Understanding Half-Angle Identities

This video tutorial explains how to use the half-angle identity to find the exact value of sine for 22.5 degrees (or π/8 radians). It begins by introducing the half-angle identity and determining the angle 'a' needed for the calculation. The tutorial then applies the formula, simplifies the expression, and verifies the result using a calculator. The process involves understanding the trigonometric identities and simplifying complex fractions to achieve the exact value.

What is the purpose of using a half-angle identity in trigonometry?

• To solve linear equations.

• To approximate the value of a trigonometric function.

• To find the exact value of a trigonometric function for a given angle.

• To convert angles from degrees to radians.

Which of the following is the correct half-angle identity for sine?

• sin(a/2) = ±√(1 + sin(a))/2

• sin(a/2) = ±√(1 - sin(a))/2

• sin(a/2) = ±√(1 - cos(a))/2

• sin(a/2) = ±√(1 + cos(a))/2

If the angle a is 45 degrees, what is the angle a/2?

• 30 degrees

• 60 degrees

• 22.5 degrees

• 15 degrees

In which quadrant is the angle 22.5 degrees located?

• Third quadrant

• Second quadrant

• First quadrant

• Fourth quadrant

What is the cosine of 45 degrees?

• 1

• √3/2

• 1/2

• √2/2

What is the next step after finding the cosine of 45 degrees in the half-angle identity?

• Multiply the expression by 2.

• Simplify the expression by clearing the denominator.

• Find the sine of 45 degrees.

• Convert the angle to radians.

What is the simplified form of the expression for sine 22.5 degrees?

• 1/2 * √(2 + √2)

• √(2 - √2)/2

• √(2 + √2)/2

• 1/2 * √(2 - √2)

What is the square root of 4?

• 0

• 2

• 4

• 1

Why is it important to verify the calculated value using a calculator?

• To ensure the accuracy of the exact value.

• To convert the value to radians.

• To check if the angle is in the correct quadrant.

• To find the cosine of the angle.

What is the decimal approximation of sine 22.5 degrees?

• 0.5

• 0.8660

• 0.3827

• 0.7071