Question

use identities to find values of the sine and cosine functions for the angle measure.
θ, given that \\( \cos 2 \theta = \frac { 40 } { 41 } \\) and \\( 0 ^ { \circ } < \theta < 90 ^ { \circ } \\)
\\( \sin \theta = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Use the double - angle identity for cosine

The double - angle identity for cosine is \(\cos2\theta = 1 - 2\sin^{2}\theta\).
We are given that \(\cos2\theta=\frac{40}{41}\), so we substitute this into the identity:
\(\frac{40}{41}=1 - 2\sin^{2}\theta\)

Step2: Solve for \(\sin^{2}\theta\)

First, rearrange the equation \(\frac{40}{41}=1 - 2\sin^{2}\theta\) to isolate \(\sin^{2}\theta\).
Subtract \(1\) from both sides: \(\frac{40}{41}-1=- 2\sin^{2}\theta\).
Since \(1=\frac{41}{41}\), then \(\frac{40 - 41}{41}=-2\sin^{2}\theta\), which simplifies to \(-\frac{1}{41}=-2\sin^{2}\theta\).
Divide both sides by \(- 2\): \(\sin^{2}\theta=\frac{1}{82}\).

Step3: Solve for \(\sin\theta\)

Take the square root of both sides. Since \(0^{\circ}<\theta<90^{\circ}\), \(\sin\theta>0\).
So \(\sin\theta=\sqrt{\frac{1}{82}}=\frac{1}{\sqrt{82}}\).
Rationalize the denominator: \(\sin\theta=\frac{1\times\sqrt{82}}{\sqrt{82}\times\sqrt{82}}=\frac{\sqrt{82}}{82}\).

Answer:

\(\frac{\sqrt{82}}{82}\)