Trigonometric Equations and Solutions

The video tutorial explains how to solve a trigonometric equation over the interval from 0 to 2 pi. It begins with a substitution using a double angle identity for sine, followed by simplification and factoring of the equation. The tutorial then finds solutions for cosine and sine, using a calculator for decimal approximations. Reference angles are used to determine all solutions within the specified interval, resulting in four solutions, two exact and two approximate.

What is the interval over which the equation 3 sin 2x + 4 cos x = 0 is solved?

• π to 2π

• 0 to 2π

• 0 to π

• 0 to 4π

Which identity is used to substitute sin 2x in the equation?

• sin 2x = 2 sin x cos x

• sin 2x = sin^2 x

• sin 2x = 1 - cos^2 x

• sin 2x = 2 cos x sin x

What is the greatest common factor used to factor the equation?

• 2 cos x

• 4 cos x

• 3 sin x

• 6 sin x

What are the solutions for 2 cos x = 0 in the interval 0 to 2π?

• π and 2π

• 0 and π

• π/4 and 3π/4

• π/2 and 3π/2

How is the equation sin x = -2/3 solved?

• Using algebraic manipulation

• Using trigonometric identities

• Using a calculator for decimal approximations

• Using a unit circle

What is the approximate value of the reference angle for sin x = -2/3?

• 0.5 radians

• 0.729 radians

• 2 radians

• 1.5 radians

Which quadrant is used to find the additional solution for sin x = -2/3?

• Third quadrant

• Fourth quadrant

• Second quadrant

• First quadrant

What is the approximate value of the angle found by subtracting the reference angle from 2π?

• 6 radians

• 2.5 radians

• 3.87 radians

• 5.55 radians

What is the approximate value of the angle found by adding the reference angle to π?

• 4 radians

• 2.5 radians

• 5.55 radians

• 3.87 radians

How many solutions are there for the equation over the interval 0 to 2π?

• Two

• Three

• Four

• Five