Friday, 9 Aug 2024

# How To Use Sohcahtoa?

Table of Contents

# How To Use Sohcahtoa?

Sohcahtoa: SOHCAHTOA is a mnemonic device that is used in mathematics to remember the definitions of the three most common trigonometric functions. This lesson will explain each one and give examples, and you’ll have the opportunity to take a quiz at the end to solidify what you’ve learned.

Sine, cosine, and tangent are the three main functions in trigonometry. They’re all based on ratios obtained from a right triangle. Before we can discuss what ratios work for which function, we need to label the right triangle.

The ratios that allow you to determine the sine, cosine, and tangent of a right triangle are: Opposite is the side opposite the angle in question, adjacent is the side next to the angle in question, and the hypotenuse is the longest side of a right triangle. The hypotenuse is always opposite the right angle.

• The sine of an angle is equal to the side opposite the angle divided by the hypotenuse.
• The cosine of an angle is equal to the side adjacent to the angle divided by the hypotenuse.
• The tangent of an angle is equal to the side opposite the angle divided by the side adjacent to the angle.

SOHCAHTOA. A way of remembering how to compute the sine, cosine, and tangent of an angle. SOH stands for Sine equals Opposite over Hypotenuse. CAH stands for Cosine equals Adjacent over Hypotenuse. TOA stands for Tangent equals Opposite over Adjacent.

SOH = Sine is Opposite over Hypotenuse. CAH = Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.

SineCosine, and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. Before getting stuck into the functions, it helps to give a name to each side of a right triangle: … “Adjacent” is adjacent (next to) to the angle θ “Hypotenuse” is the long one.

## Sohcahtoa Calculator

Sohca…what? Just an easy way to remember how Sine, Cosine, and Tangent work:

 Soh… Sine = Opposite / Hypotenuse …cah… Cosine = Adjacent / Hypotenuse …toa Tangent = Opposite / Adjacent

### Right Triangle

OK, let’s see what this is all about.

Firstly, the names Opposite, Adjacent, and Hypotenuse come from the right triangle:

• “Opposite” is opposite to the angle θ
• “Adjacent” is adjacent (next to) to the angle θ
• “Hypotenuse” is the long one

Adjacent is always next to the angle

And Opposite is opposite the angle

### Sine, Cosine and Tangent

And SineCosine, and Tangent are the three main functions in trigonometry.

They are often shortened to sincos, and tan.

The calculation is simply one side of a right-angled triangle divided by another side … we just have to know which sides, and that is where “sohcahtoa” helps.

For a triangle with an angle θ, the functions are calculated this way:

 Sine: soh sin(θ) = opposite / hypotenuse Cosine: cah cos(θ) = adjacent / hypotenuse Tangent: toa tan(θ) = opposite / adjacent

### Example: what are the sine, cosine, and tangent of 30°?

A 30° triangle has a hypotenuse (the long side) of length 2, an opposite side of length 1, and an adjacent side of √3, like this:

Now we know the lengths, we can calculate the functions:

 Sine soh… sin(30°) = 12 = 0.5 Cosine …cah… cos(30°) = 1.732…2 = 0.866… Tangent …toa tan(30°) = 11.732… = 0.577…

(get your calculator out and check them!)

## Sohcahtoa Worksheet

A way to remember the definitions of the three most common trigonometry functions: sin, cos, and tan.

Used as a memory aid for the definitions of the three common trigonometry functions sine, cosine, and tangent. When spoken it is usually pronounced a bit like “soaka towa”.

 SOH Sine = Opposite over Hypotenuse. sin x = O H CAH Cosine = Adjacent over Hypotenuse. cos x = A H TOA Tangent = Opposite over Adjacent. tan x = O A

## How To Use Sohcahtoa

These ratios can be difficult to remember. You might easily get confused and not remember which side goes where. SOHCAHTOA is a mnemonic device helpful for remembering what ratio goes with which function.

• SOH = Sine is Opposite over Hypotenuse
• CAH = Cosine is Adjacent over Hypotenuse
• TOA = Tangent is Opposite over Adjacent

With these properties, you can solve almost any problem related to finding either a side length or angle measure of a right triangle. SohCahToa can ensure that you won’t get them wrong.

## What Is Sohcahtoa

A way of remembering how to compute the sine, cosine, and tangent of an angle.

