations. y = 3cos(xπ)
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.10sin x cos x = 6cos x
For the following exercises, prove the identities. tan(2x): 2 sin xcos x 2 cos²x - 1
For the following exercises, prove the identity. sin(6x)+sin(4x) sin(6x) = sin(4x) = tan (5x) cotx
For the following exercises, construct functions that model the described behavior.Daily temperatures in the desert can be very extreme. If the temperature varies from 90°F to 30°F and the average
Prove the conjecture made in the previous exercise.Data from previous exercise Explore and discuss the graphs of and g(x) = − log2 (x). Make a conjecture based on the result.
Refer to the previous exercise. Suppose the light meter on a camera indicates an EI of −2, and the desired exposure time is 16 seconds. What should the f-stop setting be?
For the following exercises, simplify the given expression. cos(−x)sin x cot x + sin2 x
For the following exercises, find all solutions exactly that exist on the interval [0,2π). csc2 t = 3
For the following exercises, simplify the given expression.sin(−x)cos(−2x)−sin(−x)cos(−2x)
For the following exercises, find all solutions exactly that exist on the interval [0,2π).cos2 x = 1/4
For the following exercises, find the exact value.cos (7π/12)
For the following exercises, find all solutions exactly that exist on the interval [0,2π).2sin θ = −1
For the following exercises, find the exact value.tan (3π/8)
For the following exercises, find all solutions exactly that exist on the interval [0,2π).tan x sin x + sin(−x) = 0
For the following exercises, find all solutions exactly that exist on the interval [0,2π).9sin ω − 2 = 4 sin2 ω
For the following exercises, find the exact value.2sin (π/4) sin (π/6)
For the following exercises, find all solutions exactly that exist on the interval [0,2π).1 − 2tan(ω) = tan2 (ω)
For the following exercises, find all exact solutions to the equation on [0, 2π). cos2 x − sin2 x − 1 = 0
For the following exercises, use basic identities to simplify the expression. sec x cos x + cos x − 1/sec x
For the following exercises, find all exact solutions to the equation on [0, 2π).cos2 x = cos x 4sin2 x + 2sin x − 3 = 0
For the following exercises, use basic identities to simplify the expression. sin3 x + cos2 x sin x
For the following exercises, determine if the given identities are equivalent. (1 cos²x)(1+ cos²x) cos²x sin² x + sec²x-1=-
For the following exercises, find all exact solutions to the equation on [0, 2π).cos(2x) + sin2 x = 0
For the following exercises, find all exact solutions to the equation on [0, 2π).2sin2 x − sin x = 0
For the following exercises, determine if the given identities are equivalent.tan3 x csc2 x cot2 x cos x sin x = 1
For the following exercises, find all exact solutions to the equation on [0, 2π).Rewrite the expression as a product instead of a sum: cos(2x) + cos(−8x).
For the following exercises, find all exact solutions to the equation on [0, 2π).Find all solutions of tan(x) − √3 = 0.
For the following exercises, find the exact value. cos (25π/12)
For the following exercises, find all exact solutions to the equation on [0, 2π).Find the solutions of sec2 x − 2sec x = 15 on the interval [0, 2π) algebraically; then graph both sides of the
For the following exercises, find the exact value. sin(70°)cos(25°) − cos(70°)sin(25°)
For the following exercises, find all exact solutions to the equation on [0, 2π).Find sin(2θ), cos(2θ), and tan(2θ) given cot θ = −3/4 and θ is on the interval [π/2 ,π].
For the following exercises, find the exact value. cos(83°)cos(23°) + sin(83°)sin(23°)
For the following exercises, find all exact solutions to the equation on [0, 2π).Find sin (θ/2), cos (θ/2), and tan(θ/2) given cos θ = (7/25) and θ is in quadrant IV.
For the following exercise, simplify the expression. 1 tan ta 8 - tan(x) tan(x) 1- tan
For the following exercises, find all exact solutions to the equation on [0, 2π).Rewrite the expression sin4 x with no powers greater than 1.
For the following exercises, prove the identity.cos(3x) − cos3 x = − cos x sin2 x − sin x sin(2x)
For the following exercises, prove the identity.tan3 x − tan x sec2 x = tan(−x)
For the following exercises, prove the identity.Plot the points and find a function of the form y = Acos(Bx + C) + D that fits the given data. x y 0 -2 1 2 2 -2 3 2 4 -2 5 2
For the following exercises, prove the identity.sin(3x) − cos x sin(2x) = cos2 x sin x − sin3 x
For the following exercises, find the exact value. tan (sin−1 (0) + sin−1 (1/2))
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. cos(5x)+cos (3x) sin(5x)+sin(3x)
For the following exercises, find the exact value.sin (7π/8)
For the following exercises, prove the identities. sin(2x) = 2 tan x 1 + tan x
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.1 − cos(2x) = 1 + cos(2x)
For the following exercises, construct functions that model the described behavior. A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of
For the following exercises, prove or disprove the statements. If a, ß, and y are angles in the same triangle, then prove or disprove: tan a + tan ß+ tan y = tan a tan ẞtan y.
For the following exercises, prove the following sum-to-product formulas.cos x + cos y = 2cos (x + y/2) cos (x − y/2)
For the following exercises, prove the identities. cos(2a) 1 - tan² a 1 + tan² a
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.sec2 x = 7
For the following exercises, prove the identity.A woman is standing 300 feet away from a 2,000- foot building. If she looks to the top of the building, at what angle above horizontal is she looking?
For the following exercises, find the exact value.sec (3π/8)
For the following exercises, prove the identity.Two frequencies of sound are played on an instrument governed by the equation n(t) = 8 cos(20πt)cos(1,000πt). What are the period and frequency of
For the following exercises, use Figure 1 to find the desired quantities. sin(2β), cos(2β), tan(2β), sin(2α), cos(2α), and tan(2α) α 25 24 Figure 1 B
For the following exercises, use Figure 1 to find the desired quantities.sin (β/2) , cos (β/2) , tan (β/2) , sin (α/2) , cos (α/2) , and tan (α/2) α 25 24 Figure 1 B SO
For the following exercises, prove the identity.The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that
For the following exercises, prove the identity.A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the
For the following exercises, prove the identity.Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in
For the following exercises, prove the identity.cot x cos(2x) = − sin(2x) + cot x
For the following exercises, rewrite the expression with no powers.cos2 x sin4 (2x)
For the following exercises, rewrite the expression with no powers.tan2 x sin3 x
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer. cos (π/3) sin (π/4)
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.2sin (2π/3) sin (5π/6)
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.2cos (π/5) cos (π/3)
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.sin (π/12) − sin (7π/12)
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.cos (5π/12) + cos (7π/12)
For the following exercises, change the functions from a product to a sum or a sum to a product. sin(9x)cos(3x)
For the following exercises, change the functions from a product to a sum or a sum to a product.cos(7x)cos(12x)
For the following exercises, change the functions from a product to a sum or a sum to a product.sin(11x) + sin(2x)

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