How to Use the Mnemonic 'SOHCAHTOA' in Trigonometry

By: Marie Look  | 
Right trangle with sides labeled in relation to a specific angle
SOHCAHTOA is an essential mnemonic when you start learning trig, whether you're trying to find the opposite leg, the adjacent leg or the measures of a triangle's acute angles. zizou7 / Shutterstock

The mnemonic device SOHCAHTOA helps budding mathematicians remember the trigonometric functions sine (sin), cosine (cos) and tangent (tan), which they need to solve for triangles' missing sides and angles.

But to really understand how this memory tool is useful, it's necessary to first refresh yourself on the basics of right-angled triangles, which are the main focus in trigonometry.


What Are Right Triangles?

A right-angled triangle, also known as a right triangle, has one angle that is exactly 90 degrees. The side opposite this right angle is the longest side, called the hypotenuse.

The other two sides are referred to as the opposite side and the adjacent side, depending on their relationship to the angle being measured or referenced.


Trigonometric Functions and Ratios

Mathematicians need a way to solve for and relate the angles of a triangle to the length of its sides. To do this, they use the fundamental math functions called trigonometric functions, which have applications across science, engineering and everyday life.

Defined based on the ratios of the side lengths in a right-angled triangle, the trigonometric functions are:


  • Sine (sin θ): The ratio of the length of the opposite side to the hypotenuse (O/H).
  • Cosine (cos θ): The ratio of the length of the adjacent side to the hypotenuse (A/H).
  • Tangent (tan θ): The ratio of the length of the opposite side to the adjacent side (O/A).

These ratios depend on the angle θ, an acute angle (less than 90 degrees) in the triangle.

What Does SOHCAHTOA Stand for?

Here’s how the mnemonic device SOHCAHTOA helps mathematicians and math students remember the trigonometric functions and ratios.

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent


Using the Trigonometric Ratios

Mathematicians have to calculate the unknown side lengths or angles in a right triangle all the time. To do this, they apply the trigonometric functions.

For example, if you know the value of angle θ, you can find the two sides of a right-angled triangle.


Trigonometric Function Example

Suppose you have a right triangle with:

  • Angle θ = 30 degrees
  • Adjacent side = a = 5

You want to find the length of the opposite side b.

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:

tan(θ) = opposite/adjacent

You know that:

  • θ = 30
  • Adjacent side a = 5

So, using the tangent function looks like this:

tan(30) = b/5

You know from trigonometric tables or by using a calculator that:

tan(30) = 1/(√3) ≈ 0.577


0.577 = b/5

Now, to find b:

b = 5 x 0.577
b ≈ 2.885

The length of the opposite side b is approximately 2.885 units.


Using the Inverse Trigonometric Ratios

If you know the lengths of two sides of a triangle, you can find the angle θ using the inverse trigonometric functions:

  • sin-1 (also called arcsin) for sine
  • cos-1 (also called arccos) for cosine
  • tan-1 (also called arctan) for tangent

Inverse Trigonometric Function Example

Suppose you have a right triangle with:


  • Adjacent side (to angle θ) = a = 5
  • Opposite side (to angle θ) = b = 3

You want to find the angle θ.

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:

tan(θ) = opposite/adjacent = b/a = 3/5

To find the angle θ, you can use one of the inverse trigonometric functions:

θ = tan-1(3/5)

Using a calculator, you can then determine:

θ ≈ tan-1(0.6) ≈ 30.96

In other words, the angle θ is approximately 30.96 degrees, making it also an acute angle.


Pythagorean Theorem

Another essential concept in trigonometry is the Pythagorean theorem, which Pythagoras, a philosopher in ancient Greece devised. It states:

a2 + b2 = c2

where a and b are the lengths of the legs (opposite and adjacent sides), and c is the hypotenuse.


The Pythagorean theorem is useful when you know the two sides of a right triangle and are trying to find a missing side.


Solving for Missing Sides and Angles

To solve for missing sides, you can rearrange the trigonometric equations. For example:

sin⁡θ = opposite/hypotenuse ⟹ opposite = sin⁡θ × hypotenuse

Using a calculator, you can compute the sine, cosine or tangent of the angle θ, and then multiply by the known side length to find the unknown side.


Example Calculation

Consider a right-angled triangle where the hypotenuse is 10 units, and the angle θ is 30 degrees. To find the opposite side, you would use the sine function:

sin⁡30° = opposite/10 ⟹ opposite = 10 × sin⁡30°

Since sin⁡30° =0.5:

opposite = 10 × 0.5
10 × 0.5 = 5 units


Trigonometric Identities

Trig identities are equations involving trigonometric functions that are true for every value of the variable. They include fundamental identities like this basic Pythagorean identity:

sin⁡2(θ) + cos2(θ) = 1

Other common identities include angle sum and difference identities, double angle identities, half-angle identities, reciprocal identities and co-function identities. These identities are useful when you're trying to simplify trigonometric expressions and solving equations.


Real-life Applications

It's easy to view trigonometry as just a theoretical concept, but that's far from the case. It actually has practical applications across many different fields, including engineering, physics and astronomy, and even in everyday problem-solving scenarios like navigation and construction.

Engineers use trigonometric ratios to analyze forces in structures like bridges and buildings, or to design mechanical systems in engines. With these ratios, physicists can determine the trajectory, range and height of objects in projectile motion, and astronomers can calculate the necessary angles for precisely aligning their telescopes to specific celestial objects or events.


We created this article in conjunction with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.