Problem 88 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The cosine of half an angle is found by adding the cosine of the angle itself to one, dividing the result by two, and then taking the square root of the result. The value can be either positive or negative depending on the quadrant of the half-angle.

Step 1: Cosine Half-Angle Identity

In trigonometry, the cosine of half an angle can be found by using the half-angle identity. The cosine half-angle identity is an equation that expresses \(\cos(\frac{θ}{2})\) (the cosine of half an angle) in terms of \(\cos(θ)\) (the cosine of the angle itself).

Step 2: Half-Angle Identity for Cosine Formula

The cosine half-angle identity is given by the formula \(\cos(\frac{θ}{2}) = ±\sqrt{\frac{1 + \cos(θ)}{2}}\)

Step 3: Description of the Formula in words

To find the cosine of half an angle, you add the cosine of the angle itself to one, divide the result by two, and then take the square root of the result. The value can be either positive or negative depending on the quadrant in which the half-angle lies.