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} $.