## Q. What is SohCahToa ? / Define SohCahToa.

**SohCahToa **definition : **SohCahToa** is a very useful mnemonic to calculate trigonometric functions or to define the trigonometric functions. **SohCahToa** help us find side lengths of a right angle.

There are three parts in the **SohCahToa** mnemonic.Let’s break the **SohCahToa** mnemonic.

1. **Soh** stands for, **S**ine**θ** = **O**pposite / **H**ypotenuse

2.** Cah** stands for,

**C**os

**θ**=

**A**djacent /

**H**ypotenuse

3.

**stands for,**

**Toa****T**an

**θ**=

**O**pposite /

**A**djacent

In a sentence,

**SohCahToa**mnemonic is used to remember sine,cosine and tangent functions.

### Q. How to Remember SohCahToa mnemonic easily ?

To remember **SohCahToa mnemonic** easily you can memorize the following funny sentences.

1.”**S**tudying **O**ur **H**omework **C**an **A**lways **H**elp **T**o **O**btain **A**chievement.”(SohCahToa)

2. “Oscar Has A Hold On Angie.”(OH AH OA)

3. “Tommy On A Ship Of His Caught A Herring” (probably more common in Great Britain than the United States).(ToaSohCah)

## Uses of **SohCahToa** in triangles

**Q**: When to use **SohCahToa**?

**A**nswer: You can use **SohCahToa** only on right triangles.You can use **SohCahToa** on various conditions in a Triangle. When you are given a right triangle,

- where one of the side lengths and angle
**θ**is given and you are asked to find the sides length - where two of the side lengths are given and you are asked to find the third side.
- where two of the side lengths are given and you are asked to find the angle
**θ**.

**SohCahToa** finding sides in a right angle.

Let us first recognize the the sides of a right angle,

- Opposite
- Adjacent
- Hypotenuse

The **Opposite** of any right angle is opposite of the angle θ , the **Hypotenuse** of any right angle is opposite of the right angle which is the longest side and the **Adjacent **of any right angle is the side where both angle θ and right angle are adjacent.

**Must Remember**: Trigonometric ratios(sine,cosine,tangent) will always hold true for right triangles regardless of size.

**SohCahToa** questions

**Q**: Is **SohCahToa** only for right triangles ?

**A**: Yes, **SohCahToa** is only applicable to right triangles. The trigonometric ratios won’t work for oblique triangle.

**Q**: Hypotenuse of a right triangle definition.

**Definition of Hypotenuse**: **Hypotenuse** of any right angle is opposite of the right angle or 90 degree angle, and is the longest side.

**Q**: **Adjacent **side of a right triangle definition.

** Definition of Adjacent:** The

**Adjacent**of any right angle is the side where both angle θ and right angle or 90 degree angle are adjacent.

## **SohCahToa** without angle problems

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

Answer: [the **Hypotenuse** of any right angle is opposite of the right angle , so **Hypotenuse** = 6,

The **Opposite** of any right angle is opposite of the angle θ, so ** Opposite** = 3, and

**Adjacent**= 2]

**S**ine**θ** = **O**pposite / **H**ypotenuse = 3/6 = 1/2**C**os**θ** = **A**djacent / **H**ypotenuse = 2/6 = 1/3**T**an**θ** = **O**pposite / **A**djacent = 3/2

**SohCahToa** Inverse Trigonometric Functions

### Inverse of sine function

1. if sineθ = x ,then => sin^{-1}(sineθ) = sin^{-1}(x) : so, θ = sin^{-1}(x) [Here ~~sin~~^{-1}(~~sin~~θ) = θ];

sineθ = Opposite / Hypotenuse

=>sin^{-1}(sineθ) = sin^{-1}(Opposite / Hypotenuse)

=>θ = sin^{-1}(Opposite / Hypotenuse)

