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Find the general solution of the equation sin(x) + √3cos(x) = 0.
Find the general solution of the equation sin(x) + √3cos(x) = 0.
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Practice Questions
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Q1
Find the general solution of the equation sin(x) + √3cos(x) = 0.
x = (2n+1)π/3
x = nπ
x = (2n+1)π/4
x = nπ + π/6
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The general solution is x = (2n+1)π/3, where n is an integer.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the general solution of the equation sin(x) + √3cos(x) = 0.
Solution:
The general solution is x = (2n+1)π/3, where n is an integer.
Steps: 8
Show Steps
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.
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