Understanding Half Angle Identities and Power Reducing Formulas

The video tutorial explains how to determine the exact value of sin 112.5° using a half angle identity. It compares different identities, including power reducing formulas, and applies them to solve the problem. The tutorial uses a reference triangle and unit circle to find cosine 225°, simplifies the expression, and determines the sign of sin 112.5° based on its quadrant. The process is detailed and aims to provide a clear understanding of trigonometric identities and their applications.

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

• To find the exact value of cos(112.5°) using a reference triangle.

• To find the exact value of sin(112.5°) using a half angle identity.

• To determine the cosine of 112.5° using a unit circle.

• To calculate the tangent of 112.5° using trigonometric identities.

Which identity is used to express sin^2(a) in terms of cosine?

• sin^2(a) = 1 + sin(2a)

• sin^2(a) = 1 - cos^2(a)

• sin^2(a) = 1 + cos(2a)

• sin^2(a) = 1 - cos(2a)

How is the cosine of 225° determined in the video?

• By using the sine function.

• By applying the Pythagorean theorem.

• By sketching a reference triangle and using the unit circle.

• Using a calculator.

What is the cosine of 225° as found in the video?

• √2/2

• 1/2

• -√2/2

• -1/2

What is the purpose of using a reference triangle in trigonometry?

• To determine the sine and cosine of angles using known ratios.

• To calculate the tangent of angles.

• To simplify complex trigonometric expressions.

• To find the exact value of angles.

In which quadrant does the angle 112.5° lie?

• Second quadrant

• First quadrant

• Fourth quadrant

• Third quadrant

Why is the sine of 112.5° positive?

• Because it lies in the third quadrant.

• Because it lies in the first quadrant.

• Because it lies in the fourth quadrant.

• Because it lies in the second quadrant where y-coordinates are positive.

What is the final simplified expression for sin(112.5°) as derived in the video?

• √(2 - √2) / 2

• √(2 + √2) / 2

• √(2 - √2) / 4

• √(2 + √2) / 4

What is the significance of the result obtained using different methods?

• The results are the same, showing the reliability of different methods.

• The results are irrelevant to the problem.

• The results are different, indicating errors in calculation.

• The results are inconsistent.

What is the relationship between half angle identities and power reducing formulas?

• They are only applicable to angles greater than 180°.

• They can be used interchangeably in some cases.

• They are completely different and unrelated.

• They are used for different types of trigonometric problems.