Question

Could the Double-Angle formula be used to find sin 22.5°? Why or why not?

Give an example problem that uses either the double or half angle formula.
Write down the Pythagorean Identity sin^2x + cos^2x = 1. Divide each side of the equation by sin^2x and describe the simplifications that can be done.

Answer 1

The Double-Angle formula can indeed be used to find sin 22.5° by applying the half-angle formula. An example is provided to demonstrate the use of the formula. Additionally, when dividing the Pythagorean Identity by sin²x, it simplifies to 1 + cot²x = csc²x.

Step-by-step explanation:

Yes, the Double-Angle formula can be used to find sin 22.5°. This is because 22.5° is half of 45°, and by using the half-angle formula for sine, we can find the value. For instance, sin(45°/2) = ±√((1 - cos 45°)/2). Since 22.5° is in the first quadrant, where sine is positive, we only consider the positive root.

Example problem using the half-angle formula:

Find sin 22.5° using the half-angle formula.

Solution:

sin(22.5°) = sin(45°/2) = √((1 - cos(45°))/2) = √((1 - √2/2)/2)

When we divide the Pythagorean Identity sin²x + cos²x = 1 by sin²x, we get the following simplifications:

1 + (cos²x/sin²x) = 1/sin²x

1 + cot²x = csc²x