Solving Trig Equations with Calculators, Part II — Question 1

Question 1 - Solution

Step 1: Isolate the cosine term

$$2\cos(x) + 1 = 0 \Rightarrow \cos(x) = -\frac{1}{2}$$

Step 2: Solve the equation using inverse cosine

$$\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \Rightarrow \text{Reference angle} = \frac{\pi}{3}$$

Since cosine is negative, the solutions are in Quadrants II and III:

- Quadrant II: $x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$ - Quadrant III: $x = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$

$$\boxed{x = \frac{2\pi}{3},\ \frac{4\pi}{3}}$$