Solution 4.4:3a

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The right-hand side of the equation is a constant, so the equation is in fact a normal trigonometric equation of the type
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The right-hand side of the equation is a constant, so the equation is in fact a normal trigonometric equation of the type <math>\cos x = a\,</math>.
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<math>\text{cos }x=a</math>.
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In this case, we can see directly that one solution is
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In this case, we can see directly that one solution is <math>x = \pi/6\,</math>. Using the unit circle, it follows that <math>x = 2\pi - \pi/6 = 11\pi/6\,</math> is the only other solution between <math>0</math> and <math>2\pi\,</math>.
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<math>x={\pi }/{6}\;</math>. Using the unit circle, it follows that
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<math>x=2\pi -{\pi }/{6}\;={11\pi }/{6}\;</math>
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is the only other solution between
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<math>0</math>
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and
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<math>\text{2}\pi </math>.
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[[Image:4_4_3_a.gif|center]]
[[Image:4_4_3_a.gif|center]]
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We obtain all solutions to the equation if we add multiples of
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We obtain all solutions to the equation if we add multiples of <math>2\pi</math> to the two solutions above,
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<math>\text{2}\pi </math>
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to the two solutions above:
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{{Displayed math||<math>x = \frac{\pi}{6} + 2n\pi\qquad\text{and}\qquad x = \frac{11\pi}{6} + 2n\pi\,,</math>}}
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<math>x=\frac{\pi }{6}+2n\pi </math>
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where ''n'' is an arbitrary integer.
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and
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<math>x=\frac{11\pi }{6}+2n\pi </math>
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where
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<math>n</math>
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is an arbitrary integer.
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Current revision

The right-hand side of the equation is a constant, so the equation is in fact a normal trigonometric equation of the type \displaystyle \cos x = a\,.

In this case, we can see directly that one solution is \displaystyle x = \pi/6\,. Using the unit circle, it follows that \displaystyle x = 2\pi - \pi/6 = 11\pi/6\, is the only other solution between \displaystyle 0 and \displaystyle 2\pi\,.

We obtain all solutions to the equation if we add multiples of \displaystyle 2\pi to the two solutions above,

\displaystyle x = \frac{\pi}{6} + 2n\pi\qquad\text{and}\qquad x = \frac{11\pi}{6} + 2n\pi\,,

where n is an arbitrary integer.

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