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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

For the following exercises, change the functions from a product to a sum or a sum to a product.cos(6x) + cos(5x)
For the following exercises, find all exact solutions on the interval [0, 2π). tan x + 1 = 0
For the following exercises, find all exact solutions on the interval [0, 2π). 2sin(2x) + √2 = 0
For the following exercises, find all exact solutions on the interval [0, 2π). 2sin2 x − sin x = 0
For the following exercises, find all exact solutions on the interval [0, 2π).cos2 x − cos x − 1 = 0
For the following exercises, find all exact solutions on the interval [0, 2π).2sin2 x + 5 sin x + 3 = 0
For the following exercises, find all exact solutions on the interval [0, 2π).cos x − 5sin(2x) = 0
For the following exercises, prove the identities provided. tan (x+4) = tan x + 1 1 - tan x
For the following exercises, find all exact solutions on the interval [0, 2π).1/sec2 x + 2 + sin2 x + 4cos2 x = 0
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [0, 2π). Round to four decimal places. √3
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [0, 2π). Round to four decimal places.csc2 x −
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.tan2 x sin x
For the following exercises, solve exactly on the interval [0, 2π).Use the quadratic formula if the equations do not factor.cot2 x = −cot x
For the following exercises, graph each side of the equation to find the zeroes on the interval [0,2π).20cos2 x + 21cos x + 1 = 0
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table. Х У 0 1 1 6 2 11 3 6 4 1 5 6
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table. X y 0 1 1 -2 2 -2 3 4 -5 -2 I 5 1
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table. x y -3 3+2√2 -2 3 -1 2√2-1 0 1 1 3-2√2 2 1 3 -1-2V/2
For the following exercises, graph each side of the equation to find the zeroes on the interval [0,2π).sec2 x − 2sec x = 15
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.cos2 x sin(2x)
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.sin2 x + cos2 x = 0
For the following exercises, find a function of the form y = abx + csin (π/2 x) that fits the given data. x 0 y 4 1 0 2 16 3 -40
For the following exercises, prove or disprove the statements. tan(u + v) = tan u + tan v 1 tan u tan v
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.sin(3x) − sinx/sin x
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.tan2 (x/2) sin x
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.sin(2x) − sin x = 0
For the following exercises, find a function of the form y = abx cos (π/2 x) + c that fits the given data. X y 0 11 1 2 3 1 3 3 دیا
For the following exercises, find a function of the form y = abx cos (π/2 x) + c that fits the given data. ४ y 0 4 1 2 1 -11 3 1
For the following exercises, prove or disprove the statements. tan(u - v) = tan u - tan v 1 + tan utan v
For the following exercises, algebraically find an equivalent function, only in terms of sin x and/or cos x, and then check the answer by graphing both equations.sin(4x)
For the following exercises, prove or disprove the statements. tan(x + y) 1 + tan x tan x tan x + tan y 1 - tan² xtan² y
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.cos(2x) − cos x = 0
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.A man with his eye level 6 feet above the ground is standing 3 feet away from
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.sin x cos(15x) − cos x sin(15x)
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities. 2 tan x 2 - sec² x sin² x cos² x
For the following exercises, algebraically find an equivalent function, only in terms of sin x and/or cos x, and then check the answer by graphing both equations.cos(4x)
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.Using the ladder from the previous exercise, if a 6-foot-tall construction
For the following exercises, prove the following sum-to-product formulas.sin x − sin y = 2 sin (x − y/2) cos ( x + y/2)
For the following exercises, prove or disprove the statements.If α, β, and γ are angles in the same triangle, then prove or disprove sin(α + β) = sin γ.
