Question: Find the general solution of the equation sin(x) = -1/2.
Options:
x = 7π/6 + 2nπ
x = 11π/6 + 2nπ
x = 7π/6, 11π/6
Both 1 and 2
Correct Answer: Both 1 and 2
Solution:
The general solutions are x = 7π/6 + 2nπ and x = 11π/6 + 2nπ.
Find the general solution of the equation sin(x) = -1/2.
Practice Questions
Q1
Find the general solution of the equation sin(x) = -1/2.
x = 7π/6 + 2nπ
x = 11π/6 + 2nπ
x = 7π/6, 11π/6
Both 1 and 2
Questions & Step-by-Step Solutions
Find the general solution of the equation sin(x) = -1/2.
Step 1: Understand the equation sin(x) = -1/2. We need to find the angles x where the sine value is -1/2.
Step 2: Recall the unit circle. The sine function is negative in the third and fourth quadrants.
Step 3: Identify the reference angle. The reference angle for sin(x) = 1/2 is π/6 (30 degrees).
Step 4: Find the angles in the third and fourth quadrants where sine is -1/2. These angles are: π + π/6 = 7π/6 (third quadrant) and 2π - π/6 = 11π/6 (fourth quadrant).
Step 5: Write the general solutions. Since sine is periodic with a period of 2π, we add 2nπ (where n is any integer) to each solution.
Step 6: Combine the results. The general solutions are x = 7π/6 + 2nπ and x = 11π/6 + 2nπ.
Trigonometric Equations – The question tests the ability to solve basic trigonometric equations, specifically involving the sine function.
General Solutions – It assesses understanding of how to express the general solution of trigonometric equations, including the periodic nature of sine.
Unit Circle – Knowledge of the unit circle is necessary to identify the angles where sine takes specific values.
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