Question

Write the general formula for all the solutions to cos θ 2= 2 2 based on the smaller angle. θ=enter your response here​, k is any integer ​(simplify your answer. use angle measures greater than or equal to 0 and less than 4π. type an exact​ answer, using π as needed. use integers or fractions for any numbers in the expression. type an expression using k as the​ variable.) write the general formula for all the solutions to cos θ 2= 2 2 based on the larger angle. θ=enter your response here​, k is any integer ​(simplify your answer. use angle measures greater than or equal to 0 and less than 4π. type an exact​ answer, using π as needed. use integers or fractions for any numbers in the expression. type an expression using k as the​ variable.)

Answer 1

The general solutions for 'cos θ = √2/2' are θ = π/4 + 2kπ for the smaller angle and θ = 7π/4 + 2kπ for the larger angle, with k as any integer.

Step-by-step explanation:

The original question appears to contain typos and may be incorrect as it is currently written. The expression 'cos θ2= 2 2' does not make mathematical sense. Assuming the intended question is to write the general solution for the equation 'cos θ = √2/2', we can proceed with a solution.

The equation 'cos θ = √2/2' has solutions at θ = π/4 and θ = 7π/4 for the interval from 0 to 2π because the cosine function is positive in the first and fourth quadrants. Since cosine is periodic with period 2π, this means that the general solution for θ based on the smaller angle (which is the angle in the first quadrant), and allowing for all rotations, is:

θ = π/4 + 2kπ

where k is any integer.

To write the general solution for θ based on the larger angle (which is the angle in the fourth quadrant), we express it as:

θ = 7π/4 + 2kπ

Again, k represents any integer.

By ensuring that k is an integer, θ will always be in the range from 0 to 4π, meeting the requirements of the task.