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| Math Central | Quandaries & Queries |
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Question from Abeth, a teacher: True or False: Since cot (theta) = cos (theta)/sin (theta), if cot (theta) = 1/2, then cos (theta) =1 and sin (theta)=2. |
Abeth,
If $\cos \theta = 1$ then $\sin \theta$ can't be $\frac12$ since you know that for any $\theta, \sin^2 \theta + \cos^2 \theta = 1.$ Hence is $\cos \theta = 1$ then $\sin \theta = 0.$
If $\cot \theta = \frac12$ then you know that $\large \frac{\cos \theta}{\sin \theta} = \frac12$ and hence
\[\sin \theta = 2 \cos \theta.\]
To find $\sin \theta \mbox{ and } \cos \theta$ square both sides of the equation above and use the fact that $ \sin^2 \theta + \cos^2 \theta = 1.$ Make sure you verify your answers.
Penny
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