Math CentralQuandaries & Queries


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.
My answer before was true, but not my answer is false. Can you give me a solution on this matter. thanks.


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.


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