Trigonometric Concepts and Theorems

The video tutorial explains how to find the exact value of an expression involving sine and inverse tangent functions. It begins by setting up the problem and defining the angle theta using inverse tangent. The tutorial then models the angle on a coordinate plane, sketches a reference triangle, and calculates the hypotenuse using the Pythagorean theorem. Finally, it evaluates sine of the double angle using trigonometric identities, resulting in a simplified expression.

What is the initial expression given in terms of x?

• Sine of inverse tangent of x

• Tangent of two times inverse sine of x

• Sine of two times inverse tangent of x

• Cosine of inverse tangent of x

In which interval must the angle theta lie?

• 0 to 360 degrees

• 0 to 180 degrees

• -90 to 90 degrees

• -180 to 180 degrees

What quadrant is angle theta located in if x is greater than zero?

• Third quadrant

• Second quadrant

• Fourth quadrant

• First quadrant

What is the opposite side of the reference triangle labeled as?

• 2

• x

• x/2

• 2x

Which theorem is used to determine the hypotenuse of the triangle?

• Sine Rule

• Cosine Rule

• Pythagorean Theorem

• Tangent Rule

What is the expression for the hypotenuse in terms of x?

• Square root of (x^2 + 4)

• Square root of (x^2 - 4)

• Square root of (4 - x^2)

• Square root of (4 + x^2)

What identity is used to evaluate sine of two theta?

• Pythagorean identity

• Tangent double angle identity

• Cosine double angle identity

• Sine double angle identity

What is the expression for sine theta using the reference triangle?

• 2 divided by the square root of (4 + x^2)

• 2 divided by x

• x divided by the square root of (4 + x^2)

• x divided by 2

What is the final simplified expression for the given problem?

• 4x divided by (x^2 + 4)

• 4x divided by (4 - x^2)

• 4x divided by (4 + x^2)

• x divided by (4 + x^2)

Why can't the fours be simplified in the final expression?

• Because they are part of a product

• Because they are part of a sum

• Because they are part of a difference

• Because they are part of a quotient