Multi-purpose theta calculator: convert between degrees, radians, gradians, and turns. Find all 6 trig values, reference angle, quadrant, coterminal angles, and unit circle visualization.
Theta (θ) is the universal symbol for an angle. Whether you need to convert between degrees, radians, gradians, and turns, find the reference angle, determine the quadrant, or evaluate all six trigonometric functions — this all-in-one theta calculator handles it all from a single input.
The four main angle units serve different purposes: degrees (360 per revolution) are standard in everyday geometry and navigation, radians (2π per revolution) are the natural unit of calculus and physics, gradians (400 per revolution) are used in European surveying, and turns (1 per revolution) simplify frequency and rotation problems. Converting fluently between them is a fundamental skill.
This calculator also computes sin θ, cos θ, tan θ and their reciprocals (csc, sec, cot), identifies the quadrant and sign pattern, finds the reference angle, lists coterminal and related angles, and shows the angle on a unit circle diagram. Twelve presets cover the most common special angles, and a conversion reference table shows all standard values at a glance.
Angle conversion, quadrant determination, reference angle computation, and evaluating six trig functions by hand requires careful attention to signs, quotients, and formulas. This calculator combines all of these into a single tool, eliminating errors and saving time.
It is particularly useful during exams or homework sessions where you need quick verification, and for engineers switching between degree and radian systems in different software tools.
Degrees → Radians: rad = deg × π/180. Radians → Degrees: deg = rad × 180/π. Gradians: grad = deg / 0.9. Turns: turn = deg / 360. Reference angle: subtract from the nearest axis multiple (0°, 90°, 180°, 270°, 360°).
Result: 135° = 2.3562 rad = 150 grad = 0.375 turns. Quadrant II. Reference angle = 45°. sin = 0.7071, cos = −0.7071, tan = −1.
135° is in Quadrant II (sin positive, cos negative). Reference angle = 180° − 135° = 45°. sin(135°) = sin(45°) = √2/2 ≈ 0.7071. cos(135°) = −cos(45°) = −√2/2.
The radian is defined as the angle subtended by an arc equal in length to the radius. This deceptively simple definition has profound consequences: it makes the radian the only angle unit where the small-angle approximation sin(θ) ≈ θ holds exactly in the limit, making it indispensable for calculus, physics, and engineering. The circumference formula C = 2πr is essentially the statement that a full angle is 2π radians.
In physics, all angular quantities (angular velocity, angular momentum, torque) use radians. In signal processing, angular frequency ω (in rad/s) determines oscillation behavior. Every formula in mathematical physics assumes radians unless explicitly stated otherwise.
The coordinate plane is divided into four quadrants by the x and y axes. As an angle θ increases from 0° to 360°, the terminal side sweeps through all four quadrants. The sign of each trig function depends on which quadrant the terminal side is in. The mnemonic "All Students Take Calculus" (ASTC) gives the positive functions: All in Q1, Sin in Q2, Tan in Q3, Cos in Q4.
Understanding quadrants is essential for solving trig equations, because each equation typically has two solutions per period — one in each of the quadrants where the relevant function has the correct sign.
The 360-degree circle has ancient Babylonian origins — they used a base-60 number system, and 360 is close to the number of days in a year. The gradian was introduced during the French Revolution as part of the metric system (along with the meter and kilogram), dividing a right angle into 100 parts. The radian was formalized in the 18th century by Roger Cotes and Leonhard Euler. The "turn" (or revolution) is the most intuitive unit — one full rotation — and is used in computing and engineering contexts where fractional rotations are natural.
Degrees divide a circle into 360 parts; radians measure the angle by the length of arc on a unit circle. One full revolution = 360° = 2π radians. Radians are preferred in calculus because they simplify derivative and integral formulas.
The reference angle is the acute angle (0°–90°) between the terminal side of the angle and the x-axis. It allows you to use the trig values of acute angles for any angle, with appropriate sign adjustments based on the quadrant.
Coterminal angles have the same terminal side — they differ by multiples of 360° (or 2π radians). For example, 45° and 405° are coterminal, as are 45° and −315°.
Gradians (also called gons) are used primarily in European surveying. A right angle = 100 grad, making it easy to compute slopes and grades as percentages.
Use the mnemonic "All Students Take Calculus": All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4. Use this as a practical reminder before finalizing the result.
In radians, the derivative of sin(x) is cos(x). In degrees, the derivative of sin(x°) is (π/180)·cos(x°) — the extra constant complicates every formula. Radians make x and sin(x) commensurable near 0.