Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)

Question

cos²(𝜃) − cos(𝜃) − 6 = 0

I am really struggling with this portion of my homework I would really appreciate it if someone could explain this in steps and then show the final answer. It will help me understand how to solve the rest of the problems. Thank you!

Answer 1

Well suppose cos θ = x

Then x² - x - 6 = 0

(x - 3) (x + 2) = 0

x = 3 or -2

As cos θ has to be between -1 and 1, these solutions are not valid, so there is no solution.

You can check this by plotting cos² (x) - cos (x) - 6 on Desmos or a graphing calculator - you will see that the function never touches or crosses the x-axis (ie is never zero)