Simplify each expression.
±√[(1 - cos (3θ/5))/2]

Verified step by step guidance

1. Recognize that the expression inside the square root, \(\frac{1 - \cos\left(\frac{3\theta}{5}\right)}{2}\), matches the form of the half-angle identity for sine: \(\sin^2\left(\frac{x}{2}\right) = \frac{1 - \cos x}{2}\).
2. Identify \(x\) in the half-angle formula as \(\frac{3\theta}{5}\), so the expression inside the square root can be rewritten as \(\sin^2\left(\frac{3\theta}{10}\right)\).
3. Since the square root of \(\sin^2\) is the absolute value of sine, and the original expression includes a \(\pm\) sign, write the simplified form as \(\pm \sin\left(\frac{3\theta}{10}\right)\).
4. Note that the \(\pm\) sign accounts for the fact that sine can be positive or negative depending on the angle, so the simplified expression is \(\pm \sin\left(\frac{3\theta}{10}\right)\).
5. Thus, the original expression \(\pm \sqrt{\frac{1 - \cos\left(\frac{3\theta}{5}\right)}{2}}\) simplifies to \(\pm \sin\left(\frac{3\theta}{10}\right)\).

Key Concepts

Half-Angle Identities

Half-angle identities express trigonometric functions of half angles in terms of the original angle. For cosine, the identity cos(2α) = 1 - 2sin²(α) can be rearranged to find sin(α) in terms of cos(2α). This is essential for simplifying expressions like √[(1 - cos(θ))/2].

Square Root and ± Sign in Trigonometric Expressions

When taking the square root of a squared trigonometric function, the result includes a ± sign to account for both positive and negative roots. Determining the correct sign depends on the angle's quadrant, which affects the function's sign.

Angle Multiplication and Simplification

Understanding how to manipulate and simplify expressions involving multiples or fractions of angles, such as (3θ/5), is crucial. This includes applying identities correctly and recognizing how angle transformations affect trigonometric values.