Building blocks

Triangles solution

Here are three triangles with some side lengths and angles marked. Express the missing side lengths in terms of $\sin \theta$, $\cos \theta$ and $\tan \theta.$

• How can using side length ratios for $\sin \theta$, $\cos \theta$ and $\tan \theta$ help you?

From side $EF$ we have $\tan \theta = \dfrac{\sin \theta}{\cos \theta}.$
We can start by relabelling the triangles’ sides using $\sec \theta$, $\cosec \theta$ and $\cot \theta.$
Using Pythagoras’ theorem for each triangle gives us three results relating pairs of functions $\cos^2 \theta + \sin^2 \theta = 1 \quad \cot^2 \theta + 1 = \cosec^2 \theta \quad \text{and} \quad 1 + \tan^2 \theta = \sec^2 \theta.$