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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.
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}

Now let's try some practice problems.

Problem 1

Given triangle, D, E, F, find D, E.
Round your answer to the nearest hundredth.
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.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 2

Given triangle, D, O, G, find D, G.
Round your answer to the nearest hundredth.
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.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 3

Given triangle, T, R, Y, find T, Y.
Round your answer to the nearest hundredth.
A right triangle T R Y. Angle R T Y is a right angle. Angle T R Y is thirty-seven degrees. Side T Y is unknown. Side R T is three units.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Challenge problem

In the triangle below, which of the following equations could be used to find z?
A right triangle M I X. Angle I M X is a right angle. Angle I X M is twenty-eight degrees. Side I X is unknown. Side I M is twenty units.
Choose all answers that apply:
Choose all answers that apply:

Want to join the conversation?

  • starky seed style avatar for user PP
    please can someone just explain q.2 really slowly and simply? Thanks :) it's totally confusing me!
    (27 votes)
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    • starky ultimate style avatar for user Seunghyun Nam
      - You are given the side OPPOSITE the 72 degree angle, which is 8.2.
      - You are solving for the HYPOTENUSE.
      Therefore you need the trig function that contains both the OPPOSITE and the HYPOTENUSE, which would be SINE, since sin = OPPOSITE / HYPOTENUSE.

      "Let's input the value into the equation."
      sin (deg) = opposite/hypotenuse
      sin (72) = 8.2/DG

      "Since we're solving for DG, the hypotenuse, we have to move it so that it is on the numerator. Thus, you multiply both sides of the equation by DG"
      DG sin(72) = 8.2

      "Again because we're solving for DG, we have to isolate DG so that it alone is on the left side of the equation. To do so, we have to move sin(72) to the other side, or in other words divide both sides of the equation by sin(72)."
      DG = 8.2/sin(72)

      "Now use the calculator"
      8.2/sin(72) = 8.621990.....

      "Round you're answer to the nearest hundred, and you get your answer."
      8.62

      Hope this helped :)
      (5 votes)
  • blobby green style avatar for user joelmazda6.rx8
    I don't understand. How angles that have simple ratios can be discovered? For example, how do you know if sin 30 is 1/2 ?
    (11 votes)
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    • blobby green style avatar for user Andrea O'Keefe
      When using similar triangles, their sides are proportional. If two triangles have two congruent angles, then the triangles are similar. So, if you have a 30-60-90 triangle then the sine ratio is defined as the ratio of the length of the side opposite to the length of the hypotenuse. Doesn't matter how "big" the triangle, those sides will always have the ratio of 1/2.
      (7 votes)
  • blobby green style avatar for user rogirlash
    why is my calculator giving me the wrong answer?
    (6 votes)
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  • duskpin sapling style avatar for user Sophie
    I really need help with problem 3
    (5 votes)
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    • eggleston blue style avatar for user Bubble Tea's Incarnation 🧋
      Hi! I am not really sure what you need help with but I will try to explain it the best I can.

      To find TY, the side you are looking for, you need to use tan.

      You use tan because of SOH CAH TOA, to use tan you use the opposite and adjacent which you have in the problem.

      Look at the 37 degree. The side across from it, or the opposite, is what you are trying to find so it will be your x or unknown value. You know the adjacent side, it is three.

      So you will set up your equation like this

      tan(37)=x/3

      The 37 comes from the degree you used as a reference point. The x comes from TOA, so you put the opposite side over the adjacent. The opposite side is x in this case and the adjacent is 3 in this case. You then find the value of tan(37) using your calculator and multiply it by three (you are using basic algebra here, treat tan(37) like a number), and you are done!

      Hope that helps!
      (2 votes)
  • male robot donald style avatar for user aadityasuri1
    I don't understand that how does one calculate a tangent without a calculator
    (4 votes)
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  • blobby green style avatar for user tom-moy forbes
    can trig ratios be used on acute triangles
    (5 votes)
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    • leaf green style avatar for user nc120321
      Yes it can. Trig ratios can apply to non-right triangles. When you get to the law of sines and cosines, you will see that you can find the measures of angles and the lengths of sides on obtuse and acute triangles.
      (2 votes)
  • duskpin seed style avatar for user SaraJo Gardner
    What if you have only the hypotenuse and one angle then what do you do?
    (2 votes)
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    • hopper cool style avatar for user Marian
      It'd depend on what are you find, the adjacent or the opposite. To find the opposite you'd use the SIN(angle)= opposite you want to find/hypotenuse you have; you'd just have to do the equation with the O/H inverse. Same to find the adjacent, but with COS.

      Hope it helps you. :)
      (5 votes)
  • blobby green style avatar for user ElaDevani
    Can someone please help me on problem 3, I would greatly appreciate it :) (given TRY, find TY)
    (3 votes)
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    • spunky sam blue style avatar for user Cornelius
      This problem is a tangent problem. At least, tan is the simplest for this one. I will do the working for tangent.
      The given angle is 37 deg. We want to find the opposite angle, and are given the adjacent length, 3. So plug in our formula: tan*37=?/3. We multiply both sides by 3: 3*tan*37=?. You can punch that in your calculator (make sure it is in degree mode). I will not tell you the answer, but I basically did already.
      Hope that was a help.
      (4 votes)
  • blobby green style avatar for user zthosier
    For Question 1, I got 180.78 instead of 6.97
    (2 votes)
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  • blobby green style avatar for user Elena
    Can someone please help me understand problem number 2? I do not know how to start.
    (2 votes)
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    • aqualine ultimate style avatar for user hannahmorrell
      Hi Elena! These problems seem very daunting, but if you take them step by step, they are pretty simple, actually. First, analyze the things you know. One angle measures 72 degrees, and the side opposite that is 8.2. The side you are solving for is the hypotenuse. So, since the sides you're dealing with are the opposite (relative to the angle you know) and the hypotenuse, you have to use the sine function.
      sin(72)= opp/hyp
      sin(72)= 8.2/?
      ?sin(72)=8.2
      ?=8.2/sin(72)
      ?=8.622
      Hope this helps!
      (4 votes)