SOHCAHTOA

Here you will learn how to use the ratio of the sides of a right triangle, SOHCAHTOA, and right triangle trigonometry to find unknown sides and angles in right angled triangles. You will learn how to label the sides of right-angled triangles and understand what each of the side ratios, sin, cos and tan are, and what their inverses are (\sin^{-1}, \cos^{-1}, \tan^{-1}) in order to problem solve.

Students first learn about right triangle trigonometry in geometry and expand their knowledge as they learn about the unit circle in algebra 2 and precalculus.

What is SOHCAHTOA?

SOHCAHTOA is a mnemonic device that gives you an easy way to remember the three main trigonometric ratios of the sides of a right triangle. The trig ratios are used to find the missing sides of the triangle (right triangle) and the missing acute angles.

The names of the three ratios are:

• Sine (Sin)
• Cosine (Cos)
• Tangent (Tan)

When writing the trigonometric ratios, you must write them with respect to one of the acute angles of the right triangle.

Let’s use angle at A as the acute angle by which to write the 3 main trigonometric ratios. Label this angle \theta. Since all the ratios will be written with respect to angle \theta, the side opposite \theta is called the ‘opposite’ side

The side that is part of angle \theta or next to angle \theta is called the ‘adjacent’ side.

The side across from the right angle is always the ‘hypotenuse’.

What happens if you use the angle at C instead of the angle at A to write the trigonometric ratios?

Notice how the adjacent side and the opposite side switch since the angle being used is the angle at C, labelled \theta.

Now that you understand how to label the sides of the right triangle, let’s write the ratios.

• The sine or sin ratio with respect to angle \theta is the opposite side length divided by the hypotenuse side length.
  
  \sin(\theta)=\cfrac{\text{opposite}}{\text{hypotenuse}}~(\mathrm{SOH})

• The cosine or cos ratio with respect to angle \theta is the adjacent side length divided by the hypotenuse side length.
  
  \cos(\theta)=\cfrac{\text{adjacent}}{\text{hypotenuse}}~(\mathrm{CAH})

• The tangent or tan ratio with respect to angle \theta is the opposite side length divided by the adjacent side length.
  
  \tan(\theta)=\cfrac{\text{opposite}}{\text{adiacent}}~(\mathrm{TOA})
  
  

You can also remember the abbreviation for SOHCAHTOA using triangles.

Sine, Cosine and Tangent are therefore trigonometric functions as the sine, cosine, or tangent of an angle \theta is equal to the division of the two respective sides O\div{H}, A\div{H} or O\div{A}.

Each SOHCAHTOA triangle can be used to determine the missing side, or missing angle of a right-angled triangle. Cover up the letter you want to find and calculate the product or division of the other two letters using one of the three formulae highlighted in the table.

What is SOHCAHTOA?

Common Core State Standards

How does this relate to high school math?

• High School Geometry – Similarity, Right Triangles, and Trigonometry HSG-SRT.C.6
  Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

• High School Geometry – Similarity, Right Triangles, and Trigonometry HSG-SRT.C.8
  Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

How to use SOHCAHTOA to find the missing side of a right triangle

In order to use SOHCAHTOA to find the missing side length of a right triangle:

1. Label the sides of the right triangle with respect to one of the acute angles.
2. Determine the trigonometric ratio to use and write the formula with the correct subject.
3. Substitute known values into the trig ratio to find the missing side.
4. Calculate the unknown side.

SOHCAHTOA – missing side examples

Example 1: find a missing side (sine function)

Calculate the side of the right triangle labeled x.

1. Label the sides of the right triangle with respect to one of the acute angles.

Using the 40^{\circ} angle, the side marked x is the opposite side and the side marked 8\mathrm{~cm} is the length of the hypotenuse because it is across from the right angle.

2Determine the trigonometric ratio to use and write the formula with the correct subject.

The hypotenuse (H) is known, and the opposite side (O) is what you want to find.

The adjacent side (A) is unknown and so not useful for this problem.

Looking at SOHCAHTOA, the sine function (S) is the ratio of the opposite side (O) and the hypotenuse (H).

You therefore use the sine function:

Using the formula triangle, cover up the opposite side (O) as you want to calculate this length. This gives the formula:

3Substitute known values into the formula to find the missing side.

