# Solving for a side in right triangles with trigonometry

Learn how to use trig functions to find an unknown side length in a right triangle.
We can use trig ratios to find unknown sides in right triangles.

### Let's look at an example.

Given triangle, A, B, C, find A, C.

### Solution

Step 1: Determine which trigonometric ratio to use.
Let's focus on angle start color goldD, B, end color goldD since that is the angle that is explicitly given in the diagram.
Note that we are given the length of the start color purpleC, h, y, p, o, t, e, n, u, s, e, end color purpleC, and we are asked to find the length of the side start color blueD, o, p, p, o, s, i, t, e, end color blueD angle start color goldD, B, end color goldD. The trigonometric ratio that contains both of those sides is the sine.
The trig ratios are defined for angle A below.
In these definitions, it is important to understand that the terms opposite, adjacent, and hypotenuse indicate the length of the corresponding sides relative to the given angle, in this case, A.
The mnemonic device S, start color blueD, O, end color blueD, start color purpleC, H, end color purpleC C, start color maroonC, A, end color maroonC, start color purpleC, H, end color purpleC T, start color blueD, O, end color blueD, start color maroonC, A, end color maroonC can help us remember these definitions.
Step 2: Create an equation using the trig ratio sine and solve for the unknown side.
\begin{aligned}\sin( \goldD{ B}) &= \dfrac{ \blueD{\text{ opposite}} \text{ } }{\purpleC{\text{ hypotenuse} }} ~~~~~~~~\small{\gray{\text{Define sine.}}}\\\\ \sin (\goldD{50^\circ})&= \dfrac{\blueD{AC}}{\purpleC6}~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Substitute.}}} \\\\\\\\ 6\sin ({50^\circ})&= {{AC}} ~~~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Multiply both sides by }6.}}\\\\\\\\ 4.60&\approx AC~~~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Evaluate with a calculator.}}} \end{aligned}
Here's a video of Sal solving a triangle using the trigonometric ratios.

# Now let's try some practice problems.

### Problem 1

Given triangle, D, E, F, find D, E.

Step 1: Determine which trigonometric ratio to use.
Here we are given the length of the side start color maroonC, a, d, j, a, c, e, n, t, end color maroonC to angle E and are asked to find the length of the start color purpleC, h, y, p, o, t, e, n, u, s, e, end color purpleC. The trigonometric ratio that contains both of those sides is the cosine.
Step 2: Create an equation using the trig ratio cosine and solve for the unknown side.
\begin{aligned}\cos (\goldD{ E}) &= \dfrac{\maroonC{\text{ adjacent}} }{\purpleC{\text{ hypotenuse} }}~~~~~~~~\small{\gray{\text{Define cosine.}}}\\\\ \\\\ \cos (\goldD{55^\circ})&= \dfrac{\maroonC{4}}{\purpleC{ED}} ~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Substitute.}}}\\\\\\\\ ED\cdot\cos ({55^\circ})&= 4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Multiply both sides by}ED.}}\\\\\\\\ ED&=\dfrac{4}{\cos (55^\circ)}~~~~~~~~~~~~~~\small{\gray{\text{Divide both sides by}\cos(55^\circ).}}\\\\\\\\ ED&\approx 6.97~~~~~~~~~~~~~~~~~~~~~~~\small{\gray{\text{Evaluate with a calculator.}}} \end{aligned}

### Problem 2

Given triangle, D, O, G, find D, G.

### Problem 3

Given triangle, T, R, Y, find T, Y.