Trigonometry
Fundamental Identities
Sum, Difference, & Double Angle Identities
Law of Sines
Law of Cosines
100
What is the value of cos(5π/6)?
cos(5π/6) = -√3/2
100
1 - cos^2(x) = ? state the identity
sin^2(x) by 1st Pythagorean Identity
100
Condense: sin(43)cos(22) - sin(22)cos(43) State the identity used.
sin(43 - 22) = sin(21) Difference of Sine Identity
100
Solve the triangle if A = 40 degrees, B = 30, and b = 10.
C = 110 a = 10sin(40)/sin(30) c = 10isn(110)/sin(30)
100
Solve the triangle if A = 55 degrees, b = 12, c = 7
a = √144 + 49 - 2(12)(7)cos(55) = 9.8 B = sin^-1[12sin(55)/9.8] = 89.3 C = sin^-1[7sin(55)/9.8] = 35.7
200
Solve for both values of x: tan(x) = 1
x= π/4 and 5π/4
200
sin(-x)sin(π/2 - x) = -sin(x)cos(x) Which two identities allow you to go from the left to the right and side?
Odd-Even Identity & Cofunction Identity
200
Prove sin(2u) = 2sin(u)cos(u) state the identity used.
sin(2u) = sin(u + u) = sin(u)cos(u) +sin(u)cos(u) = 2sin(u)cos(u) Sum of Sine Identity
200
Solve the triangle if B = 70, b = 14, and c = 9
C= sin^-1[9sin(70)/14] = 37.2 A = 180 - 70 - 37.2 = 72.8 a = 14sin(72.8)/sin(70) = 14.2
200
Find the area of the triangle if A = 47 degrees, b = 32ft, and c = 19ft
Area = (1/2)(32)(19)sin(47) = 222.33 ft^2
300
Find cos(x) and sec(x) if tan(x) = √3
tan(x) = √3 when x = π/3 cos(π/3) = 1/2 sec(π/3) = 1/cos(π/3) = 2
300
Find sec^2(x) if tan^2(x) = 6 and cos x > 0
1 + tan^2(x) = 1 + 6^2 = 37 = sec^2(x) sec x = √37
300
Find the exact value of sin(15) state the identity used.
sin(15) = sin(45 - 30) = sin(45)cos(30) - sin(30)cos(45) = (√2/2)(√3/2) - (1/2)(√2/2) = (√6 + √2)/4 Difference of Sine Identity
300
If you have to find the altitude(height) of an isosceles triangle and are given the base of the triangle in cm and one of the angles in the isosceles triangle, which of the following could you use to solve the problem: a) Pythagorean Theorem and Law of Sines b) Law of Sines, Law of Cosines, and Pythagorean Theorem c) Only the Law of Sines d) Not enough information
a)Pythagorean Theorem & Law Of Sines
300
In which of the following cases would you use Law of Cosines to solve the triangle: a) AAS b) SSS c) SAS d) ASA e) you cannot use the Law of Cosines for any of these cases
b) SSS c) SAS
400
Find cos(x), tan(x), and cot(x) if sin(x) = -1
sin(x) = -1 when x = 3π/2 cos(3π/2) = 0 tan(3π/2) = undefined cot(3π/2) = 0
400
Prove: sin^3(x) + sin(x)cos^2(x) = sin(x) State the Identity used.
sin(x)[sin^2(x) + cos^2(x)] factor out sin (x) = sin(x)[1] 1st Pythagorean Identity = sin(x)
400
Find the exact value of sin(5π/12) state the identity used
sin(5π/12) = sin((5π/6)(1/2)) = √[1 - cos(5π/6)]/2 = √[1 - (-√3/2)]/2 = √[1 + √3/2]/2 Half-Angle Sine Identity
400
In which of the following cases would you use Law of Sines to solve the triangle: a) AAS b) SSS c) SAS d) ASA e) you cannot use the Law of Sines for any of these cases
a) AAS d) ASA
400
You swam across the lake in P.E. instead of running around it. Your friend measures an angle of 70 degrees when he is standing 654 ft away from where you dove into the lake and 156 ft away from where you got out of the lake on the opposite side. How far did you swim, assuming you swam in a straight line from A to B?
distance = √ (654)^2 + (156)^2 - (2)(654)(156)cos(70) = 618.275 feet
500
Fill out the entire first quadrant of the unit circle.
π/6 π/4 π/3 cos(x) √3/2 √2/2 ½ sin(x) ½ √2/2 √3/2 tan(x) √3/3 1 √3
500
Solve 2sin^2(x) + sin(x) = 1 algebraically in [0,2π)
2sin^2(x) + sin(x) = 1 2sin^2(x) + sin(x) - 1 = 0 (2sinx - 1)(sinx + 1) = 0 sin(x) = 1/2 and sin(x) = -1 x = π/6, 5π/6, and 3π/2
500
Solve: sin(2x) = cos(x) algebraically in [0,2π)
sin(2x) = cos(x) 2sin(x)cos(x) = cos(x) 2sin(x)cos(x) - cos(x) = 0 cos(x)[2sin(x) - 1] = 0 cos(x) = 0 and sin(x) = 1/2 x = π/6, π/2, 5π/6, and 3π/2
500
Two observers are 600ft apart on opposite sides of a flagpole. The angles of elevation from the observers to the top of the flagpole are 19degrees and 21 degrees. Find the height of the flagpole.
side opp 19degreess = 600sin(19)/sin(140) = 303.896 flagpole height = 303.896sin(21) = 108.9 feet
500
Two airplanes flying together take off in different directions. One flies due east at 300mph and the other flies east-northeast at 340mph. How far apart are the two airplanes 2 hours after they have separated?
distance apart = √(600)^2 + (680)^2 - 2(600)(680)cos(45) = 495.379 miles
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