Using a Calculator to Solve a Trigonometric Equation Involving Secant

We can begin with some algebra.

Check that the MODE is in radians. Now use the inverse cosine function.

Since $\frac{\pi }{2}\approx 1.57$ and $\pi \approx 3.14,$ 1.8235 is between these two numbers, thus $\theta \approx \text{1}\text{.8235}$ is in quadrant II. Cosine is also negative in quadrant III. Note that a calculator will only return an angle in quadrants I or II for the cosine function, since that is the range of the inverse cosine. See Figure 2.

So, we also need to find the measure of the angle in quadrant III. In quadrant II, the reference angle is $\theta \text{}\text{}\text{'}\approx \pi -\text{1}\text{.8235}\approx \text{1}\text{.3181}\text{.}$ The other solution in quadrant III is $\theta \text{}\text{}\text{'}\approx \pi +\text{1}\text{.3181}\approx \text{4}\text{.4597}\text{.}$

The solutions are $\theta \approx 1.8235\pm 2\pi k$ and $\theta \approx 4.4597\pm 2\pi k.$