## 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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