Simplify each expression.
±√[(1 + cos 20α)/2]

Verified step by step guidance

1. Recognize that the expression \( \pm \sqrt{\frac{1 + \cos 20\alpha}{2}} \) is related to the half-angle identity for cosine.
2. Recall the half-angle identity: \( \cos \theta = \pm \sqrt{\frac{1 + \cos 2\theta}{2}} \).
3. Identify that in the given expression, \( 20\alpha \) is playing the role of \( 2\theta \) in the identity.
4. Set \( 2\theta = 20\alpha \), which implies \( \theta = 10\alpha \).
5. Conclude that the expression simplifies to \( \pm \cos 10\alpha \) using the half-angle identity.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all real numbers and is periodic with a period of 2π. Understanding the properties of the cosine function is essential for simplifying expressions involving cosine, such as the one in the question.

Half-Angle Identity

The half-angle identities are formulas that express trigonometric functions of half an angle in terms of the functions of the original angle. For cosine, the half-angle identity states that cos(θ/2) = ±√[(1 + cos θ)/2]. This identity is particularly useful for simplifying expressions like ±√[(1 + cos 20α)/2] by recognizing that it represents cos(10α).

Square Root and Absolute Value

The square root function returns the principal (non-negative) root of a number, while the ± symbol indicates that both the positive and negative roots are considered. In trigonometric simplifications, understanding how to handle square roots and the implications of the ± sign is crucial for accurately expressing the results, especially when dealing with angles and their trigonometric values.