Home › Math & Conversions › Core Math & Algebra › SOHCAHTOA SOHCAHTOA Calculator (Sine, Cosine & Tangent) Learn and apply SOHCAHTOA: compute sin, cos, and tan for any angle, or solve a right triangle from one angle and one side with clear step-by-step explanations. Core Math & Algebra Interactive SOHCAHTOA tool Use this panel in two ways: Trig ratios from angle gives you \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) directly. Right triangle solver applies SOHCAHTOA to find missing side lengths from one acute angle and one known side of a right triangle. Trig ratios from angle Right triangle (SOHCAHTOA) Switch between pure trig ratios and geometric right-triangle mode. Angle \(\theta\) Enter a number; you can also type common angles like 30, 45, 60. Angle unit degrees (°) radians (rad) Switch between degrees and radians; the calculator converts internally. Quick angles 30° 45° 60° 90° Common special angles for exact SOHCAHTOA triangles. Compute sin, cos, tan Clear Trig ratios at \(\theta\) In this mode, we assume a right triangle with one 90° angle. The angle \(\theta\) is one of the acute angles (\(0^\circ < \theta < 90^\circ\)). SOHCAHTOA applies to the sides relative to this angle. Acute angle \(\theta\) Must be between 0° and 90° (or 0 and \(\pi/2\) radians). Angle unit degrees (°) radians (rad) Choose the unit you are using for \(\theta\). Side length unit (optional label) Used only for display; does not affect calculations. Known side (relative to \(\theta\)) Hypotenuse Opposite side Adjacent side Known side length Must be a positive number; same unit as any other sides. Solve right triangle Clear Example: 3–4–5 triangle Right triangle solution Side lengths Trig ratios at \(\theta\) SOHCAHTOA summary What is SOHCAHTOA? SOHCAHTOA is a compact mnemonic for remembering the definitions of the three primary trigonometric ratios in a right triangle, relative to an acute angle \(\theta\): SOH: \(\displaystyle \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\) CAH: \(\displaystyle \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) TOA: \(\displaystyle \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\) Here, “opposite” means the side directly across from the angle \(\theta\), “adjacent” is the side that touches \(\theta\) but is not the hypotenuse, and the “hypotenuse” is the longest side, opposite the right angle. Using SOHCAHTOA to find missing sides To use SOHCAHTOA effectively: Draw or imagine the right triangle and clearly mark the angle \(\theta\) you are using. Label each side as opposite , adjacent , or hypotenuse relative to \(\theta\). Decide which ratio uses the known side and the unknown side: SOH, CAH, or TOA. Set up the equation and solve for the unknown side length. The right-triangle panel of this calculator automates these steps once you select the known side type and enter its value. Degrees vs radians Trigonometric functions accept angles in either degrees or radians . Many school problems use degrees, while calculus and programming typically use radians. You can convert between them via \(\text{radians} = \text{degrees} \times \dfrac{\pi}{180}, \quad \text{degrees} = \text{radians} \times \dfrac{180}{\pi}.\) This calculator has a unit selector for both angle modes and performs the conversion internally, so the formulas for SOHCAHTOA stay consistent. Special SOHCAHTOA triangles Certain angles produce especially nice ratios. Two classic examples are: The 30°–60°–90° triangle , with side ratios \(1 : \sqrt{3} : 2\). For example, if the hypotenuse is 2, then the shorter leg is 1 and the longer leg is \(\sqrt{3}\). The 45°–45°–90° triangle , with side ratios \(1 : 1 : \sqrt{2}\). Both legs are equal. The quick-angle buttons use these special triangles so you can see the corresponding sin, cos, and tan values and use them as reference points. SOHCAHTOA – FAQ Can I use SOHCAHTOA on any triangle? No. SOHCAHTOA only applies to right triangles , where one angle is exactly 90°. For non-right triangles you need other tools such as the Law of Sines or the Law of Cosines. What happens if the angle is 0° or 90°? At 0° and 90°, some ratios become undefined (for example, \(\tan 90^\circ\)). The right-triangle solver therefore requires \(0^\circ < \theta < 90^\circ\). The angle-only trig mode can still evaluate sin, cos, and tan at boundary angles but will indicate when a ratio is undefined or numerically unstable. Why does \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)? From SOHCAHTOA, \(\sin\theta = \frac{\text{Opp}}{\text{Hyp}}\) and \(\cos\theta = \frac{\text{Adj}}{\text{Hyp}}\). Dividing these expressions gives \(\dfrac{\sin\theta}{\cos\theta} = \dfrac{\text{Opp}/\text{Hyp}}{\text{Adj}/\text{Hyp}} = \dfrac{\text{Opp}}{\text{Adj}} = \tan\theta.