Solve the equation 3sin(θ) - cos(θ) = 1 for 0° ≤ θ ≤ 360°.

Steps to Solve

1. Rearranging the Equation

Start by rearranging the equation to isolate the trigonometric functions: $$ 3\sin \theta - \cos \theta = 1 \implies 3\sin \theta = \cos \theta + 1 $$

2. Expressing in Terms of $\sin \theta$

Next, express $\cos \theta$ in terms of $\sin \theta$. Use the Pythagorean identity: $$ \cos^2 \theta = 1 - \sin^2 \theta \implies \cos \theta = \sqrt{1 - \sin^2 \theta} $$

3. Substituting and Squaring

Substitute this expression back into the equation: $$ 3\sin \theta = \sqrt{1 - \sin^2 \theta} + 1 $$ Square both sides to eliminate the square root: $$ (3\sin \theta - 1)^2 = 1 - \sin^2 \theta $$

4. Expanding the Equation

Expand the left side: $$ (3\sin \theta)^2 - 2(3\sin \theta)(1) + 1 = 1 - \sin^2 \theta $$ This gives: $$ 9\sin^2 \theta - 6\sin \theta + 1 = 1 - \sin^2 \theta $$

5. Combining and Rearranging

Combine like terms to form a standard quadratic equation: $$ 9\sin^2 \theta + \sin^2 \theta - 6\sin \theta = 0 $$ This simplifies to: $$ 10\sin^2 \theta - 6\sin \theta = 0 $$

6. Factoring the Equation

Factor out the common term: $$ 2\sin \theta(5\sin \theta - 3) = 0 $$ This gives the solutions: $$ \sin \theta = 0 \quad \text{or} \quad \sin \theta = \frac{3}{5} $$

7. Finding Angles Corresponding to $\sin \theta = 0$

The angles where $\sin \theta = 0$ in the interval $[0°, 360°]$ are: $$ \theta = 0°, 180° $$

8. Finding Angles Corresponding to $\sin \theta = \frac{3}{5}$

For $\sin \theta = \frac{3}{5}$, use the arcsine function: $$ \theta = \arcsin\left(\frac{3}{5}\right) \approx 36.87° $$ The sine function is positive in the first and second quadrants, giving: $$ \theta \approx 36.87°, 180° - 36.87° \approx 143.13° $$

9. Listing All Solutions

The final solutions for the equation $3\sin \theta - \cos \theta = 1$ in the interval $[0°, 360°]$ are: $$ \theta \approx 0°, 36.87°, 143.13°, 180° $$