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Math & Conversions Angle Calculator Angle Calculator (Right Triangle Solver) This calculator solves for the missing angles and sides of any right-angled triangle. Enter the minimum two known values (two sides, or one side and one acute angle) in the corresponding fields below. Enter Known Values (Minimum 2) Side a (Opposite $\alpha$) Side b (Opposite $\beta$) Side c (Hypotenuse) Angle $\alpha$ (degrees) Angle $\beta$ (degrees) Angle $\gamma = 90^\circ$ Solve Triangle Solved Triangle Side Results Angle Results Step-by-Step Solution The Core Formulas of Right Triangle Trigonometry A right triangle is fully defined by three key relationships: 1. Pythagorean Theorem (Sides) Used to find the length of any side when the other two sides are known: $$a^2 + b^2 = c^2$$ Where $c$ is the hypotenuse (the side opposite the $90^\circ$ angle), and $a$ and $b$ are the legs. 2. Angle Sum Rule The sum of the internal angles of any triangle is $180^\circ$ (or $\pi$ radians). Since the right angle $\gamma$ is $90^\circ$, the two acute angles ($\alpha$ and $\beta$) must sum to $90^\circ$: $$\alpha + \beta + 90^\circ = 180^\circ \quad \text{or} \quad \alpha + \beta = 90^\circ$$ 3. SOHCAHTOA (Sides and Angles) The trigonometric ratios relate the angles to the ratio of side lengths: $$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$ To find a missing angle ($\theta$) when all sides are known, we use the inverse functions (arcsin, arccos, arctan): $\theta = \sin^{-1}(\frac{O}{H})$. Frequently Asked Questions (FAQ) What is SOHCAHTOA? SOHCAHTOA is a mnemonic device used to remember the three basic trigonometric ratios in a right triangle: SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ What is the sum of angles in a triangle? The sum of the interior angles of any planar triangle, including a right triangle, is always $180$ degrees ($\pi$ radians). What is the Law of Sines and Law of Cosines? The Law of Sines and Law of Cosines are trigonometric rules used to solve **non-right** triangles (oblique triangles). For right triangles, SOHCAHTOA and the Pythagorean theorem are generally sufficient and simpler. What is the required minimum input to solve a right triangle? To solve a right triangle (i.e., find all missing sides and angles), you need to know exactly two non-angle parts (two sides) or one side and one acute angle. Knowing the right angle ($90^\circ$) is already assumed. Trigonometry Basics Pythagorean Theorem $$c^2 = a^2 + b^2$$ Tangent (TOA) $$\tan(\alpha) = \frac{a}{b}$$ Inverse Sine (SOH) $$\alpha = \sin^{-1}\left(\frac{a}{c}\right)$$ Related Geometry Tools Right Triangle Calculator Pythagorean Theorem Calculator Triangle Calculator (General) Law of Sines Calculator Law of Cosines Calculator Radians to Degrees Converter
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