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 \(\tan \theta.\)

three right-angled triangles with angle theta marked and one side of length 1.
  • How can using side length ratios for \(\sin \theta\), \(\cos \theta\) and \(\tan \theta\) help you?

  • How can using similar triangles help you?

In triangle \(OAB\) we are given the hyptenuse and we want to find the opposite and adjacent sides. We know that \(\sin \theta = \dfrac{AB}{OB}\) and \(\cos \theta = \dfrac{OA}{OB}\), so using \(OB=1\) gives us \(AB =\sin \theta\) and \(OA= \cos \theta.\)

If we use \(\sin \theta = \dfrac{CD}{OD}\) in triangle \(OCD\) we obtain \(OD = \dfrac{1}{\sin \theta}.\) Using \(\tan \theta = \dfrac{AB}{OA}\) gives us \(OC =\dfrac{1}{\tan \theta}.\)

We can also find the side lengths in triangle \(OEF\) using these approaches.

Note that triangle \(OCD\) is similar to triangle \(OAB\) and the scale factor for side length is \(\dfrac{1}{\sin \theta}\). This gives us \(OD = \dfrac{1}{\sin \theta}\) and \(OC=\dfrac{\cos \theta}{\sin \theta}.\)

What happens if we use a similar approach for triangle \(OEF\)?

Using trig ratios we obtain the following side lengths.

triangles with all sides obtained from trig ratios

Using similar triangles we obtain these side lengths.

triangles with all sides obtained using similar triangles

What do you notice if you compare the results of the two approaches?

From side \(EF\) we have \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}.\)

We now have six trigonometric functions. From the side lengths in the triangles above, what relationships can you find between these functions?

We can start by relabelling the triangles’ sides using \(\sec \theta\), \(\cosec \theta\) and \(\cot \theta.\)

three triangles with sides cos theta, sin theta, 1, cot theta, cosec theta, 1, and 1, tan theta, sec theta respectively

Using Pythagoras’ theorem for each triangle gives us three results relating pairs of functions \[\cos^2 \theta + \sin^2 \theta = 1 \quad \cot^2 \theta + 1 = \cosec^2 \theta \quad \text{and} \quad 1 + \tan^2 \theta = \sec^2 \theta.\]