Understanding Half-Angle Identities and Cosine Calculations

This video tutorial explains how to use the half-angle identity to find the exact value of cosine for pi/12 radians or 15 degrees. It covers determining the angle, applying the formula, simplifying the expression, and verifying the result using a calculator. The tutorial emphasizes understanding the sign of the cosine function based on the angle's quadrant and provides a step-by-step approach to solving the problem.

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

• To find the exact value of trigonometric functions for specific angles.

• To convert angles from degrees to radians.

• To solve quadratic equations.

• To determine the area of a triangle.

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

• cos(a/2) = ±√(1 - sin(a))/2

• cos(a/2) = ±√(1 + sin(a))/2

• cos(a/2) = ±√(1 - cos(a))/2

• cos(a/2) = ±√(1 + cos(a))/2

In which quadrant is the angle 15 degrees located, and what is the sign of cosine in that quadrant?

• Quadrant I, positive

• Quadrant II, negative

• Quadrant III, negative

• Quadrant IV, positive

What is the value of cosine 30 degrees?

• √3/2

• 1/2

• 1

• 0

How do you simplify the complex fraction under the square root when calculating cosine 15 degrees?

• Add 1 to both the numerator and denominator.

• Subtract 1 from both the numerator and denominator.

• Divide the numerator and denominator by 2.

• Multiply the numerator and denominator by 2.

What is the simplified form of the exact value of cosine 15 degrees?

• 1/2 * √(2 + √3)

• 1/2 * √(2 - √3)

• √(2 + √3)/4

• √(2 - √3)/4

What is the purpose of verifying the exact value of cosine 15 degrees using a calculator?

• To ensure the calculated value matches the decimal approximation.

• To convert the value from radians to degrees.

• To check the value of cosine 30 degrees.

• To find the sine of 15 degrees.

Which mode should the calculator be in to verify the cosine of 15 degrees?

• Scientific mode

• Graphing mode

• Radian mode

• Degree mode

What is the decimal approximation used for in the verification process?

• To find the sine of the angle.

• To compare with the exact value.

• To convert the angle to radians.

• To determine the tangent of the angle.

What is the final step in verifying the exact value of cosine 15 degrees?

• Comparing the decimal approximation with the exact value.

• Finding the tangent of 15 degrees.

• Converting the angle to radians.

• Calculating the sine of 15 degrees.