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Wednesday, 30 July 2014

CHAPTER 7- 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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Find all solutions of the equation tex2html_wrap_inline67 .
Solution: Let u=2A; the equation is then equivalent to tex2html_wrap_inline71 , for which the solutions are (see EXAMPLE 1):

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Hence the solutions for A are:

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Find all solutions of the equation tex2html_wrap_inline81 that lie in the interval tex2html_wrap_inline83
Solution: The left hand side of the equation can be factored as:

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hence either tex2html_wrap_inline85 or tex2html_wrap_inline87 . For tex2html_wrap_inline89 ,

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while

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and

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The solution set of the original equation is then

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Solve the equation tex2html_wrap_inline65 . Restrict solutions to the interval tex2html_wrap_inline67 .
Solution: The substitution tex2html_wrap_inline69 yields the equation tex2html_wrap_inline71 , which is quadratic in u. We use the quadratic formula to solve for tex2html_wrap_inline69 :

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If tex2html_wrap_inline77 the calculator gives x=1.1191 as the acute solution, so we deduce that the other solution is tex2html_wrap_inline81 . The equation tex2html_wrap_inline83 has no solutions, since tex2html_wrap_inline85 for all x. The solution set is therefore tex2html_wrap_inline89 .
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