Answers to Trig Frog's Quiz
Below, thou wilt find the answers and explanations to Trig Frog's quiz. If thou has missed a problem, be sure that thou understands the explanation. If thou truly comprehends the material, thou shouldst score at least a 75% on Trig Frog's quiz.
1. Answer: (£6 + £2)/4
sin75° = sin(45° + 30°)
sin(A + B) = sinAcosB + cosAsinB Sum and Difference Formulae.
sin(45° + 30°) = sin45°cos30° + cos45°sin30°
= (£2/2)(£3/2) + (£2/2)(1/2) Obtain these values from the Unit Circle.
= £6/4 + £2/4 Perform the operations.
= (£6 + £2)/4 Simplify.
2. Answer: -£2/2
cos135° = cos(90° + 45°)
cos(A + B) = cosAcosB - sinAsinB Sum and Difference Formulae.
cos(90° + 45°) = cos90°cos45° - sin90°sin45°
= (0)(£2/2) - (1)(£2/2) Obtain these values from the Unit Circle.
= 0 - £2/2 Perform the operations.
= -£2/2 Simplify.
3. Answer: 2 - £3
tan195° = tan(225° - 30°)
tan(A - B) = (tanA - tanB)/(1 + tanAtanB) Sum and Difference Formulae.
tan(225° - 30°) = (tan225° - tan30°)/(1 + tan225°tan30°)
= (1 - £3/3)/(1 + (1)(£3/3)) Obtain these values from the Unit Circle.
= (1 - £3/3)(1 - £3/3)/(1 + £3/3)(1 - £3/3) Multiply by the conjugate.
= (1 - 2£3/3 + 1/3)/(1 - 1/3) Perform the operations.
= (4/3 - 2£3/3)/(2/3) Perform the operations.
= 2 - £3/3 Simplify.
4. Answer: 0
cos2A, A = pi/4
cos2A = cos^2 A - sin^2 A Half Angle Formulae
cos2(pi/4) = cos^2 pi/4 - sin^2 pi/4
= (£2/2)^2 - (£2/2)^2 Obtain these values from the Unit Circle.
= 0 Perform the operations.
5. Answer: -£3
tan2A, tan = 60°
tan2A = (2tanA)/(1 - tan^2 A) Half Angle Formulae.
tan2(60°) = (2tan60°)/(1 - tan^2 60°)
= 2(£3)/(1 - (£3)^2) Obtain these values from the Unit Circle.
= 2£3/(1 - 3) = 2£3/-2 = -£3 Perform the operations and simplify.
