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Right triangle trigonometry review

Review right triangle trigonometry and how to use it to solve problems.

What are the basic trigonometric ratios?

A right triangle A B C where angle A C B is the right angle. Angle B A C is the angle of reference. Side A B is labeled hypotenuse. Side B C is labeled opposite. Side A C is labeled adjacent.
sine, left parenthesis, angle, A, right parenthesis, equalsstart fraction, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, divided by, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, end fraction
cosine, left parenthesis, angle, A, right parenthesis, equalsstart fraction, start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff, divided by, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, end fraction
tangent, left parenthesis, angle, A, right parenthesis, equalsstart fraction, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, divided by, start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff, end fraction
Want to learn more about sine, cosine, and tangent? Check out this video.

Practice set 1: Solving for a side

Trigonometry can be used to find a missing side length in a right triangle. Let's find, for example, the measure of A, C in this triangle:
A right triangle A B C. Angle A C B is a right angle. Angle A B C is forty degrees. Side A C is unknown. Side A B is seven units.
We are given the measure of angle angle, B and the length of the start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, and we are asked to find the side start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd to angle, B. The trigonometric ratio that contains both of those sides is the sine:
sin(B)=ACABsin(40)=AC7B=40,AB=77sin(40)=AC\begin{aligned} \sin(\angle B)&=\dfrac{\blueD{AC}}{\goldD{AB}} \\\\ \sin(40^\circ)&=\dfrac{AC}{7}\quad\gray{\angle B=40^\circ, AB=7} \\\\ 7\cdot\sin(40^\circ)&=AC \end{aligned}
Now we evaluate using the calculator and round:
A, C, equals, 7, dot, sine, left parenthesis, 40, degrees, right parenthesis, approximately equals, 4, point, 5
Problem 1.1
A right triangle A B C. Angle A C B is a right angle. Angle B A C is sixty-five degrees. Side B C is unknown. Side A B is six units.
B, C, equals
  • 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

Round your answer to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

Practice set 2: Solving for an angle

Trigonometry can also be used to find missing angle measures. Let's find, for example, the measure of angle, A in this triangle:
A right triangle A B C. Angle A C B is a right angle. Angle B A C is unknown. Side A C is six units. Side A B is eight units.
We are given the length of the side start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff to the missing angle, and the length of the start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10. The trigonometric ratio that contains both of those sides is the cosine:
cos(A)=ACABcos(A)=68AC=6,AB=8A=cos1(68)\begin{aligned} \cos(\angle A)&=\dfrac{\purpleC{AC}}{\goldD{AB}} \\\\ \cos(\angle A)&=\dfrac{6}{8}\quad\gray{AC=6, AB=8} \\\\ \angle A&=\cos^{-1}\left(\dfrac{6}{8}\right) \end{aligned}
Now we evaluate using the calculator and round:
angle, A, equals, cosine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, approximately equals, 41, point, 41, degrees
Problem 2.1
A right triangle A B C. Angle A C B is a right angle. Angle B A C is unknown. Side B C is two units. Side A B is six units.
angle, A, equals
  • 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
degrees
Round your answer to the nearest hundredth.

Want to try more problems like this? Check out this exercise.

Practice set 3: Right triangle word problems

Problem 3.1
Howard is designing a chair swing ride. The swing ropes are 5 meters long, and in full swing they tilt in an angle of 29, degrees. Howard wants the chairs to be 2, point, 75 meters above the ground in full swing.
How tall should the pole of the swing ride be?
Round your final answer to the nearest hundredth.
  • 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
meters
The design of the chair swing ride. The pole of the swing is a rectangle with a short base and a long height. At the top of the pole, there are swing ropes that extend from the pole at an angle of twenty-nine degrees. The rope extends for 5 meters where there is a chair that is two point seventy-five meters off the ground.

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

  • blobby green style avatar for user John Thommen
    This is not correct. The path of the swing is an arc so at the point where it is parallel to the support pole it would closer to the ground than at the point of full swing which is 2.75 meters. To make this example correct the 2,75 meters needs to be applied to the point where the swing is parallel to the supporting pole.
    (0 votes)
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  • blobby green style avatar for user veroaghe
    Shouldn't we take in account the height at which the MIB shoots its laser. I'm guessing it would be somewhere from his shoulder. Maybe the answer wouldn't differ that much but it might make it a little more challenging to figure out. You would even be able to calculate the height the agent is holding his gun at with stretched arms when you know the angle he's keeping his arms at, his arm length and the length from his shoulders to the ground.
    (6 votes)
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    • duskpin ultimate style avatar for user AHsciencegirl
      Good point, let's estimate :D.

