Question

find all angles, $0^{circ} leq \theta < 360^{circ}$, that satisfy the equation below, to the nearest tenth of a degree.
$cos(\theta) = \frac{sqrt{2}}{2}$

Explanation:

Step1: Recall cosine values of special angles

We know that \(\cos(45^\circ)=\frac{\sqrt{2}}{2}\) and also, cosine is positive in the first and fourth quadrants.

Step2: Find the angle in the fourth quadrant

To find the angle in the fourth quadrant, we use the formula \(360^\circ - \theta\), where \(\theta\) is the reference angle. Here, the reference angle is \(45^\circ\), so the angle in the fourth quadrant is \(360^\circ - 45^\circ = 315^\circ\).

Step3: Check the range

We need to find angles in the range \(0^\circ\leq\theta < 360^\circ\). The angles \(45^\circ\) (first quadrant) and \(315^\circ\) (fourth quadrant) satisfy the equation \(\cos(\theta)=\frac{\sqrt{2}}{2}\) within this range.

Answer:

\(45.0^\circ\) and \(315.0^\circ\)