\) This identity is built into the calculator checks to ensure internal consistency. How accurate is this SOHCAHTOA calculator? The tool uses double-precision floating-point arithmetic, the same standard used by scientific software and calculators. Results are rounded for display, but internal operations have more precision than most classroom or design tasks require. Frequently Asked Questions Is SOHCAHTOA valid in any quadrant? The SOHCAHTOA side definitions come from a right triangle picture, which naturally lives in the first quadrant where all sides and ratios are positive. When you extend sine, cosine, and tangent to other quadrants on the unit circle, the numerical definitions stay the same, but the signs of sin, cos, and tan change according to the quadrant. The angle-only panel includes a summary of sign patterns by quadrant. Does the calculator keep track of opposite/adjacent automatically? Yes. When you choose which side is known (opposite, adjacent, or hypotenuse) and enter the acute angle, the right-triangle solver applies the appropriate SOH, CAH, or TOA relationship and solves for the missing sides using the standard right-triangle formulas. Can I use this tool in programming or engineering work? Absolutely. The calculator mirrors the behaviour of common math libraries for sin, cos, and tan while providing a geometric interpretation via SOHCAHTOA. It is ideal for quick checks of angle conversions, right-triangle dimensions, and sign conventions before you implement formulas in code or design documents. Related trigonometry & core math tools Strengthen your trigonometry and algebra toolkit with these calculators from the same Core Math & Algebra family. Pythagorean theorem Ellipse area Fibonacci number Continued fraction Dice roll probability Exponent calculator Set theory basics Interval notation Base converter All Math & Conversions tools SOHCAHTOA quick checklist Confirm you are working with a right triangle (one angle is 90°). Select an acute reference angle \(\theta\) and label opposite, adjacent, and hypotenuse. Pick the ratio that matches the sides you know and need: SOH, CAH, or TOA. Keep track of angle units (degrees vs radians) consistently. For sanity checks, verify that \(\sin^2\theta + \cos^2\theta \approx 1\) and \(\tan\theta \approx \sin\theta

Subcategories in Home › Math & Conversions › Core Math & Algebra › SOHCAHTOA SOHCAHTOA Calculator (Sine, Cosine & Tangent) Learn and apply SOHCAHTOA: compute sin, cos, and tan for any angle, or solve a right triangle from one angle and one side with clear step-by-step explanations. Core Math & Algebra Interactive SOHCAHTOA tool Use this panel in two ways: Trig ratios from angle gives you \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) directly. Right triangle solver applies SOHCAHTOA to find missing side lengths from one acute angle and one known side of a right triangle. Trig ratios from angle Right triangle (SOHCAHTOA) Switch between pure trig ratios and geometric right-triangle mode. Angle \(\theta\) Enter a number; you can also type common angles like 30, 45, 60. Angle unit degrees (°) radians (rad) Switch between degrees and radians; the calculator converts internally. Quick angles 30° 45° 60° 90° Common special angles for exact SOHCAHTOA triangles. Compute sin, cos, tan Clear Trig ratios at \(\theta\) In this mode, we assume a right triangle with one 90° angle. The angle \(\theta\) is one of the acute angles (\(0^\circ < \theta < 90^\circ\)). SOHCAHTOA applies to the sides relative to this angle. Acute angle \(\theta\) Must be between 0° and 90° (or 0 and \(\pi/2\) radians). Angle unit degrees (°) radians (rad) Choose the unit you are using for \(\theta\). Side length unit (optional label) Used only for display; does not affect calculations. Known side (relative to \(\theta\)) Hypotenuse Opposite side Adjacent side Known side length Must be a positive number; same unit as any other sides. Solve right triangle Clear Example: 3–4–5 triangle Right triangle solution Side lengths Trig ratios at \(\theta\) SOHCAHTOA summary What is SOHCAHTOA? SOHCAHTOA is a compact mnemonic for remembering the definitions of the three primary trigonometric ratios in a right triangle, relative to an acute angle \(\theta\): SOH: \(\displaystyle \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\) CAH: \(\displaystyle \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) TOA: \(\displaystyle \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\) Here, “opposite” means the side directly across from the angle \(\theta\), “adjacent” is the side that touches \(\theta\) but is not the hypotenuse, and the “hypotenuse” is the longest