To solve without a calculator, we can use the identity cos(180° - θ) = -cosθ, which means that:
cos 20° = cos(180° - 20°) = -cos 160°
Substituting this into the original equation gives:
-3(-cos 160°) - 6 = 19 cos θ
Simplifying this expression gives:
3 cos 160° - 6 = 19 cos θ
Using the fact that cos 160° = cos(-20°), we can rewrite this as:
3 cos (-20°) - 6 = 19 cos θ
Now, using the identity cos(-θ) = cosθ, we get:
3 cos 20° - 6 = 19 cos θ
Adding 6 to both sides and dividing by 19, we obtain:
cos θ = (-3 cos 20° + 6)/19
Using a table of trigonometric values, we can find that cos 20° is approximately 0.9397. Substituting this value into the equation above, we get:
cos θ ≈ (-3 × 0.9397 + 6)/19
cos θ ≈ -0.628
Since -1 ≤ cos θ ≤ 1, there are no angles between 0° and 360° that satisfy the equation to the nearest 10th of a degree. Therefore, the answer is that there are no solutions.