The general solution for trigonometric equations contains an infinite number of solutions that can be expressed as:
x = θ + 2πn, where n is an integer.
This applies to both sine and cosine functions.
For example:
Find the general solution to the equation cos(x) = -1/2.
Step 1: Find the reference angle
cos(x) = -1/2 has solutions in the second and third quadrants, where the reference angle θ is 60 degrees.
Step 2: Write the general solution
cos(x) = -1/2 has two solutions in each quadrant. Therefore, the general solution to this equation is:
x = 120 degrees + 360n or x = 240 degrees + 360n.
Where n is an integer, which can take values such as -2, -1, 0, 1, 2, ... etc.
By substituting these values of n into the general solution, we can find all of the possible solutions for the given trigonometric equation.
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