What are the solutions of the equation cos(x) + sin(x) = 1?

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Question: What are the solutions of the equation cos(x) + sin(x) = 1?

Options:

Correct Answer: x = 0

Solution:

The only solution is x = 0.

What are the solutions of the equation cos(x) + sin(x) = 1?

What are the solutions of the equation cos(x) + sin(x) = 1?

Step 1: Start with the equation cos(x) + sin(x) = 1.
Step 2: Recall that the maximum value of cos(x) is 1 and the maximum value of sin(x) is also 1.
Step 3: Since both cos(x) and sin(x) are at most 1, the only way their sum can equal 1 is if one of them is 1 and the other is 0.
Step 4: Check when cos(x) = 1. This happens at x = 0.
Step 5: If cos(0) = 1, then sin(0) = 0. So, cos(0) + sin(0) = 1 + 0 = 1.
Step 6: Check if there are any other angles where cos(x) + sin(x) = 1. For any other angle, either cos(x) or sin(x) will be less than 1, making their sum less than 1.
Step 7: Conclude that the only solution to the equation is x = 0.

Trigonometric Equations – The question tests the understanding of solving trigonometric equations involving sine and cosine functions.
Unit Circle – Understanding the unit circle is essential for determining the values of sine and cosine at specific angles.
Range of Functions – Recognizing that the maximum value of cos(x) + sin(x) is √2, which is greater than 1, is crucial for identifying possible solutions.