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### Unit 5: Lesson 3

Solving triangles using the trigonometric ratios

# 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.
A right triangle A B C. Angle A C B is a right angle. Angle A B C is fifty degrees. Side A C is unknown. Side A B is six units.

### Solution

Step 1: Determine which trigonometric ratio to use.
Let's focus on angle start color #e07d10, B, end color #e07d10 since that is the angle that is explicitly given in the diagram.
A right triangle A B C. Angle A C B is a right angle. Angle A B C is fifty degrees and is highlighted. Side A C is unknown. Side A B is six units.
Note that we are given the length of the start color #aa87ff, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #aa87ff, and we are asked to find the length of the side start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd angle start color #e07d10, B, end color #e07d10. The trigonometric ratio that contains both of those sides is the sine.
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}

## Now let's try some practice problems.

### Problem 1

Given triangle, D, E, F, find D, E.
A right triangle D E F. Angle D F E is a right angle. Angle D E F is fifty degrees. Side D E is unknown. Side E F is four units.

### Problem 2

Given triangle, D, O, G, find D, G.
A right triangle D O G. Angle D O G is a right angle. Angle D G O is seventy-two degrees. Side D G is unknown. Side D O is eight point two units.

### Problem 3

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