Cosine Equations and Identities

The video tutorial explains how to solve the trigonometric equation cosine 2X = cosine X within the interval from 0 to 2π. It begins by identifying the need to use a double angle identity to express the equation in terms of a single angle, X. The equation is then transformed into a quadratic form and factored. The solutions for cosine X are found using the unit circle, identifying the X values that satisfy the equation within the specified interval. The tutorial concludes with a summary of the solutions found.

What is the interval within which we need to solve the equation cosine 2x = cosine x?

• 0 to 2π, closed on both ends

• 0 to 2π, open on both ends

• 0 to 2π, closed on 0 and open on 2π

• 0 to 2π, open on 0 and closed on 2π

Which double angle identity is used to transform the equation cosine 2x = cosine x?

• cosine 2x = 2 sine squared x - 1

• cosine 2x = cosine squared x - sine squared x

• cosine 2x = 2 cosine squared x - 1

• cosine 2x = 1 - 2 sine squared x

What form does the equation take after using the double angle identity?

• Cubic equation

• Quadratic equation

• Exponential equation

• Linear equation

What are the factors of the quadratic equation 2 cosine squared x - cosine x - 1 = 0?

• (2 cosine x - 1)(cosine x + 1)

• (2 cosine x + 1)(cosine x + 1)

• (cosine x + 1)(cosine x - 1)

• (2 cosine x + 1)(cosine x - 1)

What are the possible values of cosine x from the factored equation?

• cosine x = -1/2 and cosine x = 0

• cosine x = 0 and cosine x = 1

• cosine x = -1/2 and cosine x = 1

• cosine x = 1/2 and cosine x = -1

In which quadrants is cosine x negative?

• 1st and 2nd quadrants

• 1st and 4th quadrants

• 2nd and 3rd quadrants

• 3rd and 4th quadrants

At what angles is cosine x equal to -1/2?

• 90 degrees and 270 degrees

• 120 degrees and 240 degrees

• 60 degrees and 300 degrees

• 180 degrees and 360 degrees

At what angle is cosine x equal to 1?

• 270 degrees

• 0 degrees

• 90 degrees

• 180 degrees

What is the significance of the interval being closed on 0 and open on 2π?

• It includes both 0 and 2π

• It excludes both 0 and 2π

• It includes 0 but excludes 2π

• It excludes 0 but includes 2π

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

• x = 2π/3, 4π/3, and 0

• x = π/4, 3π/4, and π

• x = π/3, 5π/3, and π

• x = π/2, 3π/2, and 2π