A Demonstration of an application of a half-angle identity

May 2014
113
6
Allentown PA USA
Dear My Math Forum Community:

Express this as a single trigonometric function:

(1 - cos 59.74 degrees)/sin 59.74 degrees

The correct answer is: tan 29.87 degrees.

Could someone please provide me with a step-by-step demonstration as to how the answer is arrived at? Thank you.:)
 
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v8archie

Math Team
Dec 2013
7,709
2,677
Colombia
\begin{align}
\frac{1 - \cos 2\theta}{\sin 2\theta} &= \frac{1 - (\cos^2 \theta - \sin^2 \theta)}{2\sin \theta \, \cos \theta} \\
&= \frac{1 - (1 - 2\sin^2 \theta)}{2\sin \theta \, \cos \theta} \\
&= \frac{\cancel{2}\sin^\cancel{2} \theta}{\cancel{2}\cancel{\sin \theta} \, \cos \theta} \\
&= \frac{\sin \theta}{\cos \theta} \\
&= \tan \theta \\
\end{align}
 
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May 2014
113
6
Allentown PA USA
to V8archie

Thanks for the proof! I was so intent on specifics that I overlooked the generalities of the situation.
 

skipjack

Forum Staff
Dec 2006
21,296
2,375
Other correct answers are cot 60.13 degrees, etc.