Questions: 2 sin θ = √3

Transcript text: $2 \sin \theta=\sqrt{3}$

Solution

Solution Steps

To solve the equation $2 \sin \theta = \sqrt{3}$, we need to isolate $\sin \theta$ and then find the values of $\theta$ that satisfy the equation.

Divide both sides of the equation by 2 to isolate $\sin \theta$.
Use the inverse sine function to find the principal value of $\theta$.
Determine all possible solutions within the given range (typically $0 \leq \theta < 2\pi$ for trigonometric equations).

Step 1: Isolate $\sin \theta$

Starting with the equation: \[ 2 \sin \theta = \sqrt{3} \] we divide both sides by 2: \[ \sin \theta = \frac{\sqrt{3}}{2} \]

Step 2: Find the Principal Value

To find the principal value of $\theta$, we take the inverse sine: \[ \theta_{\text{principal}} = \arcsin\left(\frac{\sqrt{3}}{2}\right) \approx 1.0472 \text{ radians} \]

Step 3: Determine All Possible Solutions

The sine function is positive in the first and second quadrants. Therefore, the two solutions for $\theta$ in the range $0 \leq \theta < 2\pi$ are: \[ \theta_1 = \theta_{\text{principal}} \approx 1.0472 \text{ radians} \] \[ \theta_2 = \pi - \theta_{\text{principal}} \approx 2.0944 \text{ radians} \]

Final Answer

The solutions to the equation $2 \sin \theta = \sqrt{3}$ are: \[ \boxed{\theta_1 \approx 1.0472, \quad \theta_2 \approx 2.0944} \]

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