Problem 87 - Algebra and Trigonometry

Question

Use words to describe the formula for: the sine of half an angle.

Step-by-Step Solution

Answer

The sine of half an angle formula is \( sin\left(\frac{a}{2}\right) = \frac{sin(a)}{2\sqrt{ 1 - sin^2\left(\frac{a}{2}\right)}} \)

Step 1: Start with Double Angle Identity

Start with the double angle identity which is given by: \[ sin(2x) = 2 sin(x) cos(x) \]

Step 2: Isolate sin(x)

Rearrange the equation to make \( sin(x) \) the subject by dividing both sides by \( 2cos(x) \): \[ sin(x) = \frac{sin(2x)}{2 cos(x)} \]

Step 3: Apply the Pythagorean Identity

Recall that the Pythagorean identity is given as: \[ cos^2(x) + sin^2(x) = 1 \]. Transform it into \( cos(x) = \sqrt{1 - sin^2(x)} \). Now replace \( cos(x) \) in the equation from Step 2 with \( \sqrt{1 - sin^2(x)}: \[ sin(x) = \frac{sin(2x)}{2\sqrt{1 - sin^2(x)}} \]

Step 4: Apply Half Angle Formula

Now let \( x = \frac{a}{2} \) so that the 2x term can become a by replacement. The sine half angle formula becomes: \[ sin\left(\frac{a}{2}\right) = \frac{sin(a)}{2 \sqrt{ 1 - sin^2\left(\frac{a}{2}\right)}} \]