Fact Checked & Reviewed
This Trigonometry Calculator has been designed and verified for mathematical accuracy by Dr. Helena V. Sharma, P. Eng., a specialist in applied mathematics and geometric analysis.
Solve any right triangle quickly and accurately. Simply input any two non-right angle values (a side and an angle, or two sides), and this calculator will instantly find the remaining side lengths and angles using SOH CAH TOA and the Pythagorean theorem.
Right Triangle Trigonometry Calculator
Note: This calculator solves right triangles (one 90° angle). Angle A is opposite side a, Angle B is opposite side b, and side c is the Hypotenuse (opposite 90°).
Trigonometry Calculator Formulas
The core formulas for a right triangle are:
$$a^2 + b^2 = c^2$$Angle A is the reference angle:
$$\sin(A) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{c}$$ $$\cos(A) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{b}{c}$$ $$\tan(A) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b}$$Angle B is the complement of A:
$$B = 90^\circ – A$$Formula Source: Khan Academy – Basic Trigonometry | Formula Source: Wolfram MathWorld – Right Triangle
Variables Explained
- Angle A (Degrees): The measure of the first non-right angle.
- Side a (Opposite): The length of the side opposite to Angle A.
- Side b (Adjacent): The length of the side adjacent to Angle A (and opposite Angle B).
- Side c (Hypotenuse): The length of the longest side, always opposite the 90° angle.
What is Trigonometry?
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles, particularly right-angled triangles. The word itself comes from the Greek words *trigonon* (triangle) and *metron* (measure). It is fundamental to many fields, including physics, engineering, surveying, astronomy, and computer graphics.
The primary tools of trigonometry are the trigonometric functions: sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). These functions describe the ratio of side lengths relative to an interior angle in a right triangle. Understanding these ratios allows us to solve for unknown side lengths or angles, which is the core function of this calculator.
How to Use the Calculator (Example)
- Identify Known Values: Suppose you know the Opposite side ($a=7$) and the Adjacent side ($b=10$).
- Enter Data: Input ‘7’ into the “Side a (Opposite)” field and ’10’ into the “Side b (Adjacent)” field. Leave the other fields empty.
- Click Calculate: The calculator first uses the Pythagorean theorem to find the Hypotenuse ($c$). $$c = \sqrt{7^2 + 10^2} = \sqrt{49 + 100} = \sqrt{149} \approx 12.21$$
- Find Angles: It then uses the inverse tangent function to find Angle A: $$A = \arctan(\frac{a}{b}) = \arctan(\frac{7}{10}) \approx 34.99^\circ$$
- Final Angle: Finally, it calculates Angle B: $$B = 90^\circ – 34.99^\circ \approx 55.01^\circ$$
- Review Results: The final calculated values for the entire triangle are displayed in the highlighted result area.
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- Pythagorean Theorem Solver
- Law of Sines Calculator
- Unit Circle Reference Tool
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Frequently Asked Questions (FAQ)
SOH CAH TOA is a mnemonic device used to remember the three basic trigonometric ratios for right triangles: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
No, this specific tool is designed only for right triangles (triangles with a 90-degree angle). Non-right triangles require the Law of Sines or the Law of Cosines, which are different formulas.
A triangle must be fully defined to solve for all its missing components. In a right triangle, knowing only one side or one non-right angle is not enough to find the specific lengths of the remaining parts.
All input and output angles on this calculator are in degrees ($^\circ$). All internal JavaScript functions that calculate the trigonometric ratios convert degrees to radians before computation.