SOH stands for Sine equals Opposite over Hypotenuse.

CAH stands for Cosine equals Adjacent over Hypotenuse.

TOA stands for Tangent equals Opposite over Adjacent.

## Sohcahtoa Examples

Example:

Find the values of sin θ, cos θ, and tan θ in the right triangle shown.

Answer:

sin θ = 3/5 = 0.6

cosθ = 4/5 = 0.8

tanθ = 3/4 = 0.75

## What is the formula for Soh CAH TOA?

The formula for SohCAHTOA is not a single formula but rather a mnemonic device used to remember the three most common trigonometric ratios: sine, cosine, and tangent.

In SohCAHTOA, “Sine” is represented by the ratio of the length of the opposite side to the length of the hypotenuse; “Cosine” is represented by the ratio of the length of the adjacent side to the length of the hypotenuse; and “Tangent” is represented by the ratio of the length of the opposite side to the length of the adjacent side.

To use SohCAHTOA to solve trigonometric problems, one must first identify the right triangle involved and determine which angle they want to solve for. Then, they can use the appropriate SohCAHTOA ratio to find the missing side or angle.

## Real-life Applications of SOHCAHTOA

### 1. Civil Engineering

In bridge design and analysis, SOHCAHTOA is used. To calculate the forces acting on various components, such as trusses and cables. For example, let’s consider a cable-stayed bridge with a cable attached to a tower at a specific angle. Knowing the angle and the vertical force acting on the cable, you can calculate the tension in the cable using the sine function:

Tension = Vertical force / sin(angle)

The structural analysis involves determining the internal forces in a structure. SOHCAHTOA is used to resolve these forces into their horizontal and vertical components. Similarly, in load-bearing calculations, SOHCAHTOA helps to distribute loads among various structural components, ensuring stability and safety.

### 2. Mechanical Engineering

In machine design and kinematics, engineers use SOHCAHTOA to analyze mechanisms such as gears, levers, and linkages. These components often involve angular motion and forces acting at various angles, requiring trigonometric ratios for calculations. For example, to find the torque exerted by force at a specific angle, engineers use the sine function:

Torque = Force × Distance × sin(angle)

Vibration analysis is another area where SOHCAHTOA comes into play, allowing engineers to decompose oscillating forces into their horizontal and vertical components.

### 3. Electrical Engineering

In AC circuit analysis, trigonometric ratios are used to represent the phase relationship between voltage and current. Phasor diagrams are employed to visualize this relationship, with the sine and cosine functions used to calculate the magnitude of the voltage and current waveforms. Signal processing, particularly Fourier analysis, also involves using trigonometric functions to represent and analyze time-varying signals.

### FAQs Of Sohcahtoa

#### What is Sohcahtoa?

Sohcahtoa is an acronym used to remember the trigonometric ratios of sine, cosine, and tangent.

#### How do you use Sohcahtoa to find the sine of an angle?

To find the sine of an angle, divide the length of the opposite side of the triangle by the length of the hypotenuse.

#### How do you use Sohcahtoa to find the cosine of an angle?

To find the cosine of an angle, divide the length of the adjacent side of the triangle by the length of the hypotenuse.

#### How do you use Sohcahtoa to find the tangent of an angle?

To find the tangent of an angle, divide the length of the opposite side of the triangle by the length of the adjacent side.

#### What is the inverse of Sohcahtoa?

The inverse of Sohcahtoa refers to the inverse trigonometric functions, which can be used to find the angles in a right triangle given the lengths of the sides.

#### What are the three other trigonometric ratios besides sine, cosine, and tangent?

The three other trigonometric ratios are cosecant, secant, and cotangent.

#### How do you use Sohcahtoa to find the angle in a right triangle?

To find the angle in a right triangle, you can use the inverse trigonometric functions, such as arcsine, arccosine, and arctangent.

#### What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

#### How is Sohcahtoa used in real life?

Sohcahtoa is used in many fields, such as architecture, engineering, physics, and astronomy, to solve problems involving right triangles.

#### What is the relationship between Sohcahtoa and the unit circle?

The trigonometric functions of sine and cosine can be represented by the coordinates of points on the unit circle, and the tangent function can be derived from the sine and cosine functions using Sohcahtoa.