Trig Functions Calculator

About Trig Functions Calculator

The Six Trigonometric Functions

Given a right triangle with angle θ, opposite side, adjacent side, and hypotenuse:

Function	Ratio	Reciprocal
sin(θ)	opposite / hypotenuse	csc(θ) = 1/sin(θ)
cos(θ)	adjacent / hypotenuse	sec(θ) = 1/cos(θ)
tan(θ)	opposite / adjacent = sin(θ)/cos(θ)	cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)

The mnemonic SOH-CAH-TOA helps remember: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

How Does SOH-CAH-TOA Work in Practice?

SOH-CAH-TOA is just a memory trick, but it becomes genuinely useful once you walk through a real triangle. Each chunk tells you the ratio for one function: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

Worked example: Picture a right triangle where one acute angle is 35 degrees. The side opposite that angle measures 6 metres and the hypotenuse measures 10.46 metres. Using SOH: sin(35°) = opposite / hypotenuse = 6 / 10.46 = 0.5736. Check it on a calculator and sin(35°) gives 0.5736 - it matches. Now say you know the angle and the adjacent side (8.57 m) but not the opposite side. Use TOA: tan(35°) = opposite / adjacent, so opposite = adjacent × tan(35°) = 8.57 × 0.7002 = 6.0 m. Same answer, different path. That is the power of choosing the right ratio for the information you have.

A quick way to decide which ratio to use: label what you know and what you need. If the hypotenuse is involved, pick sine or cosine. If both legs are involved (no hypotenuse), pick tangent. This alone eliminates most of the guesswork when setting up trig problems.

Exact Values at Standard Angles

Degrees	Radians	sin	cos	tan
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined
120°	2π/3	√3/2	-1/2	-√3
135°	3π/4	√2/2	-√2/2	-1
150°	5π/6	1/2	-√3/2	-1/√3
180°	π	0	-1	0
270°	3π/2	-1	0	undefined
360°	2π	0	1	0

Key Trigonometric Identities

The calculator verifies these identities for every input. The table below covers the Pythagorean, quotient, reciprocal, double-angle, and sum/difference identities that come up most often in coursework and exams.

Category	Identity	Formula
Pythagorean	Basic	sin²(θ) + cos²(θ) = 1
Pythagorean	Tangent form	1 + tan²(θ) = sec²(θ)
Pythagorean	Cotangent form	1 + cot²(θ) = csc²(θ)
Quotient	Tangent	tan(θ) = sin(θ) / cos(θ)
Reciprocal	Sine-Cosecant	sin(θ) × csc(θ) = 1
Even/Odd	Cosine (even)	cos(-θ) = cos(θ)
Even/Odd	Sine (odd)	sin(-θ) = -sin(θ)
Double Angle	Sine	sin(2θ) = 2 sin(θ) cos(θ)
Double Angle	Cosine	cos(2θ) = cos²(θ) - sin²(θ)
Double Angle	Cosine (alt)	cos(2θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
Double Angle	Tangent	tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Sum	Sine	sin(A + B) = sin A cos B + cos A sin B
Sum	Cosine	cos(A + B) = cos A cos B - sin A sin B
Difference	Sine	sin(A - B) = sin A cos B - cos A sin B
Difference	Cosine	cos(A - B) = cos A cos B + sin A sin B

All three Pythagorean identities come from the same place: the Pythagorean theorem applied to a unit circle. If a point on the unit circle has coordinates (cos θ, sin θ), then cos²θ + sin²θ equals the radius squared, which is 1. Dividing both sides by cos²θ gives the tangent form; dividing by sin²θ gives the cotangent form. The Pythagorean theorem calculator can verify the underlying a² + b² = c² relationship for any right triangle.