### Inverse of cosine function

2. if cosθ = x ,then => cos^{-1}(cosθ) = cos^{-1}(x) : so , θ = cos^{-1}(x) [Here ~~cos~~^{-1}(~~cos~~θ) = θ];

cosθ = **A**djacent / **H**ypotenuse

=>cos^{-1}(cosθ) = cos^{-1}(**A**djacent / **H**ypotenuse)

=>**θ = cos ^{-1}(Adjacent / Hypotenuse)**

### Inverse of tangent function

3. if tanθ = x ,then => tan^{-1}(cosθ) = tan^{-1}(x) : so , θ = tan^{-1}(x) [Here ~~tan~~^{-1}(~~tan~~θ) = θ];

tanθ = **O**pposite / **A**djacent

=>tan^{-1}(tanθ) = tan^{-1}(** Opposite / Adjacent**)

=>

**θ =**

**t**an^{-1}(**O**pposite /**A**djacent)## SohCahToa Angles

You can use SohCahToa to find the angle(**θ**) of the right angle.Let me give an example of inverse trigonometric functions problems.

### Q. How to find the angle of a triangle given 2 sides ?

Ans : [**Hypotenuse** of any right angle is opposite of the right angle or 90 degree angle, and is the longest side.**Hypotenuse**= 4,

The **Opposite** of any right angle is opposite of the angle θ, so ** Opposite** = 2, and

**Adjacent**= ?]

The hypotenuse and opposite is given in the fig.So we can find the angle using sine function.

sineθ = Opposite / Hypotenuse

=> sin^{-1}(sineθ) = sin^{-1}(Opposite / Hypotenuse)

=> θ = sin^{-1}(Opposite / Hypotenuse)

=> θ = sin^{-1}(2 / 4)

=> θ = sin^{-1}(1 /2)

=> θ = sin^{-1}(sin 30°)

=> θ = 30°

### Q. Using Sohcahtoa find missing angle .

Ans : [The **Adjacent** of any right angle is the side where both angle θ and right angle or 90 degree angle are adjacent , so

**= 2,**

**A**djacentThe

**Opposite**of any right angle is opposite of the angle θ, so

**= 2, and**

**Opposite****= ?]**

**Hypotenuse**The Adjacent and opposite is given in the fig.So we can find the angle using tan function.

tanθ = **O**pposite / **A**djacent

=> tan^{-1}(tanθ) = tan^{-1}(Opposite / Adjacent)

=> θ = tan^{-1}(Opposite / Adjacent)

=> θ = tan^{-1}(2 / 2)

=> θ = tan^{-1}(1 )

=> θ = tan^{-1}(tan 45°)

=> θ = 45°

### Q. Using Sohcahtoa find non right angle .

Ans : [The **Adjacent** of any right angle is the side where both angle θ and right angle or 90 degree angle are adjacent , so

**= 3,**

**A**djacent**Hypotenuse**of any right angle is opposite of the right angle or 90 degree angle, and is the longest side.

**Hypotenuse**= 6, and

**= ?]**

**opposite**The **Hypotenuse** and ** Adjacent** is given in the fig.So we can find the angle using cos function.

cosθ = **A**djacent / **H**ypotenuse

=> cos^{-1}(cosθ) = cos^{-1}(**A**djacent / **H**ypotenuse)

=> θ = cos^{-1}(**A**djacent / **H**ypotenuse)

=> θ = cos^{-1}(3 / 6)

=> θ = cos^{-1}(1/2 )

=> θ = cos^{-1}(cos 60°)

=> θ = 60°

## SohCahToa with one side and one angle.

Q.One side and one angle is given find the Adjacent?

Ans : [**Hypotenuse** of any right angle is opposite of the right angle or 90 degree angle, and is the longest side.**Hypotenuse**= 6, and the angle θ = 60° ]

The **Hypotenuse** and ** Adjacent** is given in the fig.So we can find the adjacent using cos function.

cosθ = **A**djacent / **H**ypotenuse

=> cos 60° = **A**djacent / 6

=> 1/2 = **A**djacent / 6)

=> **A**djacent = (1 x 6 ) / 2

=> **A**djacent = 3

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