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.A spring attached to a ceiling is pulled down 11 cm from
For the following exercises, rewrite the expression with an exponent no higher than 1.cos4 x sin2 x
For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.sin(195°)
For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.cos(165°)
For the following exercises, solve exactly on the interval [0, 2π).Use the quadratic formula if the equations do not factor.5cos2 x + 3cos x−1 = 0
For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to
For the following exercises, find all solutions exactly on the interval 0 ≤ θ < 2π.2cos θ = 1
For the following exercises, find the amplitude, period, and frequency of the given function.The displacement h(t) in centimeters of a mass suspended by a spring is modeled by the function h(t) =
For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.A spring attached to a ceiling is pulled down 21 cm from
For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.tan4 x
For the following exercises, rewrite the expression with an exponent no higher than 1.tan2 x sin2 x
For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.cos(345°)
For the following exercises, solve exactly on the interval [0, 2π).Use the quadratic formula if the equations do not factor.3cos2 x − 2cos x − 2 = 0
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.Two springs are pulled down from the ceiling and released at the
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.sin2 (2x)
For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to
For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point.tan(−15°)
For the following exercises, solve exactly on the interval [0, 2π).Use the quadratic formula if the equations do not factor.5sin2 x + 2sin x − 1 = 0
For the following exercises, prove the identities provided. tan(a + b) tan(a - b) sin a cos a + sin bcos b sin a cos a sin bcos b
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.Two springs are pulled down from the ceiling and released at the
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.sin2 x cos2 x
For the following exercises, solve exactly on the interval [0, 2π).Use the quadratic formula if the equations do not factor.tan2 x + 5tan x − 1 = 0
For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to
For the following exercises, prove the identities provided. cos(a + b) cos a cos b 1 tan atan b -
For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.A plane flies 1 hour at 150 mph at 22° east of north, then
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. sin(9t) sin(3t) cos(9t) cos(3t)
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.A plane flies 2 hours at 200 mph at a bearing of 60°, then
For the following exercises, find a function of the form y = abx + csin (π/2 x) that fits the given data. x 0 1 y 6 29 2 96 3 379
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.tan4 x cos2 x
For the following exercises, solve exactly on the interval [0, 2π).Use the quadratic formula if the equations do not factor.−tan2 x − tan x − 2 = 0
For the following exercises, find a function of the form y = abx + csin (π/2 x) that fits the given data. X 0 y 6 1 2 34 150 3 746
For the following exercises, prove the identities provided.cos(x + y)cos(x − y) = cos2 x − sin2 y
For the following exercises, prove the identities provided. cos(x + h) - cos x h = cos x cos h - 1 h sin h h sin x-
For the following exercises, find exact solutions on the interval [0,2π).Look for opportunities to use trigonometric identities.sin2 x − cos2 x − sin x = 0
For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.2sin(8x)cos(6x) − sin(2x)
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.cos2 (2x)sin x
For the following exercises, construct an equation that models the described behavior.A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January.
For the following exercises, solve with the methods shown in this section exactly on the interval [0, 2π).cos(2x)cos x + sin(2x)sin x = 1
For the following exercises, prove the identity.sin x + sin(3x) = 4sin x cos2 x
For the following exercises, prove or disprove the identity.csc2 x(1 + sin2 x) = cot2 x
For the following exercises, prove the identity given.sin(2x) = −2 sin(−x) cos(−x)
For the following exercises, prove or disprove the identity. sec²(-x) tan² x - tan x 2 + 2 tan x 2+2 cot x 2 sin² x cos 2x
For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that
For the following exercises, construct an equation that models the described behavior.A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January.
For the following exercises, solve with the methods shown in this section exactly on the interval [0, 2π).6sin(2t) + 9sin t = 0
For the following exercises, prove the identity.2(cos3 x − cos x sin2 x)= cos(3x) + cos x
For the following exercises, prove the identity given.cot x − tan x = 2 cot(2x)
For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that
For the following exercises, construct an equation that models the described behavior.A fish population oscillates 40 above and below average during the year, reaching the lowest value in January.
For the following exercises, prove or disprove the identity. tan x sec x sin(-x) = cos² x
For the following exercises, solve with the methods shown in this section exactly on the interval [0, 2π).9cos(2θ) = 9cos2 θ − 4
For the following exercises, prove the identity given. sin (20) 1 + cos(20) -tan²0 tan³ 0 =
For the following exercises, prove the identity.2 tan x cos(3x) = sec x(sin(4x) − sin(2x))
For the following exercises, prove or disprove the identity. sec(-x) tan x + cotx = -sin(-x)
For the following exercises, construct an equation that models the described behavior.A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by
For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that
For the following exercises, solve with the methods shown in this section exactly on the interval [0, 2π).sin(2t) = cos t
For the following exercises, prove the identity.cos(a + b) + cos(a − b) = 2cos a cos b
For the following exercises, construct an equation that models the described behavior.A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by

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