As the opposite side O=x\mathrm{~cm}, the hypotenuse H=8\mathrm{~cm}, and \theta=40^{\circ},

4Calculate the unknown side.

Use your calculator to find the solution

Always check that the answer is reasonable:

As 5.14\mathrm{~cm} is shorter than the length of the hypotenuse (8\mathrm{~cm}), the answer is reasonable.

Example 2: find a missing side (cosine)

Calculate the side of the right triangle labeled x.

Label the sides of the right triangle with respect to one of the acute angles.

Using the angle labeled 36^{\circ}, the side marked x is the adjacent side (A) because it is next to the angle and the side marked as 15\mathrm{~cm} is the hypotenuse (H) because it is across from the right angle.

Determine the trigonometric ratio to use and write the formula with the correct subject.

The hypotenuse (H) is known, and the adjacent side (A) is what you want to find.

The opposite side (O) is unknown and so not useful for this problem.

Looking at SOHCAHTOA, the cosine function (C) is the ratio of the adjacent side (A) and the hypotenuse (H).

You therefore use the cosine function:

\cos(\theta)= \cfrac{A}{H}

Using the formula triangle, cover up the adjacent side (A) as you want to calculate this length. This gives the formula:

A=H\times\cos(\theta)

Calculate the unknown side.

Use your calculator to find the solution

x=15\times\cos(36)=12.13\mathrm{~cm}\text{ (2dp)}

Always check that the answer is reasonable:

The value of the side being 12.13\mathrm{~cm} makes sense with the sketched out triangle

Example 3: find a missing side (tangent)

Calculate the side labeled x .

Label the sides of the right triangle with respect to one of the acute angles.

Using the angle marked 38^{\circ} , the side marked as x is the adjacent side because it is next to the angle, and the side marked as 10\mathrm{~cm} is the opposite side because it is opposite the angle.

Determine the trigonometric ratio to use and write the formula with the correct subject.

The opposite side (O) is known, and the adjacent side (A) is what you want to find.

The hypotenuse (H) is unknown and so not useful for this problem.

Looking at SOHCAHTOA, the tangent function (T) is the ratio of the opposite side (O) and the adjacent side (A).

You therefore use the tangent function:

\tan(\theta)=\cfrac{O}{A}

Using the formula triangle, cover up the adjacent side (A) as you want to calculate this length. This gives the formula:

A=\cfrac{O}{\tan(\theta)}

Calculate the unknown side.

Use your calculator to find the solution

x=\cfrac{10}{\tan(38)}=12.80\mathrm{~cm}\text{ (2dp)}

Always check that the answer is reasonable:

The longer side is opposite the larger angle. As the sum of angles in a triangle is 180^{\circ}, the other angle in the triangle is 52^{\circ}. The longer side is opposite the larger angle so this solution is reasonable.

How to use SOHCAHTOA to find the unknown angles of right triangles

In order to find missing angles using SOHCAHTOA:

1. Label the sides of the right triangle with respect to one of the acute angles.
2. Determine the trigonometric ratio to use and write the inverse trig formula with \bf{\theta} as the subject.
3. Substitute the two known side lengths into the formula.
4. Calculate the unknown angle.

SOHCAHTOA – unknown angle examples

Example 4: find a missing angle

Calculate the angle labeled \theta.

Label the sides of the right triangle with respect to one of the acute angles.

The side marked as 5\mathrm{~cm} is opposite the angle, so this is the opposite side.

The side marked as 9\mathrm{~cm} is the hypotenuse because it is across from the right angle.

Determine the trigonometric ratio to use and write the inverse trig formula with \bf{\theta} as the subject.

Looking at SOHCAHTOA, the sine function (S) is the ratio of the opposite side (O) and the hypotenuse (H).

As you are finding the angle, use the inverse sine function:

\theta=\sin^{-1}\left(\cfrac{O}{H}\right)

Note, you can use the formula for the sine function (\sin(\theta)=\cfrac{O}{H}), but you will need to rearrange it by finding the inverse sine of each side to obtain \theta=\sin^{-1}\left(\cfrac{O}{H}\right).

Calculate the unknown angle.

\theta=\sin^{-1}\left(\cfrac{5}{9}\right)=33.75^{\circ}\text{ (2dp)}

Remember we can get \sin^{-1}( on the calculator by pressing SHIFT or 2 nd and then sin. Don’t forget to close the bracket as well.

