Trigonometry allows us to find sides of triangles that we would not normally be able to find, by taking advantage of the sine, cosine, and tangent ratios.

Let’s restate the sine, cosine, and tangent ratios before we start on examples:

Definition:  Let \(\theta \) be one of the acute angles of a right triangle.  Then

\(\large \sin \theta  = \LARGE \frac{{opposite  leg}}{{hypotenuse}}\)

\(\large \cos \theta  = \LARGE \frac{{adjacent  leg}}{{hypotenuse}}\)

\(\large \tan \theta  = \LARGE \frac{{opposite  leg}}{{adjacent  leg}}\)

Now that we have stated the three trigonometric ratios, we can look at our examples (there are also three reciprocal trigonometric ratios: cosecant, secant, and cotangent.  For now though, we will only use the three main ratios).

Example:  Find the missing side \(x\). Round to the nearest tenth.

Solution:  Since \(x\) is the opposite leg, and \(16\) is the adjacent leg, we can use the tangent ratio to set up the equation

\(\tan 62^\circ  = \Large \frac{x}{{16}}\)

Multiplying both sides by 16 gives

\(x = 16 \cdot \tan 62^\circ  \approx 30.1\), rounded to the nearest tenth

The value \(30.1\) was obtained by computing \(16\tan 62^\circ \) in a calculator. 

MAKE SURE your calculator is in degree mode!!!

Example:  Find the missing side \(x\). Round to the nearest tenth.

Solution: Here \(x\) is the opposite leg, and \(17\) is the hypotenuse.  Then we can use the sine ratio to set up the equation

\(\sin 15^\circ  = \Large \frac{x}{{17}}\)

\(x = 17\sin 15^\circ \)

\(x \approx 4.4\), rounded to the nearest tenth

Remember to keep your calculator in degree mode!  And remember to set up the appropriate trigonometric ratio for the situation!

Below you can download some free math worksheets and practice.