Pixwords 3 lettres Pixwords 6 lettres Pixwords 10 lettres Pixwords 14 lettres Pixwords 18 lettres pixwords Français
How to Understand the Basic of Trigonometric SolutionsItsMyAcademy.com

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