## ** 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:

**EXERCISE 1 **
Find all solutions of the equation .

Solution.

**EXERCISE 2 **
Find all solutions of the equation that
lie in the interval .

Solution.

**EXERCISE 3 **
Find all solutions of the equation in the
interval .

Solution.

**EXERCISE 4 **
Solve the equation . Restrict
solutions to the interval .

Solution.

**
**

**
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*Luis Valdez-Sanchez *

Tue Dec 3 17:39:00 MST 1996

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