how do you find all the solutions when cos x = .75
To find all the solutions when cos x = 0.75, follow these steps:
1. Recall the unit circle:
- The cosine function represents the x-coordinate of a point on the unit circle.
- The unit circle is a circle with a radius of 1, centered at the origin (0,0) in a coordinate system.
- The angle made by the radius line and the positive x-axis is denoted by x.
2. Use the inverse cosine function (cos⁻¹) to solve for x:
- cos⁻¹(0.75) = x
- This will give you the principal value of x, which is the primary solution.
- In this case, cos⁻¹(0.75) will yield a value between 0 and π radians (or 0 and 180 degrees).
- Note that x can also have multiple solutions due to the periodic nature of the cosine function.
3. Determine the general solutions for x:
- To find additional solutions, consider the symmetry and periodicity of the cosine function.
- Cosine has a period of 2π radians (or 360 degrees), meaning that cos(x + 2π) = cos(x).
- So, to find all solutions, add and subtract integral multiples of 2π to the principal value obtained earlier.
- The general solutions for x are:
- x = cos⁻¹(0.75) + 2πn, where n is an integer.
- x = -cos⁻¹(0.75) + 2πn, where n is an integer.
4. Evaluate the solutions:
- Substitute different values for n into the general solutions obtained in step 3 to find specific solutions for x.
- Each specific solution will represent an angle in radians or degrees, depending on the unit used.
By following these steps, you can find all the solutions for the equation cos x = 0.75.