Find all solutions to 7cosθ - 1 = 9cosθ.

well, it's clear that

cosθ = -1/2

There will be solutions in QII and QIII

To find all the solutions to the equation 7cosθ - 1 = 9cosθ, we need to isolate the cosine term on one side of the equation. Here's how:

1. Start with the given equation: 7cosθ - 1 = 9cosθ.

2. To isolate the cosine term, subtract 9cosθ from both sides of the equation: 7cosθ - 9cosθ - 1 = 0.

3. Combine the cosine terms on the left side: (7 - 9)cosθ - 1 = 0.

4. Simplify: -2cosθ - 1 = 0.

5. Add 1 to both sides: -2cosθ = 1.

6. Now, divide by -2 to solve for cosθ: cosθ = 1/(-2).

7. Since the cosine value is equal to 1 divided by -2, which is -1/2, we look for angles whose cosine value is -1/2.

8. To find these angles, we can use the inverse cosine function (also called arccosine or cos^(-1)): θ = arccos(-1/2).

9. Using a calculator or trigonometric tables, we find that the principal value of arccos(-1/2) is approximately 120 degrees or π/3 radians.

10. However, we need to find all the solutions, so we add or subtract multiples of the period of the cosine function (which is 2π or 360 degrees) to find other valid solutions.

11. The general solutions for θ can be found by adding or subtracting integer multiples of the period: θ = 120° + 360°n or θ = (π/3) + 2πn, where n is an integer.

These are all the solutions to the equation 7cosθ - 1 = 9cosθ.