Understanding Trigonometric Expressions

The video tutorial explains how to find the exact value of the expression involving cosine of two times the inverse tangent of 3x. It begins by defining the problem and setting the context for the angle theta. The tutorial then sketches a reference triangle to determine the hypotenuse using the Pythagorean theorem. Next, it evaluates cosine of two theta using a double angle identity and simplifies the expression. The tutorial concludes by confirming the result using different identities.

What is the main goal of the problem discussed in the video?

• To solve a quadratic equation

• To calculate the area of a triangle

• To determine the exact value of a trigonometric expression

• To find the value of x

What is the range of the angle theta in the context of inverse tangent?

• -90 to 90 degrees

• 0 to 180 degrees

• -180 to 180 degrees

• 0 to 360 degrees

In which quadrant is the angle theta located when x is greater than zero?

• Second quadrant

• Third quadrant

• First quadrant

• Fourth quadrant

What is the length of the hypotenuse in the reference triangle?

• 3x

• 9x^2

• Square root of (1 + 9x^2)

• 1

Which identity is used to evaluate cosine of two theta?

• Half angle identity

• Pythagorean identity

• Sum of angles identity

• Double angle identity

What is the expression for cosine squared theta in terms of the reference triangle?

• 1 divided by 3x

• 1 divided by (1 + 9x^2)

• 3x divided by (1 + 9x^2)

• 9x^2 divided by (1 + 9x^2)

What is the expression for sine squared theta in terms of the reference triangle?

• 1 divided by (1 + 9x^2)

• 1 divided by 3x

• 9x^2 divided by (1 + 9x^2)

• 3x divided by (1 + 9x^2)

What is the final expression for cosine of two theta?

• (1 - 9x^2) / (1 + 9x^2)

• (1 + 9x^2) / (1 - 9x^2)

• (9x^2 - 1) / (1 + 9x^2)

• (1 + 9x^2) / (9x^2 - 1)

Why does the expression not simplify further?

• The numerator is a sum of squares

• Both numerator and denominator are sums of squares

• The denominator is a difference of squares

• The numerator is a difference of squares and the denominator is a sum of squares

What is the significance of using different double angle identities for cosine?

• They are used for different angles

• They give different results

• They simplify the expression differently

• They all yield the same result