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

Guida verificata passo dopo passo

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: \(\sin^2\left(\frac{x}{2}\right) = \frac{1 - \cos x}{2}\).

Identify \(x\) in the half-angle formula as \(\frac{3\theta}{5}\), so the expression inside the square root can be rewritten as \(\sin^2\left(\frac{3\theta}{10}\right)\).

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 \(\pm \sin\left(\frac{3\theta}{10}\right)\).

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 \(\pm \sin\left(\frac{3\theta}{10}\right)\).

Thus, the original expression \(\pm \sqrt{\frac{1 - \cos\left(\frac{3\theta}{5}\right)}{2}}\) simplifies to \(\pm \sin\left(\frac{3\theta}{10}\right)\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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].

Video consigliato:

Percorso guidato

05:06

Double Angle Identities

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.

Video consigliato:

Percorso guidato

2:20

Imaginary Roots with the Square Root Property

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.

Video consigliato:

Percorso guidato