Trigonometry Calculator

Solve sin, cos, tan, inverse trig functions, and missing triangle sides and angles in degrees or radians.

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Calculation Examples

Calculation Case Result
sin(30°) 0.5
cos(60°) 0.5
Right triangle: angle 45°, hypotenuse 10 Opposite = Adjacent = 7.071
Right triangle: opposite 3, adjacent 4 Angle = arctan(0.75) = 36.87°, hypotenuse = 5
Mihail Doncenco
Mihail Doncenco Site Creator & Calculator Expert

Founder and developer of CalcMate. Builds and maintains every calculator on this site, focusing on formula accuracy and clear, practical results.

How to Use the Trigonometry Calculator

Choose degree or radian mode, enter an angle, and the calculator returns all six trigonometric functions: sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent. To solve a right triangle, enter two known values (one side and one angle, or two sides) and it finds the remaining sides and angles.

Inverse functions (arcsin, arccos, arctan) turn a known side ratio back into an angle.

How Are Calculations Performed?

The calculator uses the right-triangle ratios \(\sin\theta = \text{opposite}/\text{hypotenuse}\), \(\cos\theta = \text{adjacent}/\text{hypotenuse}\), and \(\tan\theta = \text{opposite}/\text{adjacent}\). The reciprocal functions follow as \(\csc\theta = 1/\sin\theta\), \(\sec\theta = 1/\cos\theta\), and \(\cot\theta = 1/\tan\theta\). Inverse functions return the angle for a given ratio, for example \(\theta = \arcsin(\text{opposite}/\text{hypotenuse})\). The identity \(\sin^2\theta + \cos^2\theta = 1\) works as a built-in consistency check. For triangles without a right angle, the tool uses the Law of Sines, \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\), and the Law of Cosines, \(c^2 = a^2 + b^2 - 2ab\cos C\).Trigonometry Function Diagram: SOH-CAH-TOA and Unit Circle

Useful Tips 💡

  • Check the angle mode before every calculation. sin(30°) equals 0.5, but sin(30 radians) equals about -0.988, and that mismatch silently ruins everything built on it.
  • Inverse functions return only one principal value. Arcsin and arctan give angles between -90° and 90°, and arccos gives angles between 0° and 180°. If the angle in your triangle lies outside that range, add 180° or 360° as the geometry requires.

📋Steps to Calculate

  1. Select degree or radian mode to match your input.

  2. Enter the angle to compute all six trig functions, or enter two known triangle values (sides or angles) to solve for the remaining parts.

  3. Click "Calculate" to receive sine, cosine, tangent, reciprocal functions, and inverse angle results at once.

Mistakes to Avoid ⚠️

  1. Using degrees in radian mode, or the reverse. Typing 30 in radian mode gives sin = -0.988 instead of 0.5, and the answer will not match the triangle you meant.
  2. Expecting inverse functions to give every solution. Arcsin(0.5) returns 30°, but 150° also has a sine of 0.5. In triangle problems, check which solution fits the geometry.
  3. Swapping adjacent and opposite sides in SOH-CAH-TOA. The adjacent side touches the angle and is not the hypotenuse, and the opposite side faces the angle.
  4. Applying right-triangle ratios to a triangle without a 90° angle. For those, use the Law of Sines or the Law of Cosines instead.

Practical Applications📊

  1. Solve a right triangle: with a 45° angle and a hypotenuse of 10, both legs equal 10 x sin 45° = 7.07 units.

  2. Check a wheelchair ramp slope: the US ADA 2010 Standards cap the running slope at 1:12. A ramp rising 0.5 m over a 6 m run has an angle of arctan(1/12) = 4.76°, which is a grade of 8.33%.

  3. Measure a height without climbing: standing 50 m from the base of a tower and measuring a 35° angle of elevation gives a height of 50 x tan 35° = 35.0 m, plus the height of your eye above the ground.

  4. Read an AC waveform: mains labeled 120 V is an RMS value, so the voltage peaks at 120 x 1.414 = about 170 V and follows V(t) = 170 sin(2π x 60 x t) in the US.

Questions and Answers

How do I find a missing angle using inverse trig functions?

Use the inverse function that matches the two sides you know: \(\theta = \arcsin(O/H)\) for opposite and hypotenuse, \(\theta = \arccos(A/H)\) for adjacent and hypotenuse, and \(\theta = \arctan(O/A)\) for opposite and adjacent. For example, opposite 3 and adjacent 4 give \(\theta = \arctan(3/4) = \arctan(0.75) \approx 36.87^\circ\). Inverse functions return one principal value only: arcsin and arctan give angles between \(-90^\circ\) and \(90^\circ\), and arccos gives angles between \(0^\circ\) and \(180^\circ\). For an obtuse angle, check the quadrant and adjust the result, for example \(180^\circ - 30^\circ = 150^\circ\).

