Trigonometric Equations and Identities

This video tutorial from Moomoomath covers part 3 of a series on double angles, focusing on solving equations using double angle formulas for sine and cosine. The instructor explains how to substitute and factor equations to find solutions, using examples to illustrate the process. The video also touches on the use of different forms of the cosine double angle formula and provides guidance on solving these equations within the interval 0 to 2π.

What is the double angle formula for sine?

• sin(2θ) = 1 - 2sin²θ

• sin(2θ) = 2sinθcosθ

• sin(2θ) = sin²θ - cos²θ

• sin(2θ) = 2cos²θ - 1

What is the first step in solving the equation sin(2x) + sin(x) = 0?

• Add sin(x) to both sides

• Use the identity sin(2x) = 2sin(x)cos(x)

• Divide both sides by sin(x)

• Subtract sin(x) from both sides

After factoring out sin(x), what is the resulting equation?

• 2cos(x) + 1 = 0

• 2sin(x)cos(x) = 0

• sin(x)(2cos(x) + 1) = 0

• sin(x) + cos(x) = 0

Where is sin(x) equal to zero on the unit circle?

• π/4 and 3π/4

• 0 and π

• π/3 and 2π/3

• π/2 and 3π/2

What is the cosine double angle formula that involves only cosine?

• cos(2θ) = cos²θ - sin²θ

• cos(2θ) = 2cos²θ - 1

• cos(2θ) = 1 - 2sin²θ

• cos(2θ) = 2sinθcosθ

What is the first step in solving the equation cos(2x) = cos(x) + 2?

• Use the identity cos(2x) = 2cos²(x) - 1

• Divide both sides by 2

• Subtract cos(x) from both sides

• Add 1 to both sides

What is the result of factoring the equation 2cos²(x) - cos(x) - 3 = 0?

• (2cos(x) - 3)(cos(x) + 1) = 0

• (cos(x) - 3)(2cos(x) + 1) = 0

• (2cos(x) + 3)(cos(x) - 1) = 0

• (cos(x) + 3)(2cos(x) - 1) = 0

What is the solution for cos(x) = -1 on the interval 0 to 2π?

• 0

• π

• π/2

• 2π

Which of the following is not a valid solution for the equation 2cos(x) = 3?

• No solution

• x = 2π/3

• x = π/3

• x = 3π/2

What is the range of values for which the inverse cosine function is defined?

• 0 to 2π

• 0 to π

• -π/2 to π/2

• -1 to 1