Trigonometric Functions and Identities

The video tutorial explains how to find the exact value of the expression sine of two times the inverse tangent of five twelfths. It begins by defining the inverse tangent and its relation to angle theta. The tutorial then constructs a reference triangle to determine the hypotenuse using the Pythagorean theorem. Finally, it evaluates sine 2 theta using the double angle identity, resulting in the exact value of 120/169.

What is the range of the angle theta when using the inverse tangent function?

• -180 to 180 degrees

• 0 to 360 degrees

• -90 to 90 degrees

• 0 to 180 degrees

In which quadrant is the angle theta located if the tangent value is positive?

• Second quadrant

• Fourth quadrant

• First quadrant

• Third quadrant

What are the lengths of the opposite and adjacent sides in the reference triangle for theta?

• 5 and 13

• 5 and 12

• 12 and 13

• 12 and 5

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

• 12

• 10

• 13

• 5

Which theorem is used to verify the length of the hypotenuse?

• Cosine Rule

• Pythagorean Theorem

• Sine Rule

• Tangent Rule

What is the double angle identity for sine?

• sin(2a) = 2cos(a)sin(a)

• sin(2a) = sin^2(a) - cos^2(a)

• sin(2a) = sin(a) + cos(a)

• sin(2a) = 2sin(a)cos(a)

What is the sine of theta in the reference triangle?

• 12/13

• 5/12

• 13/5

• 5/13

What is the cosine of theta in the reference triangle?

• 5/13

• 13/5

• 5/12

• 12/13

What is the exact value of the original expression?

• 12/13

• 120/169

• 13/5

• 5/12

Does the fraction 120/169 simplify further?

• Yes, it simplifies to 5/12

• No, it does not simplify

• Yes, it simplifies to 12/13

• Yes, it simplifies to 13/5