Questions: Assignment 9.3: Double-Angle, Half-Angle Formulas If sin x = 1/4, x in quadrant I, then find the exact answers for the following (without finding x): sin (2 x)= cos (2 x)= tan (2 x)=

Solution Steps

1. Use the double-angle formulas for sine, cosine, and tangent.
2. For \(\sin(2x)\), use the formula \(\sin(2x) = 2 \sin(x) \cos(x)\).
3. For \(\cos(2x)\), use the formula \(\cos(2x) = \cos^2(x) - \sin^2(x)\).
4. For \(\tan(2x)\), use the formula \(\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}\).
5. Since \(\sin(x) = \frac{1}{4}\), find \(\cos(x)\) using the Pythagorean identity \(\cos(x) = \sqrt{1 - \sin^2(x)}\).
6. Calculate \(\tan(x)\) as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

Using the double-angle formula for sine, we have: \[ \sin(2x) = 2 \sin(x) \cos(x) \] Substituting the values: \[ \sin(2x) = 2 \left(\frac{1}{4}\right) \left(0.9682\right) \approx 0.4841 \]

Using the double-angle formula for cosine, we have: \[ \cos(2x) = \cos^2(x) - \sin^2(x) \] Substituting the values: \[ \cos(2x) = (0.9682)^2 - \left(\frac{1}{4}\right)^2 \approx 0.8750 \]

Using the double-angle formula for tangent, we have: \[ \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \] First, we calculate \(\tan(x)\): \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \approx \frac{0.25}{0.9682} \approx 0.2582 \] Now substituting into the tangent formula: \[ \tan(2x) = \frac{2 \cdot 0.2582}{1 - (0.2582)^2} \approx 0.5533 \]

Final Answer