How does the calculator solve for missing values in right triangles?

It picks the SOH-CAH-TOA ratio that connects what you know to what you need. With an angle \(\theta\) and the hypotenuse \(c\), the legs are \(\text{opposite} = c\sin\theta\) and \(\text{adjacent} = c\cos\theta\). With two sides, it finds the third using \(a^2 + b^2 = c^2\) and then the angles using inverse functions such as \(\theta = \arctan(\text{opposite}/\text{adjacent})\). The two acute angles must add up to 90°, so all three interior angles sum to 180°, and the calculator uses that as a check. For example, \(\theta = 30^\circ\) and \(c = 10\) give opposite \(= 5\) and adjacent \(\approx 8.66\).

What does SOH-CAH-TOA mean and how do I apply it?

SOH-CAH-TOA is a mnemonic for the three main right-triangle ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent. In symbols, \(\sin\theta = O/H\), \(\cos\theta = A/H\), and \(\tan\theta = O/A\). To use it, mark the reference angle \(\theta\), label the side across from it (O), the side that touches it but is not the hypotenuse (A), and the hypotenuse (H). Then choose the ratio that contains your two quantities. If you know the hypotenuse and need the opposite side, use SOH: \(O = H\sin\theta\).

What is a trigonometry calculator and what can it solve?

A trigonometry calculator computes sin, cos, tan, csc, sec, and cot for any angle, and solves right or oblique triangles from two known values for a right triangle or three for an oblique one. It accepts degrees or radians, returns arcsin, arccos, and arctan to recover angles from ratios, and uses the Pythagorean theorem plus the Laws of Sines and Cosines for full triangle solutions. Students use it from secondary school through university, and professionals use it wherever angle and distance must be computed precisely.

What is the difference between degrees and radians?

Degrees and radians are two units for angles: a full turn is 360° or \(2\pi\) radians (about 6.2832), so one radian is about 57.2958°. Convert with \(\text{radians} = \text{degrees} \times \pi/180\) and \(\text{degrees} = \text{radians} \times 180/\pi\). Common reference points are \(90^\circ = \pi/2\), \(180^\circ = \pi\), and \(45^\circ = \pi/4\). Degrees dominate navigation, surveying, and introductory geometry. Radians are the standard in calculus and physics because the derivative of \(\sin x\) is \(\cos x\) only when \(x\) is in radians, and most programming languages expect radians by default.

What is the Pythagorean identity and how is it used in trigonometry?

The Pythagorean identity states that \(\sin^2\theta + \cos^2\theta = 1\) for every angle \(\theta\). It comes straight from the Pythagorean theorem on the unit circle: a point \((\cos\theta, \sin\theta)\) lies at distance 1 from the origin, so \(\cos^2\theta + \sin^2\theta = 1^2\). Use it to simplify expressions, prove other identities, and find one function when the other is known. For example, if \(\sin\theta = 0.6\) in the first quadrant, then \(\cos\theta = \sqrt{1 - 0.36} = 0.8\). Dividing by \(\cos^2\theta\) or \(\sin^2\theta\) gives the two related identities \(1 + \tan^2\theta = \sec^2\theta\) and \(1 + \cot^2\theta = \csc^2\theta\).

What is the unit circle and how does it relate to trigonometry?

The unit circle is a circle of radius 1 centered at the origin, and for an angle \(\theta\) measured counterclockwise from the positive x-axis, the point on it is \((\cos\theta, \sin\theta)\). This definition extends sine and cosine to all angles, including negative angles and angles above 360°. Key points are \((1, 0)\) at \(0^\circ\), \((0, 1)\) at \(90^\circ\), \((-1, 0)\) at \(180^\circ\), and \((0, -1)\) at \(270^\circ\). The unit circle explains why \(\sin\) and \(\cos\) repeat every \(360^\circ\) (or \(2\pi\) radians), how their signs change across the four quadrants, and why they appear in Fourier analysis and complex numbers.

What is trigonometry and why is it important in mathematics?

Trigonometry is the branch of mathematics that links the angles of a triangle to its side lengths, and by extension describes circular and periodic motion. The three main ratios, sine, cosine, and tangent, let you calculate distances and angles that you cannot measure directly. It shows up in structural engineering (force components along an angled beam), navigation (bearing and distance), physics (waves, optics, oscillations), computer graphics (rotation and projection), and signal processing, where Fourier analysis breaks any periodic signal into sine and cosine components.
Disclaimer: This calculator is designed to provide helpful estimates for informational purposes. While we strive for accuracy, financial (or medical) results can vary based on local laws and individual circumstances. We recommend consulting with a professional advisor for critical decisions.