Sohcahtoa in Right Triangle Trigonometry

In Mathematics, there are different terms and definitions that are difficult to remember. Trigonometry is a part of Mathematics with many hard terms and those terms can be easily understood using the mnemonic phrase SOHCAHTOA. The mnemonic phrase SOHCAHTOA is a way of remembering three primary trigonometric ratios. Their names and abbreviations are sine (sin), cosine (cos), and tangent (tan). These three primary trigonometric ratios can be easily calculated using SOHCAHTOA.

In SOHCAHTOA,

• SOH stands for Sine equals opposite over hypotenuse

Sine = \[\frac{Opposite}{Hypotenuse}\]

• CAH stands for Cos equals adjacent over hypotenuse 

Cosine = \[\frac{Adjacent}{Hypotenuse}\]

• TOA stands for Tangent equals opposite over adjacent.

Tangent = \[\frac{Opposite}{Adjacent}\]

SOH CAH TOA Formula 

In a right-angled triangle, the formula for SOH CAH TOA is given as:

• SOH = Sine is opposite side over hypotenuse side.

• CAH = Cosine is adjacent side over hypotenuse side.

• TOA = Tangent is opposite side over adjacent side.

How To Do Trigonometry SOHCAHTOA?

In trigonometry, the right angle is a very special triangle with unique properties that are not found in the triangle. The mnemonic SOHCAHTOA is better to use here.

Let us see how to do trigonometry SOHCAHTOA in the right triangle.

In the right triangle ABC given above, angle C is a 90-degree angle and this angle is known as the right angle. 

The side ‘c’ that is the longest side and opposite to the right angle is known as the hypotenuse side.

The side ‘b’ that is next to the right angle is known as the adjacent side whereas the side ‘a’ that is opposite to the right angle is known as the opposite side.

For any angle, say angle A, there are three basic trigonometric functions: These are 

• The sine of A

• The cosine of A

• The tangent of A

Now, here we will consider the mnemonic Soh - Cah -Toa.

The sine A is the ratio of the opposite sides of angle A over hypotenuse.

The cosine of A is the ratio of the adjacent sides of angle A over hypotenuse.

The tangent of A is the ratio of the opposite side of angle A divided by the hypotenuse.

Let us understand the trigonometric SOHCAHTOA with an example.

Trigonometric SOHCAHTOA Example

Let's say, you have a right-angled triangle and one of the angles of a right-angled triangle is 30 degrees. What is the length of the opposite side if the length of the hypotenuse side of a right-angled triangle is 4 units?

As we know, the value of sin 30 degrees is 12. This means the ratio between the opposite side and hypotenuse side is 1: 2., sine Using the sine formula, we get:

Sin 30⁰ = Soh = \[\frac{\text{Opposite Side}}{\text{Hypotenuse Side Side}}\]

\[\frac{1}{2}\] = \[\frac{\text{Opposite Side}}{4}\]

2 х Opposite side = 4 

Opposite Side = \[\frac{4}{2}\] = 2

Therefore, the length of the opposite side is 2 units.

What is the SOHCAHTOA Used For?

SOHCAHTOA is a mnemonic way of remembering the three basic primary functions of trigonometry ratios that are used to find the unknown sides and angles of a right-angled triangle.

The term SOH, CAH, TOA in SOHCAHTOA is used to find the height of a building or the length of the shadow. With these angles, you can easily determine the angle that the shadow is cast from. 

Did You Know

SOHCAHTOA Stands for 

S - SINE

O - Opposite

H - Hypotenuse

C - Cosine

A - Adjacent

H - Hypotenuse

T - Tangent

O - Opposite

A - Adjacent 

SOHCAHTOA Examples With Solution

1. In the Right Triangle PQR Given Below, Find the Sine, Cosine, and Tangent of Angle θ.

Considering the given angle θ in the above right-angled triangle PQR, PQ is the opposite side of angle θ, PR is the adjacent side of angle θ, and QR is the hypotenuse side of angle θ.

Accordingly,

Sin θ = SOH = \[\frac{\text{Opposite Side}}{\text{Hypotenuse Side}}\] = \[\frac{5}{13}\]

Cos θ = CAH = \[\frac{\text{Adjacent Side}}{\text{Hypotenuse Side}}\] = \[\frac{12}{13}\]

Tan θ = TOA = \[\frac{\text{Opposite Side}}{\text{Adjacent Side}}\] = \[\frac{5}{12}\]

2. In the Figure Given Below, Find the Value of Sin B, Cos C, and Tan C 

Solution:

In the Δ ABD given above:

Using the Pythagorean theorem, we get

AB² = AD² + BD²

13² = AD² + 5²

AD² = 169 -  25

AD² = 144

AD = 12 

In the Δ ACD given above:

Using the Pythagorean theorem, we get

AC² = AD² + CD²

AC² = 12² + 16²

AC² = 144 +  256

AC² = 400

AC² = 20

AC = 12 

Accordingly,

Sin B = SOH = \[\frac{\text{Opposite Side}}{\text{Hypotenuse Side}}\] = \[\frac{AD}{AB}\] =  \[\frac{12}{13}\]

Cos C = CAH = \[\frac{\text{Adjacent Side}}{\text{Hypotenuse Side}}\] = \[\frac{CD}{AC}\] = \[\frac{16}{20}\] = \[\frac{4}{5}\]

Tan C = TOA = \[\frac{\text{Opposite Side}}{\text{Adjacent Side}}\] = \[\frac{AD}{CD}\] = \[\frac{12}{16}\] = \[\frac{3}{4}\]