Solving Trig Equations with Calculators, Part II — Question 4

Question 4 - Solution

Step 1: Isolate the trigonometric function. The equation becomes

$$\cos(3x)=0.2,\qquad u=3x,\qquad 0\le u\le 6\pi.$$

Let $\alpha=\arccos(0.2)$. All solutions are

$$u=\pm\alpha+2k\pi,\qquad k\in\mathbb Z.$$

Step 2: Restrict and convert. Keep precisely the values of $u$ in $[0,6\pi]$ and divide by $3$.

Evaluating the inverse function at full precision and rounding only the final values gives

$$\boxed{x\approx 0.46,\ 1.64,\ 2.55,\ 3.73,\ 4.65,\ 5.83}.$$

These are all 6 solutions in the stated interval, in radians. Substitution of the unrounded values verifies the original equation.