Math 1060 Midterm Review Dugopolski Trigonometry Edition, Chapter and.1 Use identities to find the exact value of the function for the given value. 1) sin α = and α is in quadrant II; Find tan α. Simplify the expression. ) cot x sec x sin x ) 1 + sec x cos x cot x ) cos x - cos x sin x Determine whether the function is odd, even, or neither. 5) f(x) = cos x csc x Simplify the expression. cos (-x) 6) tan (-x) - sin x. Multiply and simplify. 7) (1 - cos x)(1 + cos x) Factor and simplify the expression. 8) sinx + sinx cotx Identify the equation as either an identity or not. sin x 9) 1 - cos x + sin x 1 + cos x = csc x 10) cotx csc x - 1 = 1 + sin x sin x. Find the exact value by using a sum or difference identity. 11) cos 75 Find the exact value by using a sum or difference identity. 1) cos 1π 1 Use the sum/difference identities to simplify the expression. Do not use a calculator. 1) cos 5 cos 0 - sin 5 sin 0 Write in terms of the cofunction of a complementary angle. 1) cos π 1
15) cot 17π 18 Find cos(a + B). 16) sin A = - 1 and sin B = 1, with A in quadrant IV and B in quadrant II. Find cos(a - B). 17) cos A = 10 10 and sin B = -, where 0 < A < 90 and 70 < B < 60. Use the identities for the cosine of a sum or a difference to write the expression as a single function of α. 18) cos (α + 90 ). Find the exact value by using a sum or difference identity. 19) sin 11π 1 Find the exact value by using a sum or difference identity. 0) tan 105 Use trigonometric identities to find the exact value. tan 5 + tan 5 1) 1 - tan 5 tan 5 Use a sum or difference identity to find the exact value. ) sin π π cos 15 15 + cos π π sin 15 15 Solve the problem. ) If cos A = 1 and sin B = 1, where 0 A π and π B π, then find sin(a - B). Using a sum or difference identity, write the following as an expression involving functions of α. ) sin (α + 5 ) Decide whether the expression is or is not an identity. 5) sin (A + B) sin (A - B) = sina - sinb Determine if the equation is an identity. cot x cot y - 1 6) cot(x + y) = cot x + cot y.5 Find the exact value by using a half-angle identity. 7) sin.5
Find the exact value by using a half-angle identity. 8) cos - π 8 9) tan 7π 8 Determine whether the positive or negative sign makes the equation correct. Do not use a calculator. 0) cos 111 = ± 1 + cos Use identities to simplify the expression. Do not use a calculator. 1) cos.5-1 ) sin.5 cos.5 Identify the equation as either an identity or not. ) - tan x + tan x = 1 ) csc x - sec x csc x + sec x = cos x 1 + sin x 5) sin x = sin x Decide whether the expression is or is not an identity. 6) tan x = 1 - cos x 1 + cos x 7) sin x = sec x + 1 sec x Solve the problem. 8) Find cos θ.sin θ = 15, θ lies in quadrant I. 17 9) Find sin θ. tan θ = 7, θ lies in quadrant III. 0) Find tan θ.sin θ = 0, θ lies in quadrant II. 9 Use the given information given to find the exact value of the trigonometric function. 1) cos θ = 1, csc θ > 0 Find sin θ.
