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):

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EXAMPLE 1: Find all solutions of the equation tex2html_wrap_inline77 .

Solution: We can graphically visualize all the angles u which satisfy the equation by noticing that tex2html_wrap_inline81 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 tex2html_wrap_inline87 that satisfy the equation: tex2html_wrap_inline89 and tex2html_wrap_inline91 . Since the period of the sine function is tex2html_wrap_inline93 , it follows that all solutions of the original equation are:

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EXERCISE 1 Find all solutions of the equation tex2html_wrap_inline95 .
Solution.

EXERCISE 2 Find all solutions of the equation tex2html_wrap_inline97 that lie in the interval tex2html_wrap_inline87.
Solution.

EXERCISE 3 Find all solutions of the equation tex2html_wrap_inline101 in the interval tex2html_wrap_inline87 .
Solution.

EXERCISE 4 Solve the equation tex2html_wrap_inline105 . Restrict solutions to the interval tex2html_wrap_inline107 .
Solution.

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Luis Valdez-Sanchez
Tue Dec 3 17:39:00 MST 1996

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