Always check that the answer is sensible:

You can estimate from the sketch that \theta is an acute angle.

As 33.7^{\circ} is less than 90^{\circ}, the answer is reasonable.

Example 5: find a missing angle

Calculate the angle labeled \theta .

Label the sides of the right triangle with respect to one of the acute angles.

The side marked as 9\mathrm{~cm} is next to the angle, so this is the adjacent side.

The side marked as 20\mathrm{~cm} is the hypotenuse because it is across from the right angle.

Calculate the unknown angle.

\theta=\cos^{-1}\left(\cfrac{9}{20}\right)=63.26^{\circ}\text{ (2dp)}

Remember we can get \cos^{-1}( on the calculator by pressing SHIFT or 2 nd and then cos. Don’t forget to close the bracket as well.

Always check that the answer is sensible:

You can estimate from the sketch that \theta is an acute angle.

As 63.3^{\circ} is less than 90^{\circ}, the answer is reasonable.

Example 6: find a missing angle

Calculate the angle labeled \theta .

Label the sides of the right triangle with respect to one of the acute angles.

The side marked as 15\mathrm{~cm} is opposite the angle \theta , so this is the opposite side.

The side marked as 7\mathrm{~cm} is the adjacent side because it is next to the angle.

Determine the trigonometric ratio to use and write the inverse trig formula with \bf{\theta} as the subject.

Looking at SOHCAHTOA, the tangent function (T) is the ratio of the opposite side (O) and the adjacent side (A).

As you are finding the angle, use the inverse tangent function:

\theta=\tan^{-1}\left(\cfrac{O}{A}\right)

Note, you can use the formula for the tangent function (\tan(\theta)=\cfrac{O}{A}), but you will need to rearrange it by finding the inverse tangent of each side to obtain \theta=\tan^{-1}\left(\cfrac{O}{A}\right).

Calculate the unknown angle.

\theta=\tan^{-1}\left(\cfrac{15}{7}\right)=64.98^{\circ}\text{ (2dp)}

Remember we can get \tan^{-1}( on the calculator by pressing SHIFT or 2 nd and then tan. Don’t forget to close the bracket as well.

Always check that the answer is sensible:

Draw an accurate sketch of the triangle and measure the angle using a protractor.

Teaching tips for SOHCAHTOA

• Spend time looking at different right-angle triangles and labeling sides in relation to the angle so that students practice working out which trigonometric function to use sine, cosine or tangent.

• Use revision cards and practice questions including real life word problems.

• Solve real life problems that require them to apply the trig ratios out of the classroom: find the height of a tree / building, estimate the length of a football pitch, work out the angle of elevation from a horizontal distance to a window / sign etc.

Easy mistakes to make

• Using the incorrect mode on the calculator
  When calculating the side lengths or angle measurements using one of the trig ratios, be sure the calculator is in ‘degree’ mode, not ‘radian’ mode. Radians are a different unit of angle measurement which will be explored in a precalculus class.

• Thinking the hypotenuse is the opposite side
  When working with a right triangle, the hypotenuse is always the side across from the right angle. The opposite side is across from the acute angle being used to write the ratio.
  
  

• Using the right angle when writing a trig ratio
  To calculate side lengths or angle measurements using the trig ratios, the angle being used to write the ratio must be one of the acute angles.
  
  For example, if trying to calculate the adjacent side:
  
  \tan(90)=\cfrac{24}{\text{adjacent}} \rightarrow This is an incorrect ratio.
  
  You cannot use the 90^{\circ} angle when writing the ratio and 24 is NOT the opposite side.
  
  The hypotenuse is always the side across from the right angle and the longest side.
  
  The correct ratio is: \cos(57)=\cfrac{\text{adjacent}}{24} OR \sin(57)=\cfrac{\text{opposite}}{24}
  
  

Related trigonometry lessons

• Trigonometry
• Trig functions
• How to find the exact value of a trig function
• Sin cos tan
• Law of sines
• Trig formula for area of a triangle
• Trig formulas
• Trig identities
• Law of cosines
• Trig tables

The next lessons are

• Circle math
• Angles of a circle
• Circle theorems

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