) sin θ = - 5, θ lies in quadrant IV Find sin θ..1 Find the exact value of the expression without using a calculator or table. ) sin-1 Find the exact value of the expression without using a calculator or table. ) csc-1(-) Find the exact value of the expression in degrees without using a calculator or table. 5) cos -1 6) arctan - Find the exact value of the composition. 7) csc sin -1 5 8) sin (arctan ()) 9) cos 1 arcsin 5 1 Find an equivalent algebraic expression for the composition. 50) sin (arccot (x)) 51) cot (arctan (x)). Find all real numbers that satisfy the equation. 5) sin x = - Find all real numbers that satisfy the equation. 5) 10 cos x + 8 = 8 cos + 7 Find all angles in degrees that satisfy the equation. Round approximate answers to the nearest tenth of a degree. 5) tan α = -.01 Solve the equation for 0 t < π. Approximate the solution to four decimal places. 55) sin t = 1
Solve the equation. 56) tan(α) + 1 = 0 for -60 α 60 Solve the equation for x. 57) y = tan (x - 1) 58) y = tan x - 1 Find the inverse of the function, and state the domain and range of the inverse function. 59) f(x) = 5 sin (x) for - π 6 x π 6 Find all real numbers that satisfy the equation. Round approximate answers to decimal places. 60) 5 = 5 sin(x) +. Find all real numbers that satisfy the equation. 61) sec x = Find all real numbers that satisfy the equation. 6) cot x = 1 Find all values of θ in [0, 60 ) that satisfy the equation. 6) cos θ = - 6) sin θ - = 0 Find all real numbers in [0, π] that satisfy the equation. 65) sin x = 66) cos x + 1 = 0 Find all angles in degrees that satisfy the equation. Round approximate answers to the nearest tenth of a degree. 67) sin α = 0.59 Find all real numbers that satisfy the equation. Round approximate answers to the nearest hundredth. 68) 10 cos x - = 0. Find all real numbers in the interval [0, π) that satisfy the equation. 69) cos x + cos x + 1 = 0 5
Find all real numbers in the interval [0, π) that satisfy the equation. 70) sin x = sin x 71) cos x = sin x 7) sin x - cos x = 0 Find all values of x in the interval [0, 60 ) that satisfy the equation. Round approximate answers to the nearest tenth of a degree. 7) sin x - 8 sin x - = 0 7) 7 cot x - 5 = 0 Solve the problem. 75) A weight is suspended on a system of springs and oscillates up and down according to P = 1 [sin(t) + sin t] 10 where P is the position in meters above or below the point of equilibrium (P = 0) and t is time in seconds. Find the time when the weight is at equilibrium. Find the exact values. Do not use a calculator. 6
Answer Key Testname: 1060 CH AND REVIEW 1) - 7 7 ) 1 ) sec x ) cos x 5) Odd 6) -csc x 7) sinx 8) 1 9) Identity 10) Identity 11) ( - 1) 1) 1) 1) sin 11π 15) tan -π 9 16) 1-5 8 17) 18) -sin α 19) 10-0 0 ( - 1) 0) - - 1) ) ) - 0 + 1 1 ) cos α + sin α 5) Identity 6) Identity 7) 1-8) 1 + 9) 1-7
Answer Key Testname: 1060 CH AND REVIEW 0) Negative 1) ) ) Not an identity ) Identity 5) Not an identity 6) Identity 7) Not an identity 8) - 161 89 9) 6 65 0) - 80 1 1) 6 ) - 10 10 ) π ) -π 6 5) 5 6) -0 7) 5 8) 5 5 9) 5 6 6 50) 1 x + 1 51) 1 x 5) {x x = - π + kπ, x = - π + kπ} 5) x = π + nπ or x = 5π + nπ 5) {α α = 116.5 + k180 } 55) 0.56,.6180 56) {-0, -10, 150, 0 } 57) x = 1 + 1 arctan y 8
Answer Key Testname: 1060 CH AND REVIEW 58) x = 1 arctan y + 1 59) f -1 (x) = 1 sin-1 x 5 ; domain = [-5, 5]; range = - π 6, π 6 60) 0.6 + kπ or.50 + kπ where k is any integer 61) x x = π 11π + kπ or x = 1 1 + kπ 6) x x = π + 6kπ or x = 15π + 6kπ 6) {150, 10 } 6) {0, 60, 10, 0 } 65) π 1, π 6, π, 7π 1, 7π 6, 1π 1, 5π, 19π 1 66) π, π 67) {α α = 1.1 + k10 or α = 7.9 + k10 } 68) {x x =.5 + kπ or x = 10.0 + kπ} 69) {π} 70) 0, π, π 6, 5π 6 71) 7) π, 5π π, π, 5π, 7π 7) {08., 1.8 } 7) {9.8, 10., 9.8, 10. } 75) 0 sec, π sec, π sec, π sec 9