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

Guida verificata passo dopo passo

Recognize that the expression inside the square root, \(\frac{1 + \cos 20\alpha}{2}\), matches the form of the cosine double-angle identity rearranged for cosine squared: \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\).

Identify the angle \(\theta\) such that \(2\theta = 20\alpha\), which gives \(\theta = 10\alpha\).

Rewrite the expression inside the square root as \(\cos^2 10\alpha\) using the identity: \(\frac{1 + \cos 20\alpha}{2} = \cos^2 10\alpha\).

Take the square root of \(\cos^2 10\alpha\), which results in \(|\cos 10\alpha|\) because the square root of a square is the absolute value of the original expression.

Include the \(\pm\) sign given in the problem, so the simplified expression is \(\pm \cos 10\alpha\), noting that the absolute value and \(\pm\) together cover all possible signs.

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 Identity for Cosine

The half-angle identity expresses the cosine of an angle in terms of the cosine of half that angle. Specifically, cos²(θ) = (1 + cos 2θ)/2, which can be rearranged to find cos θ or sin θ using square roots. This identity is essential for simplifying expressions involving (1 + cos 2α)/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 or context, which is crucial for accurate simplification.

Video consigliato:

Percorso guidato

2:20

Imaginary Roots with the Square Root Property

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves applying identities and algebraic manipulation to rewrite expressions in simpler or more recognizable forms. Recognizing patterns like half-angle formulas helps reduce complex expressions to basic trigonometric functions.

Video consigliato:

Percorso guidato