Inverse Trig Functions

Inverse trig functions find the angle when you know the ratio:

Function	Input Range	Output Range	Example
arcsin (sin⁻¹)	[-1, 1]	[-90°, 90°]	arcsin(0.5) = 30°
arccos (cos⁻¹)	[-1, 1]	[0°, 180°]	arccos(0.5) = 60°
arctan (tan⁻¹)	all real numbers	(-90°, 90°)	arctan(1) = 45°

Each inverse function has a restricted output range (principal value) to ensure a unique answer. For example, sin(30°) = sin(150°) = 0.5, but arcsin(0.5) returns only 30° (the principal value).

Sign Rules by Quadrant

Quadrant	Angle Range	sin	cos	tan
I	0° - 90°	+	+	+
II	90° - 180°	+	-	-
III	180° - 270°	-	-	+
IV	270° - 360°	-	+	-

The mnemonic "All Students Take Calculus" tells which functions are positive: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4).

Why Do Radians Exist and When Should You Use Them?

Degrees split a circle into 360 parts - a number chosen by the ancient Babylonians, likely because 360 is close to the number of days in a year and has many divisors. Radians take a completely different approach. One radian is the angle formed when the arc length equals the radius of the circle. A full circle has a circumference of 2πr, so a full turn is 2π radians. The concept was first described by the English mathematician Roger Cotes in his 1714 paper "Logometria," though the word "radian" itself did not appear in print until 1873, when James Thomson (brother of Lord Kelvin) used it on an exam at Queen's College Belfast.

Conversion: radians = degrees × π/180. Degrees = radians × 180/π.

Degrees	Radians (exact)	Radians (decimal)
30°	π/6	0.5236
45°	π/4	0.7854
60°	π/3	1.0472
90°	π/2	1.5708
180°	π	3.1416
360°	2π	6.2832

So why does anyone bother with radians when degrees feel more intuitive? The short answer: calculus. The derivative of sin(x) is cos(x), but only when x is measured in radians. If x is in degrees, the derivative picks up a factor of π/180, making every formula messier. The fundamental limit that makes this work is lim(x to 0) sin(x)/x = 1, and that limit equals 1 only when x is in radians. Leonhard Euler popularised radian measure in his 1748 work "Introductio in Analysin Infinitorum," and it became the standard for analysis and physics from that point on. In practice, use degrees for geometry, construction, and navigation. Use radians for calculus, physics equations, and programming (most math libraries including JavaScript's Math.sin expect radians).

Why Do Reciprocal Functions (csc, sec, cot) Exist?

Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively. It is fair to ask why we need three extra functions when we could just write 1/sin, 1/cos, and 1/tan. The answer is partly historical and partly practical. Before calculators existed, mathematicians worked with lookup tables. Having separate columns for csc, sec, and cot saved the error-prone step of dividing by a decimal looked up from another column. That convenience carried forward into mathematical notation.

In modern use, these functions still show up naturally in certain contexts. Secant appears in the formula for the length of a curve: the arc length integral involves the square root of 1 + (dy/dx)², which often simplifies to sec(θ). Cosecant appears in certain integration results and in optics formulas. Cotangent shows up in Fourier series and in formulas for the area of regular polygons. The Pythagorean identity 1 + tan²(θ) = sec²(θ) is used constantly in integral calculus when performing trigonometric substitution - it would be awkward to write that without sec. So while you can always replace them with 1/sin and friends, the reciprocal functions earn their keep by making formulas shorter and easier to read.

Where Are Trig Functions Used in Real Life?

Trigonometry is not just an exam topic. It shows up in dozens of professions and everyday situations. Here are four major application areas.

Surveying and construction: Surveyors use trig to measure distances and heights that cannot be measured directly. By measuring angles with a theodolite and knowing a single baseline distance, they can calculate the height of a building, the width of a river, or the area of an irregular plot of land. The triangle calculator applies these same principles to solve any triangle given partial information.

Navigation: GPS receivers calculate their position by measuring the time signals take to arrive from multiple satellites, then applying trigonometric calculations to pinpoint a location on Earth. Maritime and aviation navigation rely on spherical trigonometry to compute great-circle distances and bearing angles across the curved surface of the Earth.

