CCSS Math: HSG.SRT.C.8
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 ABC\triangle ABC, find ACAC.

Solution

Step 1: Determine which trigonometric ratio to use.
Let's focus on angle B\goldD B since that is the angle that is explicitly given in the diagram.
Note that we are given the length of the hypotenuse\purpleC{\text{hypotenuse}}, and we are asked to find the length of the side opposite\blueD{\text{opposite}} angle B\goldD B. The trigonometric ratio that contains both of those sides is the sine.
The trig ratios are defined for angle AA 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, AA.
The mnemonic device SOHS\blueD{O}\purpleC{H} CAHC\maroonC{A}\purpleC{H} TOAT\blueD{O}\maroonC{A} can help us remember these definitions.
Step 2: Create an equation using the trig ratio sine and solve for the unknown side.
sin(B)= opposite  hypotenuse        Define sine.sin(50)=AC6                       Substitute.6sin(50)=AC                         Multiply both sides by 6.4.60AC                         Evaluate with a calculator.\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 DEF\triangle DEF, find DEDE.
Round your answer to the nearest hundredth.
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Step 1: Determine which trigonometric ratio to use.
Here we are given the length of the side adjacent\maroonC{\text{adjacent}} to angle EE and are asked to find the length of the hypotenuse\purpleC{\text{hypotenuse}}. 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.
cos(E)= adjacent hypotenuse        Define cosine.cos(55)=4ED                       Substitute.EDcos(55)=4                            Multiply both sides byED.ED=4cos(55)              Divide both sides bycos(55).ED6.97                       Evaluate with a calculator.\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 DOG\triangle DOG, find DGDG.
Round your answer to the nearest hundredth.
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Problem 3

Given TRY\triangle TRY, find TYTY.
Round your answer to the nearest hundredth.
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Challenge problem

In the triangle below, which of the following equations could be used to find zz?
Choose all answers that apply:
Choose all answers that apply:

We are given the length of the side opposite angle XX, and we are asked to find the length of the hypotenuse. The trigonometric ratio that contains both of those sides is the sine. So sin(28)=20z\sin (28^\circ)=\dfrac{20}{z} .
Since the acute angles of a right triangle sum to 9090^\circ, we also know that mI=62m\angle I = 62^\circ. In relation to this angle, we are given the length of the adjacent side and are asked to find the length of the hypotenuse. The trigonometric ratio that contains both of those sides is the cosine. So, cos(62)=20z\cos (62^\circ)=\dfrac{20}{z} .
The following two equations could be used to find zz:
sin(28)=20z\sin (28^\circ)=\dfrac{20}{z}
cos(62)=20z\cos (62^\circ)=\dfrac{20}{z}