Trigonometric Ratios - sohcahtoa with examples

This article will be about the trigonometric ratios, what are they, what does SOHCAHTOA means, and what is it used for, with examples.

Trigonometric ratios

The trigonometric ratios are equations that are used to find both sides and angles of a right triangle. It is important to understand that this trigonometric ratios can only be used in a triangle that has a 90 degrees angle.

This equations are also useful in other branches besides the triangles, for example, the trigonometric ratios are used to find distances between two points or to make operations between vectors, or even to work with vector magnitudes. The trigonometric ratios are really used in physics because it is really simple to find right triangles in different situations, this is why it is so important to learn how to work with triangles and the trigonometric ratios. For example in a velocity vector, when we know the magnitude and direction of a velocity vector we can decompose that vector where the hypotenuse will be the vector itself, while the legs are the “x” and “y” component of the vector.

Soh cah toa and the trigonometric ratios

Names of the sides of a triangle

Usually it is easy to distinguish a right triangle because most of the time it is positioned on a way where we have a total horizontal and vertical side, but we always have to double check if the triangle that we have is a right triangle, because sometimes the triangle is rotated of a way where it is not easy to differentiate whether it is or not a right triangle.

That said, a triangle is a three-sided shape, but in a right triangle every side has a name, the longest side of a right triangle is called the “hypotenuse”, while the other 2 sides are called ”legs” and this two “legs” form the 90° angle. Each leg can be named of two ways, the “opposite leg” and the “adjacent leg” and this sub-names will be depending on the angle taken as reference (the angles that are not 90° are the ones that have an adjacent and an opposite legs). The side that is next to an angle will be the adjacent leg, while the side that is in front of an angle will be the opposite leg. It is important to know which angle is taken as reference because both of the legs are adjacent of an angle, and opposite of the other angle. The hypotenuse will always be the longest side and this will be right in front of the 90° angle that normally will be marked with a square.

Right triangles in the trigonometric ratios

Formulas of the trigonometric ratios

This formulas o equations are relations between the sides and the angles of a triangle, and this relations are called sino, cosino and tangent of the angle.

  • sin(Θ) =
    Opposite leg/Hypotenuse
  • cos(Θ) =
    Adyacent leg/Hypotenuse
  • tan(Θ) =
    Opposite leg/Adjacent leg

SOHCAHTOA

Learn each formula could be a little complicated, but there is a way to remember the tree of them in only one word, and this word is SOHCAHTOA. This word is formed by the first letter of the variables of the equations, as follows: Sino = Opposite / Hipotenuse, Cosin = Adjacent / Hipotenuse and Tangent = Opposite / Adjacent. As you can see, it is easier to learn a word than 3 different equations, and when we remember this word we only have to break down the word to remember each equation.

SOHCAHTOA

Trigonometric ratios examples

Example 1: Find the angle "b" of the following triangle, if we know that the leg "A" is 3 meter long and the leg "B" is 6 meters long

Example 1

In each problem we have to evaluate the data we have so we can pick correctly the formula we have to use, in this problem we have 2 data, the Leg b = 6m, and the Leg a = 3m, so based on this, we are going to use the next formula.

  • tan(Θb) =
    Opposite/adjacent
  • First we clear the angel Θb
  • Θb = tan-1(
    Opposite/adjacent
    )
  • Θb = tan-1(
    6/3
    )
  • Θb = tan-1(2)
  • Θb = 63.43°

Example 2: We have a triangle that has a 45° angle, if we know that the opposite leg to this angle is 72 centimeters long, find the length of the hypotenuse and the adjacent leg.

Example 2

Data: Leg 1 = 72cm, Angle 1 = 45°

First we are going to find the length of the adjacent leg

  • First we write the formula
  • tan(Θ) =
    opposite/adjacent
  • Now we clear the adjacent leg
  • adjacent =
    opposite/tan(Θ)
  • adjacent =
    72cm/tan(45)
  • adjacent =
    72cm/1
  • adjacent = 72 cm

And finally we are going to find the legth of the hypotenuse, to find it we could use any of the tree formulas because we have both adjacent and opposite leg

  • sin(Θ) =
    hypotenuse/hipotenusa
  • And now we clear the hypotenuse and solve
  • hypotenuse =
    opposite/sin(Θ)
  • hypotenuse =
    72/0.71
  • hypotenuse = 101.4cm

Example 3: A triangle as a hypotenuse that is 43 meters long, and if it has a 31° angle, find with this data the 2 legs and the other angle.

Example 3

First we are going to find the adjacent leg to the 31° angle

Data: angle a = 31°, hypotenuse = 43 m

  • We find the adjacent leg
  • cos(Θ a) =
    adjacent/hypotenuse
  • We clear the adjacent leg
  • adjacent = cos(Θ a) * hypotenuse
  • adjacent = cos(31) * 43
  • adjacent = 0.86 * 43
  • adjacent = 36.98
  • We find the opposite leg to the 31° angle
  • sin(Θ a) =
    opposite/hypotenuse
  • we clear the opposite leg
  • opposite = sin(Θ a) * hypotenuse
  • opposite = sin(31) * 43
  • opposite = 0.52 * 43
  • opposite = 22.36

Now we only have to find the remaining angle, that could be easily found by subtracting to 180 the sum of 31 and 90, that would be the simple path, but we are going to find it by using the trigonometric ratios

  • We write the equation (we can use any of the formulas)
  • sin(Θ b) =
    opposite/hypotenuse
  • We clear the angle
  • Θ b = sin-1(
    opposite/hypotenuse
    )
  • Now the adjacent of the angle "a" will be the opposite of this one
  • Θ b = sin-1(
    36.98/43
    )
  • Θ b = sin-1(0.86)
  • Θb = 59.31

Example 4: Two men start from the same point, but one of them starts walking north, and the other one starts walking east, after 5 hours, both men decide to rest, the man that walked north walked 13km and the other one advanced 10km, define the distance between the 2 men at the end of the walk.

Example 4

Data: Leg A = 13km, Leg B =10km

As we can see in the previous image, when we draw the distances that each man have walked, we can see that these two paths form the legs of a right triangle, where the distance between the 2 men is the hypotenuse. There is no formula to find the hypotenuse when we have only the 2 legs, but we can find one of the angles and then calculate the length of the hypotenuse.

  • We find any of the angles, in this case we are going to find the opposite angle to the 13km leg.
  • tan(Θ a) =
    opposite/adjacent
  • We clear the "a" angle
  • Θ a = tan-1 (
    opuesto/adjacent
    )
  • Θ a = tan-1 (
    13/10
    )
  • Θ a = 52.43

Now with one solved angle, we can find the distance between the two (the hypotenuse)

  • We write the formula
  • sin(Θ a) =
    opposite/hypotenuse
  • Then we clear the hypotenuse (distance)
  • distance =
    opposite/sin(Θ a)
  • distance =
    13/sin(52.43)
  • distance = 16.40

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Pythagorean theorem Special right triangles