Welcome to the professional SOH CAH TOA Calculator. This tool is designed to solve for all missing sides and angles of a right-angled triangle given any two independent variables (two sides, or one side and one non-90-degree angle). Enter the known values below to solve the triangle instantly.
SOH CAH TOA Calculator
SOH CAH TOA Calculator Formula:
SOH CAH TOA is an acronym for the three primary trigonometric ratios based on a right-angled triangle:
CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
Pythagorean Theorem: $\text{Opposite}^2 + \text{Adjacent}^2 = \text{Hypotenuse}^2$
Formula Source: For a comprehensive review of these fundamental principles, refer to these highly authoritative resources:
Variables:
The calculation is based on an acute Angle A and the sides relative to it in a right triangle:
- Angle A (Degrees): One of the two non-90 degree angles.
- Side Opposite (o): The side directly across from Angle A.
- Side Adjacent (a): The side next to Angle A, which is not the hypotenuse.
- Hypotenuse (h): The longest side, opposite the 90-degree angle.
Related Calculators:
- Pythagorean Theorem Solver
- Sine Rule Triangle Calculator
- Cosine Rule Triangle Calculator
- Triangle Area Calculator
What is SOH CAH TOA?
SOH CAH TOA is a mnemonic device used in mathematics to remember the definitions of the three most common trigonometric functions: sine, cosine, and tangent. It is exclusively applied to right-angled triangles, which are triangles containing one 90-degree angle.
The ratios define the relationship between the lengths of the triangle’s sides and the measures of its non-right angles. Understanding these ratios allows you to “solve the triangle”—that is, find the measures of all missing angles and sides—with a minimal set of known inputs, which is exactly what this calculator automates.
How to Calculate SOH CAH TOA (Example):
Assume you know the Opposite side (o) = 5 and the Adjacent side (a) = 12. Here is the step-by-step calculation to find the Hypotenuse (h) and Angle A ($\theta$):
- Find the Hypotenuse (h) using Pythagorean Theorem: $$h = \sqrt{o^2 + a^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$
- Find Angle A ($\theta$) using the Tangent (TOA) ratio: $$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{12}$$ $$\theta = \arctan\left(\frac{5}{12}\right) \approx 22.62^\circ$$
- Find the remaining Angle B: $$\text{Angle B} = 90^\circ – \text{Angle A} = 90^\circ – 22.62^\circ = 67.38^\circ$$
Frequently Asked Questions (FAQ):
What is the minimum input required to use this calculator?
You need exactly two known values that are independent of each other: either the length of any two sides, or the length of one side and the measure of one non-90-degree angle (Angle A).
Why is this called a SOH CAH TOA calculator?
The calculator uses the mathematical principles represented by the SOH CAH TOA mnemonic to derive the missing values. It’s a tool that automates the application of these fundamental trigonometric ratios.
Can I use this for non-right triangles?
No. SOH CAH TOA and the Pythagorean theorem are strictly reserved for right-angled triangles. For triangles without a 90-degree angle, you should use the Law of Sines or the Law of Cosines (see our related calculators).
What unit should the sides be in?
The units for the sides (Opposite, Adjacent, Hypotenuse) must be consistent (e.g., all in meters, or all in inches). The calculator will output the missing side lengths in that same unit. Angles must always be input in degrees.