t for tan
Can you find a geometric proof for some $\tan\theta$ trig identities?
Age
16 to 18
Challenge level
Problem
Image
This resource is from Underground Mathematics.
If we write $t = \tan \theta$, then the following equations are true.
\begin{align*}
\tan(2\theta) &= \frac{2t}{1-t^2}, \\
\sin(2\theta) &= \frac{2t}{1+t^2}, \\
\cos(2\theta) &= \frac{1-t^2}{1+t^2}.
\end{align*}
Can you use this diagram to obtain these formulae?
Image
For what range of values of $\theta$ does this argument work?
This is an Underground Mathematics resource.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
Student Solutions
Well done to Jakub from English International School Prague who has solved this problem, and written a very clear proof which you can find here.