      Men In Black are generally rather tall so it is fair to estimate the man is about two meters tall. The average arm length of an adult human is ~25 inches which equates to about 0.6 meters. If we assume that the man holds his arms directly above his head (not technically realistic but a fair assumption) then we can estimate the height of the LASER to be about 2.5 meters. If we subtract 2.5 from 324 we get 321.5. Arctan(321.5/54) = 80.465.

      That deviates from the ground angle by only 0.09%, so it probably wouldn't affect how he aimed the laser.

      If we assume he shot from shoulder height with his arms straight out, then that would be arctan(322/53.5) = ~80.567 which deviates from the ground angle by only 0.04%.
      (13 votes)
  • piceratops ultimate style avatar for user Aditya Lagoo
    What is the value of sine, cosine, and tangent?
    (3 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      The Sine, Cosine, and Tangent are three different functions. They do not have a value outright, it would be like trying to ask what the value of f(x) = x + 1 is. The trig functions give outputs in terms of the ratios of two sides of a triangle when we feed them the input of an angle measure.
      Sine outputs the ratio of opposite to hypotenuse
      Cosine outputs the ratio of adjacent to hypotenuse
      Tangent outputs the ratio of opposite to adjacent
      (10 votes)
  • piceratops seedling style avatar for user Raghunandan wable
    in question 1.1 the given answer is approx 5.44 my calculator is giving 0.91 as an answer even in degrees mode
    (0 votes)
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    • leaf orange style avatar for user mathslacker2016
      The whole trick to the question is that zero radians is an answer, and if you look closely, you see that no other answer other than 0*pi/10 will get you there, if zero is a possible answer for n. But then since sin(u) must be 20x, then you must still find an answer for every negative pi and positive pi in addition to finding the answer that will get you to zero, which is one of the possible answers. (And remember "every possible solution" must be included, including zero). So you need to pick the two answers that would get you to zero radians, plus positive and minus every other pi. It's a brutal question because the zero radians thing is a hard thing to remember, amidst so many answers that have every answer, but just happen to exclude zero radians. Harsh.
      (10 votes)
  • leafers seedling style avatar for user egeegeg
    when working out the inverse trig, is the bigger number always on the bottom?
    (2 votes)
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  • blobby green style avatar for user 91097027
    do i have to be specific
    (2 votes)
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  • blobby green style avatar for user Trevor Amrhannah Davis
    My problem is that I do not know which one is adjacent and opposite you the one closest to the angle is adjacent but if it doesn't show the angle then how am I supposed to know which one.
    (3 votes)
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  • piceratops seed style avatar for user claire-ann manning
    how do i know to use sine cosine or tangent? im taking trig and i need a good grade having to teach myself the class :( so HELP SOS!
    (1 vote)
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    • blobby green style avatar for user Thien D Ho
      Look at the formula of each one of them. They all different. So, it depend on what you look for, in order apply the properly formula. One example is: sin of 1 angle (in the right triangle) = opposite over hypotenuse. So, if you know sin of that angle, and you also know the length of the opposite. Then apply the formula of sin, you can find hypotenuse.
      (7 votes)
  • duskpin ultimate style avatar for user Nadia Richardson
    I am so confused...I try my best but I still don't get it . I need someone to Break it down further for me? I never not understand math but this one really has me stuck.Thank you.
    (4 votes)
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  • blobby green style avatar for user seonyeongs
    when solving for an angle why does cos have a -1 on top?
    (2 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      Trig functions like cos^-1(x) are called inverse trig functions. THey are the inverse functions of the normal trig functions. Where cos(x) would take in an angle and output a ratio of side lengths, cos^-1(x) takes in the ratio of adjacent/hypotenuse and gives you an angle, which is why we use it when solving for unknown angles.
      Note that cos^-1(x), (cos(x))^-1, and cos(x^(-1)) give three completely separate results.
      (4 votes)