ASSIGNMENT \#5 1) If \( \cos \theta=\frac{\sqrt{3}}{2} \), find all values of \( \theta \) where \( 0 \leq \theta \leq 360^{\circ} \). Draw a diagram to support your answer.

To solve the problem, we need to find all angles \( \theta \) in the range \( 0 \leq \theta \leq 360^{\circ} \) for which \( \cos \theta = \frac{\sqrt{3}}{2} \). ### Step 1: Identify the reference angle The cosine function is positive in the first and fourth quadrants. The reference angle \( \theta_r \) for which \( \cos \theta = \frac{\sqrt{3}}{2} \) is: \[ \theta_r = 30^{\circ} \] ### Step 2: Find all angles in the specified range 1. **First Quadrant**: The angle is simply the reference angle: \[ \theta_1 = 30^{\circ} \] 2. **Fourth Quadrant**: The angle can be found by subtracting the reference angle from \( 360^{\circ} \): \[ \theta_2 = 360^{\circ} - 30^{\circ} = 330^{\circ} \] ### Step 3: List all solutions Thus, the solutions for \( \theta \) in the range \( 0 \leq \theta \leq 360^{\circ} \) are: \[ \theta = 30^{\circ}, 330^{\circ} \] ### Step 4: Draw a diagram To support the answer, we can visualize the unit circle where the angles are located: - The angle \( 30^{\circ} \) is in the first quadrant. - The angle \( 330^{\circ} \) is in the fourth quadrant. Now, I will create a diagram to illustrate this. ### Diagram The unit circle will show the angles \( 30^{\circ} \) and \( 330^{\circ} \) with their corresponding cosine values. Let's proceed to create the diagram. (Note: As a text-based AI, I cannot draw diagrams directly, but I can describe how to draw it.) 1. Draw a circle with a radius of 1. 2. Mark the angle \( 30^{\circ} \) from the positive x-axis, which will be in the first quadrant. 3. Mark the angle \( 330^{\circ} \) from the positive x-axis, which will be in the fourth quadrant. 4. Draw the horizontal line from the center to the right (positive x-axis) and the vertical line from the center to the top (positive y-axis). 5. Label the points where the angles intersect the circle. ### Conclusion The values of \( \theta \) where \( \cos \theta = \frac{\sqrt{3}}{2} \) in the range \( 0 \leq \theta \leq 360^{\circ} \) are \( 30^{\circ} \) and \( 330^{\circ} \).