Solve the trigonometric functions of a right-angle triangle with the sohcahtoa calculator. You can also solve the area and angles of a right-angled triangle.

## What Is SOHCAHTOA?

It is the combination of three functions SOH, CAH, and TOA. The online soh cah toa calculator uses the below formulas to find the side length and angles. Therefore, let's examine these:

- SOH {Sin(θ)} = Perpendicular / Hypotenuse
- CAH {Cos(θ)} = Base / Hypotenuse
- TOA {Tan(θ)} = Perpendicular / Base

## Using SOHCAHTOA for Trig Ratios:

The mnemonic sohcahtoa is used to remember which function is used in what circumstances and is also used to find the trigonometric ratios of an acute angle of a triangle.

No doubt it's like a challenging task. Let us look at the sides of the triangle below and clarify this with the help of a diagram.

### Perpendicular ( Opposite ):

The side of a right triangle makes an angle of 90 degrees with the base.

### Base ( Adjacent ):

The side on which the right triangle stands. It is connected to the acute angle and opposite.

### Hypotenuse ( Diagonal ):

The side that is always the opposite of the right angle is known as the hypotenuse.

Suppose you have two sides and one is missing. In this case, you can take the help of a __Pythagorean theorem calculator__.

## How To Calculate The SOHCAHTOA?

Trigonometric functions took some basic angle measurements. Therefore, some popular measurements of sohcahtoa measurements are as follows:

$$ {\displaystyle \sin \theta } $$ | $$ {\displaystyle \cos \theta } $$ | $$ {\displaystyle \tan \theta =\sin \theta {\Big /}\cos \theta } $$ | |

0° = 0 radians | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {blue}{0}} }}{2}}=\;\;0} $$ | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {red}{4}} }}{2}}=\;\;1} $$ | $$ {\displaystyle \;\;0\;\;{\Big /}\;\;1\;\;=\;\;0} $$ |

30° = π/6 radians | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {teal}{1}} }}{2}}=\;\,{\frac {1}{2}}} $$ | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {orange}{3}} }}{2}}} $$ | $$ {\displaystyle \;\,{\frac {1}{2}}\;{\Big /}{\frac {\sqrt {3}}{2}}={\frac {1}{\sqrt {3}}}} $$ |

45° = π/4 radians | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {green}{2}} }}{2}}={\frac {1}{\sqrt {2}}}} $$ | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {green}{2}} }}{2}}={\frac {1}{\sqrt {2}}}} $$ | $$ {\displaystyle {\frac {1}{\sqrt {2}}}{\Big /}{\frac {1}{\sqrt {2}}}=\;\;1} $$ |

60° = π/3 radians | $${\displaystyle {\frac {\sqrt {\mathbf {\color {orange}{3}} }}{2}}} $$ | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {teal}{1}} }}{2}}=\;{\frac {1}{2}}} $$ | $$ {\displaystyle {\frac {\sqrt {3}}{2}}{\Big /}\;{\frac {1}{2}}\;\,={\sqrt {3}}} $$ |

90° = π/2 radians | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {red}{4}} }}{2}}=\;\,1} $$ | $$ {\displaystyle {\frac {\sqrt {\mathbf {\color {blue}{0}} }}{2}}=\;\,0} $$ | $$ {\displaystyle \;\;1\;\;{\Big /}\;\;0\;\;=} Undefined $$ |

## Measurements of SOHCAHTOA Ratios:

In calculus and analytic geometry, these ratios have a wide range of uses. So which are these and how these are calculated? Read on!

### Basic Ratios

#### 1. Sine:

Sine = Perpendicular / Hypotenuse

#### 2. Cosine:

Cosine = Base / Hypotenuse

#### 3. Tangent:

Tangent = Perpendicular / Base

#### 4. Secant:

Secant = Hypotenuse / Perpendicular

#### 5. Cosecant:

Cosecant = Hypotenuse / Base

#### 6. Cotangent:

Cotangent = Base / Perpendicular

### Inverse Ratios:

#### 1. Arcsine:

Arcsine = sin^{-1}x

#### 2. Arccosine:

Arccosine = cos^{-1}x

#### 3. Arctangent:

Arctangent = tan^{-1}x

#### 4. Arcsecant:

Arcsecant = sec^{-1}x

#### 5. Arccosecant:

Arccosecant = cosec^{-1}x

#### 6. Arcotangent:

Arcotangent = cot^{-1}x

Sohcahtoa triangle calculator assists you to determine all these ratios in a fraction of a second and enables you to get fast calculations.

## Example:

Suppose a right-angle triangle whose perpendicular (a) is equal to the 4cm and Angle β is 10 degrees. Calculate other ratios with the help of the sohcahtoa calculator with steps.

### Solution:

So for the final results use the sohcahtoa to find sides b, c, and after that angle and __area__.

**Find b;**

$$ b=\sqrt{(c^2-a^2)} $$

$$ b=\sqrt{(4.061706447543^2-4^2)} $$

$$ b=\sqrt{(16.497459266012-16)} $$

$$ b=0.70530792283386 $$

**Find c; **

$$ c=\sqrt{(a^2+b^2)} $$

$$ c=\sqrt{(4^2+0.70530792283386^2)} $$

$$ c=\sqrt{(16+0.49745926601221)} $$

$$ c=4.061706447543 $$

Certainly, you must demand how to find angles using sohcahtoa. Look further calculations:

**Find Angle α;**

$$ \alpha= arctan(\dfrac{a}{b}) $$

$$ \alpha= arctan(\dfrac{4}{0.70530792283386}) $$

$$ \alpha= arctan(5.6712818196177) $$

$$ α=1.3962634015955 $$

**Find Area;**

$$ area=\dfrac{a*b}{2} $$

$$ area=\dfrac{4*0.70530792283386}{2} $$

$$ area= 1.4106158456677 $$

## Working of SOHCAHTOA Calculator:

The mnemonic sohcahtoa to find angle and to remember the trig ratios use the Soh cah toa calculator that takes into service the following points:

**Input:**

- Put two out of six values in the tool
- Tap
**“Calculate”**

**Output:**

Our sohcahtoa solver gives you the following answers.

- Sides of trigonometric ratios
- Find Angle α
- Area
- Step-by-step calculations

## Other Mnemonics for Remembering Triangle Trig Ratios:

Another collective phrase commonly used to recall the trig functions is seen below:

**“Oscar Had A Heap Of Apples”**

This implies that:

- Sin(θ) = Oscar / Had
- Cos(θ) = A / Heap
- Tan(θ) = Of / Apples

## FAQs:

### Is Sohcahtoa The Same As The Pythagorean Theorem?

In order to find the missing side of the right angle triangle we use the Pythagorean theorem and in order to memorize the trigonometric functions easily we use the sohcahtoa.

## References:

From the source **Wikipedia:** Mnemonics in trigonometry, sohcahtoa how to find angle, Hexagon chart.