Trigonometric Equations and Identities

The video tutorial explains how to solve the trigonometric equation cosine 7X - cosine 3X = sine 5X within a specified interval. It begins by rewriting the equation using a sum-to-product identity, then factors the equation to find solutions. The tutorial demonstrates solving for sine 5X = 0 and sine 2X = -1/2 using the unit circle, and identifies solutions within the interval. The video concludes with a summary of the solutions found.

What is the initial equation that needs to be solved in the given interval?

• sine 7x + sine 3x = cosine 5x

• sine 7x - sine 3x = cosine 5x

• cosine 7x - cosine 3x = sine 5x

• cosine 7x + cosine 3x = sine 5x

Which identity is used to transform the left side of the equation?

• Pythagorean Identity

• Sum-to-Product Identity

• Angle Addition Identity

• Product-to-Sum Identity

Why should we avoid dividing by sine 5x when solving the equation?

• It makes the equation more complex

• It results in losing some solutions

• It simplifies the equation too much

• It introduces imaginary numbers

What substitution is made to solve the first equation?

• 5x = gamma

• 5x = theta

• 5x = beta

• 5x = alpha

What is the value of sine theta when theta equals 0 radians?

• 1

• Undefined

• 0

• -1

How is the solution for x derived from theta in the first equation?

• X = theta multiplied by 5

• X = theta minus 5

• X = theta plus 5

• X = theta divided by 5

What substitution is made to solve the second equation?

• 2x = beta

• 2x = theta

• 2x = gamma

• 2x = alpha

In which quadrants is sine theta equal to -1/2?

• 1st and 2nd

• 1st and 4th

• 3rd and 4th

• 2nd and 3rd

What is the value of x when theta equals 7π/6?

• 19π/12

• 23π/12

• 11π/12

• 7π/12

Why is the angle 23π/6 ignored in the final solution?

• It is greater than π

• It is not a multiple of π

• It is equal to π

• It is less than 0