Trigonometric Functions and Equations

The video tutorial explains how to solve the trigonometric equation 2 cosine squared x - sine x = 1 within the interval from 0 to 2π. The instructor begins by analyzing the equation, noting the presence of two trigonometric functions and the lack of common factors. A substitution is performed to express the equation in terms of sine only, transforming it into a quadratic form. The equation is then factored, and solutions are found by setting each factor to zero. Finally, the unit circle is used to determine the angles corresponding to the sine values, resulting in three solutions within the specified interval.

What is the interval within which we need to solve the equation 2 cosine squared x - sine x = 1?

• 0 to 2π, open on both ends

• 0 to 2π, open on 0 and closed on 2π

• 0 to 2π, closed on both ends

• 0 to 2π, closed on 0 and open on 2π

What substitution is used to express cosine squared x in terms of sine x?

• cosine squared x = 1 + sine squared x

• cosine squared x = 1 - sine x

• cosine squared x = 1 - sine squared x

• cosine squared x = sine x

After substitution, what form does the equation take before factoring?

• Exponential form

• Cubic form

• Quadratic form

• Linear form

What is the first step in factoring the quadratic equation obtained?

• Factor out a positive coefficient

• Multiply both sides by 2

• Add 1 to both sides

• Factor out a negative coefficient

What are the possible values of sine x after solving the factored equation?

• 1/2 and -1

• 1 and -1/2

• 1/2 and 1

• 0 and -1

Which angle corresponds to sine x = 1/2 in the first quadrant?

• π/3

• π/4

• π/6

• π/2

Which angle corresponds to sine x = 1/2 in the second quadrant?

• 7π/6

• 2π/3

• 5π/6

• 3π/4

Which angle corresponds to sine x = -1?

• 3π/2

• π

• 2π

• π/2

How many solutions are there for the equation in the given interval?

• One

• Two

• Three

• Four

What is the significance of the unit circle in solving this equation?

• It helps find cosine values

• It helps find sine values

• It helps find secant values

• It helps find tangent values