side, opposite the right angle. Using SOHCAHTOA to find missing sides To use SOHCAHTOA effectively: Draw or imagine the right triangle and clearly mark the angle \(\theta\) you are using. Label each side as opposite , adjacent , or hypotenuse relative to \(\theta\). Decide which ratio uses the known side and the unknown side: SOH, CAH, or TOA. Set up the equation and solve for the unknown side length. The right-triangle panel of this calculator automates these steps once you select the known side type and enter its value. Degrees vs radians Trigonometric functions accept angles in either degrees or radians . Many school problems use degrees, while calculus and programming typically use radians. You can convert between them via \(\text{radians} = \text{degrees} \times \dfrac{\pi}{180}, \quad \text{degrees} = \text{radians} \times \dfrac{180}{\pi}.\) This calculator has a unit selector for both angle modes and performs the conversion internally, so the formulas for SOHCAHTOA stay consistent. Special SOHCAHTOA triangles Certain angles produce especially nice ratios. Two classic examples are: The 30°–60°–90° triangle , with side ratios \(1 : \sqrt{3} : 2\). For example, if the hypotenuse is 2, then the shorter leg is 1 and the longer leg is \(\sqrt{3}\). The 45°–45°–90° triangle , with side ratios \(1 : 1 : \sqrt{2}\). Both legs are equal. The quick-angle buttons use these special triangles so you can see the corresponding sin, cos, and tan values and use them as reference points. SOHCAHTOA – FAQ Can I use SOHCAHTOA on any triangle? No. SOHCAHTOA only applies to right triangles , where one angle is exactly 90°. For non-right triangles you need other tools such as the Law of Sines or the Law of Cosines. What happens if the angle is 0° or 90°? At 0° and 90°, some ratios become undefined (for example, \(\tan 90^\circ\)). The right-triangle solver therefore requires \(0^\circ < \theta < 90^\circ\). The angle-only trig mode can still evaluate sin, cos, and tan at boundary angles but will indicate when a ratio is undefined or numerically unstable. Why does \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)? From SOHCAHTOA, \(\sin\theta = \frac{\text{Opp}}{\text{Hyp}}\) and \(\cos\theta = \frac{\text{Adj}}{\text{Hyp}}\). Dividing these expressions gives \(\dfrac{\sin\theta}{\cos\theta} = \dfrac{\text{Opp}/\text{Hyp}}{\text{Adj}/\text{Hyp}} = \dfrac{\text{Opp}}{\text{Adj}} = \tan\theta.\) This identity is built into the calculator checks to ensure internal consistency. How accurate is this SOHCAHTOA calculator? The tool uses double-precision floating-point arithmetic, the same standard used by scientific software and calculators. Results are rounded for display, but internal operations have more precision than most classroom or design tasks require. Frequently Asked Questions Is SOHCAHTOA valid in any quadrant? The SOHCAHTOA side definitions come from a right triangle picture, which naturally lives in the first quadrant where all sides and ratios are positive. When you extend sine, cosine, and tangent to other quadrants on the unit circle, the numerical definitions stay the same, but the signs of sin, cos, and tan change according to the quadrant. The angle-only panel includes a summary of sign patterns by quadrant. Does the calculator keep track of opposite/adjacent automatically? Yes. When you choose which side is known (opposite, adjacent, or hypotenuse) and enter the acute angle, the right-triangle solver applies the appropriate SOH, CAH, or TOA relationship and solves for the missing sides using the standard right-triangle formulas. Can I use this tool in programming or engineering work? Absolutely. The calculator mirrors the behaviour of common math libraries for sin, cos, and tan while providing a geometric interpretation via SOHCAHTOA. It is ideal for quick checks of angle conversions, right-triangle dimensions, and sign conventions before you implement formulas in code or design documents. Related trigonometry & core math tools Strengthen your trigonometry and algebra toolkit with these calculators from the same Core Math & Algebra family. Pythagorean theorem Ellipse area Fibonacci number Continued fraction Dice roll probability Exponent calculator Set theory basics Interval notation Base converter All Math & Conversions tools SOHCAHTOA quick checklist Confirm you are working with a right triangle (one angle is 90°). Select an acute reference angle \(\theta\) and label opposite, adjacent, and hypotenuse. Pick the ratio that matches the sides you know and need: SOH, CAH, or TOA. Keep track of angle units (degrees vs radians) consistently. For sanity checks, verify that \(\sin^2\theta + \cos^2\theta \approx 1\) and \(\tan\theta \approx \sin\theta.