Sound and signal processing: Sound waves are modelled as sine waves. A pure musical tone at concert pitch A (440 Hz) completes 440 full sine cycles every second. Audio engineers, synthesiser designers, and noise-cancellation systems all depend on the sine function to decompose complex sounds into individual frequency components through Fourier analysis.

Architecture: Architects use trigonometry to calculate roof pitch angles, determine load distribution in triangular trusses, and ensure structural members meet at correct angles. A roof designed with a 30-degree pitch uses tan(30°) to relate the rise to the run - for every metre of horizontal span, the roof rises tan(30°) = 0.577 metres.

Worked example - finding a building's height from the ground: You stand 50 metres from the base of a building. Using a clinometer (or a phone app), you measure the angle of elevation to the top as 62 degrees. Your eye level is 1.7 metres above the ground. Using TOA: tan(62°) = opposite / adjacent = building height above eye level / 50. So the height above eye level = 50 × tan(62°) = 50 × 1.8807 = 94.04 metres. Add eye level: total building height = 94.04 + 1.7 = 95.74 metres. This is exactly the method surveyors use before sophisticated equipment is available, and it still works as a quick field estimate today.

Common Mistakes with Trig Calculations

Certain errors come up repeatedly in trig coursework and on exams. Being aware of them can save a lot of frustration.

1. Calculator in the wrong angle mode. This is the single most common source of wrong answers in trigonometry. If you type sin(30) expecting 0.5 but your calculator is set to radians, it will return -0.9880 instead. Texas Instruments' own support page lists "unexpected trigonometric results" caused by wrong mode as one of the top user issues for the TI-84 family. Always check that your calculator display shows DEG or RAD matching the unit of the angle you are entering. A good habit: before starting any problem set, type sin(30) and confirm you get 0.5 (for degree mode) or sin(π/6) and confirm 0.5 (for radian mode).

2. Confusing inverse functions with reciprocal functions. The notation sin⁻¹(x) means arcsin(x) - it finds the angle whose sine is x. It does not mean 1/sin(x), which is csc(x). This confusion is entirely the fault of the notation, since in most other contexts a superscript -1 does mean "one divided by." On a calculator, the sin⁻¹ button gives arcsin. If you actually want the reciprocal, compute 1 divided by sin of the angle. Mixing these up can produce answers that are wildly off.

3. Forgetting domain restrictions on inverse functions. If you try arcsin(1.5), there is no answer because sine never exceeds 1. Similarly, arccos only accepts values from -1 to 1. If your calculation produces an input outside that range for arcsin or arccos, it usually means an earlier step has an error.

4. Losing the negative sign in quadrants II, III, and IV. Students often compute the reference angle correctly but forget to apply the sign rules for the quadrant. For example, sin(210°) is not 0.5 - it is -0.5, because 210° is in quadrant III where sine is negative. The ASTC mnemonic ("All Students Take Calculus") described above prevents this mistake if used consistently.

5. Using degrees in formulas that expect radians. Physics and engineering formulas involving angular velocity (ω), arc length (s = rθ), or simple harmonic motion all assume radians. Plugging in degrees without converting by π/180 will produce answers that are off by a large factor. If a formula does not include any degree symbols or conversion factors, it almost certainly expects radians. To quickly plot or check trig values across different angles, the graphing calculator can visualise sine, cosine, and tangent curves side by side.

For a complete visual reference of trig values on the unit circle, see the unit circle tool. For solving triangles with known sides and angles, the law of sines and law of cosines calculators apply these functions directly.

All calculations run in your browser. No data is sent to any server.

Sources

• Wolfram MathWorld - Trigonometry
• Wolfram MathWorld - Radian
• MDN - Math.sin (radians input)
• Encyclopedia Britannica - Trigonometry
• Texas Instruments - Changing Angle Modes on TI-84