Find the general solution of the equation sin(x) + √3cos(x) = 0.

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Question: Find the general solution of the equation sin(x) + √3cos(x) = 0.

Options:

Correct Answer: x = (2n+1)π/3

Solution:

The general solution is x = (2n+1)π/3, where n is an integer.

Find the general solution of the equation sin(x) + √3cos(x) = 0.

Find the general solution of the equation sin(x) + √3cos(x) = 0.

Step 1: Start with the equation sin(x) + √3cos(x) = 0.
Step 2: Rearrange the equation to isolate sin(x): sin(x) = -√3cos(x).
Step 3: Divide both sides by cos(x) (assuming cos(x) ≠ 0) to get tan(x) = -√3.
Step 4: Recall that tan(x) = -√3 corresponds to specific angles. The reference angle where tan(x) = √3 is π/3.
Step 5: Since tan is negative in the second and fourth quadrants, the angles are: x = π - π/3 and x = 2π - π/3.
Step 6: Calculate these angles: x = 2π/3 and x = 5π/3.
Step 7: The general solution for tan(x) = -√3 can be expressed as x = 2nπ + 2π/3 and x = 2nπ + 5π/3, where n is any integer.
Step 8: Combine these solutions into one expression: x = (2n + 1)π/3, where n is an integer.

Trigonometric Equations – The question tests the ability to solve equations involving sine and cosine functions.
General Solutions – It assesses understanding of how to express solutions in terms of integer multiples of π.
Quadrants of Trigonometric Functions – The problem requires knowledge of the signs of sine and cosine in different quadrants to find all solutions.