Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].

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Question: Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].

Options:

Correct Answer: π/2

Solution:

The only solution is x = π/2.

Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].

Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].

Step 1: Start with the equation cos(x) + sin(x) = 1.
Step 2: Recall that the maximum value of cos(x) and sin(x) is 1.
Step 3: Notice that for cos(x) + sin(x) to equal 1, both cos(x) and sin(x) must be less than or equal to 1.
Step 4: Check the point where both functions can add up to 1. This happens when sin(x) is at its maximum value of 1 and cos(x) is 0.
Step 5: Identify the angle where sin(x) = 1. This occurs at x = π/2.
Step 6: Verify if x = π/2 satisfies the original equation: cos(π/2) + sin(π/2) = 0 + 1 = 1.
Step 7: Since the equation holds true, x = π/2 is a solution.
Step 8: Check the interval [0, 2π] to ensure there are no other solutions. The only solution in this interval is x = π/2.

Trigonometric Equations – The question tests the ability to solve equations involving sine and cosine functions.
Interval Restrictions – The solution must be found within a specified interval, which is [0, 2π] in this case.
Understanding of Trigonometric Values – The question requires knowledge of the values of sine and cosine at specific angles.