Simplify each expression.
√[(1 + cos 76°)/2]

Verified step by step guidance

1. Recognize that the expression \( \sqrt{\frac{1 + \cos 76^\circ}{2}} \) is related to a trigonometric identity.
2. Recall the half-angle identity for cosine: \( \cos \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 + \cos \theta}{2}} \).
3. Identify that \( \theta = 76^\circ \) in the given expression, so we are looking for \( \cos \left( \frac{76^\circ}{2} \right) \).
4. Calculate \( \frac{76^\circ}{2} \) to find the angle for which we need the cosine value.
5. Express the simplified form as \( \cos \left( 38^\circ \right) \).

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. In the context of the expression given, cos 76° represents the cosine of 76 degrees, which is a specific value that can be calculated or approximated using a calculator or trigonometric tables.

Half-Angle Identity

The half-angle identities are formulas that express trigonometric functions of half angles in terms of the functions of the original angle. For example, the identity for cosine states that cos(θ/2) = √[(1 + cos θ)/2]. This identity is particularly useful for simplifying expressions involving angles that are halved, such as the expression in the question.

Square Root Function

The square root function is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In the expression √[(1 + cos 76°)/2], the square root is applied to the result of the fraction, which simplifies the expression further. Understanding how to manipulate square roots is essential for simplifying trigonometric expressions effectively.