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## Monday, 25 March 2013

### TRIGONOMETRIC EQUATIONS

Some equations which involve trigonometric functions of the unknown may be readily solved by using simple algebraic ideas (as Equation 1 below), while others may be impossible to solve exactly, only approximately (e.g., Equation 2 below):

EXAMPLE 1: Find all solutions of the equation  .
Solution: We can graphically visualize all the angles u which satisfy the equation by noticing that  is the y-coordinate of the point where the terminal side of the angle u (in standard position) intersects the unit circle (see Figure 1):
We can see that there are two angles in  that satisfy the equation:  and  . Since the period of the sine function is  , it follows that all solutions of the original equation are:

Solve for x in the following equation.

Example 1:

There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term.

We know that the therefore  The sine function is positive in quadrants I and II. The is also equal to  Therefore, two of the solutions to the problem are and

The period of the sin  function is  This means that the values will repeat every  radians in both directions. Therefore, the exact solutions are  and  where n is an integer. The approximate solutions are  and  where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Check answer .

• Left Side:
• Right Side:        0

Since the left side equals the right side when you substitute  for x, then  is a solution.

Check answer .

• Left Side:
• Right Side:        0
Since the left side equals the right side when you substitute  for x, then  is a solution.

Graphical Check:
Graph the equation

f (x) = 2 sin(x) - 1

Note that the graph crosses the x-axis many times indicating many solutions. Note that it crosses at . Since the period is , it crosses again at 0.5236+6.283=6.81 and at0.5236+2(6.283)=13.09, etc. The graph crosses at . Since the period is , it will cross again at 2.618+6.283=8.9011 and at 2.618+2(6.283)=15.18, etc.
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