# How to Understand the Basics of Trigonometric Solutions

First thing, when solving a TRIG equation, is to understand or accept that each of SINE, COSINE and TANGENT have 2 angles that will satisfy the given equation within any 360 degree range. This applies only when we have things like sin(x) or tan(x) + 2 for example.

If we have multiples, such as cos(2x), then we also multiply up the possible number of solutions. In “general”, as a rule of thumb, expect the following, within any 360 degree range:

sin(x) –> 2 solutions

sin(2x) –> 4 solutions

sin(3x) –> 6 solutions

sin(4x) –> 8 solutions

sin(5x) –> 10 solutions etc

If a trig equation can be solved analytically, these steps will do it:

1. Put the equation in terms of one function of one angle.

2. Write the equation as one trig function of an angle equals a constant.3. Write down the possible value(s) for the angle.

4. If necessary, solve for the variable.

5. Apply any restrictions on the solution.

Let us see an example:

cos(4A) – sin(2A) = 0

Here the “angles”, the arguments to the trig functions, are 4A and 2A. True, you want to solve for A ultimately. But if you can solve for the angle 4A or 2A, it is then quite easy to solve for the variable.

*. Solve sin ( x ) + 2 = 3 for 0° < x < 360°

Just as with linear equations, first isolate the variable-containing term:

sin ( x ) + 2 = 3

sin ( x ) = 1

using the reference angles: x = 90°

Let us see one more example

cos(4A) – sin(2A) = 0

2sin²(2A) + sin(2A) – 1 = 0

sin(2A) = -1 or sin(2A) = 1/2

The sine of 3pi/2 is -1, so the first possibility reduces to 2A = 3pi/2. But remember that the sine function is periodic, so write sin(2A) = ?1

2A = 3pi/2 + 2pin.

For the second possibility, sin(2A) = 1/2, there are two solutions, because sin(pi/6) and sin(5pi/6) both equal 1/2, and again we add 2pin to the angle to account for all solutions:

sin(2A) = 1/2

2A = pi/6 + 2pi.n or 5pi/6 + 2pi.n

Combining these, here are the three solutions for the original equation:

2A = 3pi/2 + 2pi.n or pi/6 + 2pin or 5pi/6 + 2pi.n

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