Sohcahtoa Calculator

SOHCAHTOA Calculator

The calculator helps to solve right-angled triangles using trigonometric ratios based on mnemonic SOHCAHTOA. It provides you with accurate values for each side and angle of a right triangle and shows complete steps for better understanding.

What Does It Actually Do?

The calculator requires only 2 inputs to calculate results:

Possible Input Combinations:

Input 1	Input 2	Valid?	Reason
Side a	Side b	✓	Use Pythagorean theorem → find hypotenuse and angles
Side a	Side c	✓	Use Pythagoras and trig ratios → find b and angles
Side b	Side c	✓	Same as above
Side a	Angle α	✓	Use SOH (sin α = a/c) → find c, then b and β
Side b	Angle α	✓	Use TOA (tan α = a/b) or CAH
Side c	Angle α	✓	Use SOH and CAH
Side a	Angle β	✓	Similar process, using sin β
Side b	Angle β	✓	Use tan β = b/a
Side c	Angle β	✓	Use sin β = b/c

Invalid Input Combinations:

Input 1	Input 2	Valid?	Reason
Angle α	Angle β	✗	Only gives full angle info; no side lengths can be determined
Angle α	None	✗	Angle alone doesn’t define triangle size
Angle β	None	✗	Same as above
Side a	None	✗	Only one side - can’t determine angles or other sides
Side b	None	✗	Same issue
Side c	None	✗	Need one more piece of information

Features:

Find missing angles (α and β)
Find missing sides (a, b, or c)
Identify relationships between the sides and angles
Apply the correct trig function (sine, cosine, or tangent) automatically
Show step-by-step calculations

What Is SOHCAHTOA?

It’s a mnemonic device that anyone can use to remember the three basic trigonometric ratios to determine the missing sides and angles of the right-angled triangle.

It is the combination of three functions: SOH, CAH, and TOA.

How to Calculate Trig Ratios Using SOHCAHTOA?

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.

SOH = Perpendicular / Hypotenuse
CAH = Base / Hypotenuse
TOA = Perpendicular / Base

sohcahtoa

SOHCAHTOA Table for Common Angles:

	sin(θ)	cos(θ)	tan(θ) = sin(θ) / cos(θ)
0° = 0 radians	√0/2 = 0	√4/2 = 1	0 / 1 = 0
30° = π/6 radians	√1/2 = 1/2	√3/2	(1/2) / (√3/2) = 1/√3
45° = π/4 radians	√2/2 = 1/√2	√2/2 = 1/√2	(1/√2) / (1/√2) = 1
60° = π/3 radians	√3/2	√1/2 = 1/2	(√3/2) / (1/2) = √3
90° = π/2 radians	√4/2 = 1	√0/2 = 0	1 / 0 = Undefined

Practical Example

Given:

A right-angled triangle
Perpendicular side a = 4 cm
Angle β = 10°

We are to find the other sides b, c, the remaining angle α, and the area using trigonometric identities and formulas.

Step 1: Find the Hypotenuse (c)

Using the sine ratio: sin(β) = a / c

c = a / sin(β) = 4 / sin(10°) ≈ 4 / 0.1736481777

c ≈ 4.0617 cm

Step 2: Find the Base (b)

Using the Pythagorean Theorem: b = √(c² - a²)

b = √(4.0617² - 4²) = √(16.4975 - 16)

b ≈ √0.4975 ≈ 0.7053 cm

Step 3: Verify Hypotenuse (Optional)

c = √(a² + b²) = √(4² + 0.7053²) = √(16 + 0.4975)

c ≈ √16.4975 ≈ 4.0617 cm

Step 4: Find Angle α

Using tangent ratio: tan(α) = a / b

α = arctan(4 / 0.7053) = arctan(5.6713)

α ≈ 79.0° (or 1.3963 radians)

Step 5: Find the Area of the Triangle

Area = ½ × a × b = ½ × 4 × 0.7053

Area ≈ 1.4106 cm²

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

FAQs

What Is The Other Mnemonics for Remembering the 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

Do I need to know the angle to use the SOHCAHTOA calculator?

Not always. You can use the calculator if you know at least one angle (besides the right angle) and one side, or if you know two sides. The SOH CAH TOA calculator will use trigonometric ratios to find the missing sides or angles, depending on what you enter.

What units does the calculator use (degrees or radians)?

By default, our SOHCAHTOA calculator uses both degrees and radians, since these are the most common formats in school problems and real-world use.

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.

Sohcahtoa Calculator