Which angle has −0.8 as its cosine value? Find all solutions between 0° and 360°.

Step-by-step solution

1. 1

   Decide where the solutions can live. Cosine equals the x-coordinate on the unit circle, so a negative cosine puts the angle to the left of the vertical axis — quadrant II or quadrant III. Expect exactly two solutions in one full turn.

2. 2

   Find the reference angle from the absolute value. The reference angle uses $+0.8$, because it measures the acute angle to the nearest part of the horizontal axis:

   $\alpha = \cos^{-1}(0.8) \approx 36.87^\circ$

   (This is the familiar $3$–$4$–$5$ angle, since $0.8 = 4/5$.)

3. 3

   Place the reference angle in quadrant II. In quadrant II the angle is $180^\circ$ minus the reference angle:

   $\theta_1 = 180^\circ - 36.87^\circ = 143.13^\circ$

   This is also what a calculator returns directly for $\cos^{-1}(-0.8)$, since the arccosine range is $[0^\circ, 180^\circ]$.

4. 4

   Place it in quadrant III. In quadrant III the angle is $180^\circ$ plus the reference angle:

   $\theta_2 = 180^\circ + 36.87^\circ = 216.87^\circ$

   Equivalently $\theta_2 = 360^\circ - \theta_1$, because cosine is an even function.

5. 5

   Convert to radians and check. Multiplying by $\pi/180$:

   $\theta_1 \approx 2.498 \text{ rad}, \qquad \theta_2 \approx 3.785 \text{ rad}$

   Evaluating back: $\cos(143.13^\circ) = -0.8000$ and $\cos(216.87^\circ) = -0.8000$, both correct to four decimal places. Adding any multiple of $360^\circ$ gives further solutions.