Questions: Find θ, 0° ≤ θ<360°, cos θ=-sqrt(3)/2 and θ in QII

Transcript text: 3. Find $\theta, 0^{\circ} \leq \theta<360^{\circ}$, $\cos \theta=-\frac{\sqrt{3}}{2}$ and $\theta$ in QII

Solution

Solution Steps

To find the angle $\theta$ in the specified range where $\cos \theta = -\frac{\sqrt{3}}{2}$ and $\theta$ is in the second quadrant (QII), we need to identify the reference angle whose cosine is $\frac{\sqrt{3}}{2}$ and then determine the corresponding angle in QII.

Step 1: Identify the Cosine Value

We are given that $ \cos \theta = -\frac{\sqrt{3}}{2} $. This indicates that the angle $ \theta $ is in the second quadrant (QII), where cosine values are negative.

Step 2: Find the Reference Angle

The reference angle $ \theta_r $ corresponding to $ \cos \theta = \frac{\sqrt{3}}{2} $ is calculated as: \[ \theta_r = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = 30^\circ \]

Step 3: Determine the Angle in QII

To find the angle $ \theta $ in the second quadrant, we use the formula: \[ \theta = 180^\circ - \theta_r \] Substituting the reference angle: \[ \theta = 180^\circ - 30^\circ = 150^\circ \]