Questions: If sin(x) = 1/8 and x is in quadrant I, find exact values for sin(2x) and cos(2x), without finding x.

Transcript text: If $\sin (x)=\frac{1}{8}$ and $x$ is in quadrant I, find exact values for $\sin (2 x)$ and $\cos (2 x)$, without finding $x$.

Solution

Solution Steps

Solution Approach

To find $\sin(2x)$ and $\cos(2x)$ given $\sin(x) = \frac{1}{8}$ and $x$ is in quadrant I, we can use the double angle formulas:

$\sin(2x) = 2 \sin(x) \cos(x)$.
$\cos(2x) = \cos^2(x) - \sin^2(x)$.

First, we need to find $\cos(x)$. Since $x$ is in quadrant I, $\cos(x)$ is positive. We use the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to find $\cos(x)$.

Step 1: Calculate $\cos(x)$

Given that $\sin(x) = \frac{1}{8}$, we can use the Pythagorean identity:

\[ \sin^2(x) + \cos^2(x) = 1 \]

Substituting $\sin(x)$:

\[ \left(\frac{1}{8}\right)^2 + \cos^2(x) = 1 \]

This simplifies to:

\[ \frac{1}{64} + \cos^2(x) = 1 \]

Rearranging gives:

\[ \cos^2(x) = 1 - \frac{1}{64} = \frac{64}{64} - \frac{1}{64} = \frac{63}{64} \]

Taking the positive square root (since $x$ is in quadrant I):

\[ \cos(x) = \sqrt{\frac{63}{64}} = \frac{\sqrt{63}}{8} \approx 0.9922 \]

Step 2: Calculate $\sin(2x)$

Using the double angle formula for sine:

\[ \sin(2x) = 2 \sin(x) \cos(x) \]

Substituting the known values:

\[ \sin(2x) = 2 \cdot \frac{1}{8} \cdot \frac{\sqrt{63}}{8} = \frac{2\sqrt{63}}{64} = \frac{\sqrt{63}}{32} \approx 0.2480 \]

Step 3: Calculate $\cos(2x)$

Using the double angle formula for cosine:

\[ \cos(2x) = \cos^2(x) - \sin^2(x) \]

Substituting the known values:

\[ \cos(2x) = \left(\frac{\sqrt{63}}{8}\right)^2 - \left(\frac{1}{8}\right)^2 = \frac{63}{64} - \frac{1}{64} = \frac{62}{64} = \frac{31}{32} \approx 0.9688 \]