Questions: If sin x = 3/8, x in quadrant I, then find (without finding x) sin(2x)= cos(2x)= tan(2x)=

Transcript text: If $\sin x=\frac{3}{8}, x$ in quadrant I , then find (without finding $x$ ) \[ \begin{array}{l} \sin (2 x)= \\ \cos (2 x)= \\ \tan (2 x)= \end{array} \]

Solution

Solution Steps

Step 1: Calculate $ \cos x $

Using the Pythagorean identity $ \sin^2 x + \cos^2 x = 1 $, we can find $ \cos x $:

\[ \cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left(\frac{3}{8}\right)^2} = \sqrt{1 - \frac{9}{64}} = \sqrt{\frac{55}{64}} = \frac{\sqrt{55}}{8} \approx 0.9270 \]

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

Using the double angle formula for sine, $ \sin(2x) = 2 \sin x \cos x $:

\[ \sin(2x) = 2 \cdot \frac{3}{8} \cdot \frac{\sqrt{55}}{8} = \frac{3\sqrt{55}}{32} \approx 0.6953 \]

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

Using the double angle formula for cosine, $ \cos(2x) = \cos^2 x - \sin^2 x $:

\[ \cos(2x) = \left(\frac{\sqrt{55}}{8}\right)^2 - \left(\frac{3}{8}\right)^2 = \frac{55}{64} - \frac{9}{64} = \frac{46}{64} = \frac{23}{32} \approx 0.7188 \]

Step 4: Calculate $ \tan(2x) $

Using the definition of tangent, $ \tan(2x) = \frac{\sin(2x)}{\cos(2x)} $:

\[ \tan(2x) = \frac{\frac{3\sqrt{55}}{32}}{\frac{23}{32}} = \frac{3\sqrt{55}}{23} \approx 0.9673 \]