Questions: Jeff said he found three angles for which cos θ = 4/5. Is that possible if 0° ≤ θ ≤ 360° ? Explain.

Transcript text: Jeff said he found three angles for which $\cos \theta=\frac{4}{5}$. Is that possible if $0^{\circ} \leq \theta \leq 360^{\circ}$ ? Explain.

Solution

Solution Steps

Step 1: Evaluate

\cos(36.87^{\circ})

We calculate $\cos(36.87^{\circ})$ : $\cos(36.87^{\circ}) \approx 0.8000$

Step 2: Evaluate

\cos(323.13^{\circ})

Next, we evaluate $\cos(323.13^{\circ})$ : $\cos(323.13^{\circ}) \approx 0.8000$

Step 3: Evaluate

\cos(0^{\circ})

We also find $\cos(0^{\circ})$ : $\cos(0^{\circ}) = 1$

Step 4: Evaluate

\cos(360^{\circ})

Finally, we evaluate $\cos(360^{\circ})$ : $\cos(360^{\circ}) = 1$

Step 5: Explanation of Results

The cosine function is positive in the first and fourth quadrants. The angles that satisfy $\cos(\theta) = \frac{4}{5}$ are approximately $36.87^{\circ}$ and $323.13^{\circ}$ . Therefore, Jeff's claim of finding three angles is incorrect, as there are only two angles in the range $0^{\circ} \leq \theta \leq 360^{\circ}$ .