Evaluate each expression without using a calculator.
sin (2 cos⁻¹ (1/5))

Verified step by step guidance

1. Recognize that the expression involves the sine of twice an inverse cosine value: \(\sin\left(2 \cos^{-1}\left(\frac{1}{5}\right)\right)\). This suggests using the double-angle identity for sine.
2. Recall the double-angle identity for sine: \(\sin(2\theta) = 2 \sin\theta \cos\theta\). Here, let \(\theta = \cos^{-1}\left(\frac{1}{5}\right)\).
3. Since \(\theta = \cos^{-1}\left(\frac{1}{5}\right)\), we know \(\cos\theta = \frac{1}{5}\). To find \(\sin\theta\), use the Pythagorean identity: \(\sin\theta = \sqrt{1 - \cos^2\theta} = \sqrt{1 - \left(\frac{1}{5}\right)^2}\).
4. Substitute \(\sin\theta\) and \(\cos\theta\) into the double-angle formula: \(\sin(2\theta) = 2 \times \sin\theta \times \cos\theta = 2 \times \sqrt{1 - \left(\frac{1}{5}\right)^2} \times \frac{1}{5}\).
5. Simplify the expression inside the square root and multiply through to express \(\sin(2 \cos^{-1}(\frac{1}{5}))\) in simplest radical form without using a calculator.

Key Concepts

Inverse Cosine Function (cos⁻¹ or arccos)

The inverse cosine function returns the angle whose cosine is a given value. For example, cos⁻¹(1/5) gives an angle θ such that cos(θ) = 1/5. Understanding this helps convert the expression into an angle measure for further trigonometric manipulation.

Double-Angle Identity for Sine

The double-angle identity states that sin(2θ) = 2 sin(θ) cos(θ). This formula allows us to rewrite sin(2 cos⁻¹(1/5)) as 2 sin(θ) cos(θ), where θ = cos⁻¹(1/5), facilitating evaluation without a calculator.

Right Triangle Interpretation for Trigonometric Values

By interpreting θ = cos⁻¹(1/5) as an angle in a right triangle with adjacent side 1 and hypotenuse 5, we can find sin(θ) using the Pythagorean theorem. This geometric approach helps compute sine and cosine values exactly without a calculator.