Trig
Compute all six trig functions (sin, cos, tan, csc, sec, cot) for any angle in degrees. Free calculator with degree-to-radian conversion and instant results.
About This Calculator
The Trigonometry Calculator computes all six fundamental trigonometric functions -- sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) -- for any angle entered in degrees. Whether you are a high school or college student studying trigonometry, an engineer solving structural or electrical problems, a physicist analyzing wave motion and oscillations, or a professional in navigation, surveying, or computer graphics, this calculator provides instant, accurate trigonometric values for any angle.
The calculator converts the input angle from degrees to radians (multiplying by pi/180) before evaluating each trigonometric function using JavaScript's built-in Math library. The primary functions sin(theta) and cos(theta) are evaluated directly, then the remaining four are derived: tan(theta) = sin(theta)/cos(theta), csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), and cot(theta) = cos(theta)/sin(theta). Results are displayed with up to 6 decimal places of precision, and the calculator also shows the angle converted to radians for reference. At angles where a function is undefined (division by zero), the result shows "undefined" -- for example, tan(90 deg) is undefined because cos(90 deg) = 0, and csc(0 deg) is undefined because sin(0 deg) = 0.
The six trig functions follow important periodic properties. Sine and cosine have a period of 360 deg (2pi radians), meaning sin(theta + 360 deg) = sin(theta) and cos(theta + 360 deg) = cos(theta). Tangent and cotangent have a shorter period of 180 deg (pi radians). Sine and cosine are bounded between -1 and 1, while tangent and cotangent can take any real value. Understanding these periodic and bounded properties is essential for solving trigonometric equations, analyzing waveforms, and working with alternating currents in electrical engineering. Our calculator handles angles of any magnitude and correctly identifies undefined values, making it a reliable tool for both learning and professional work.
Frequently Asked Questions
What are the six trigonometric functions?
The six fundamental trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Given an angle theta: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = sin(theta)/cos(theta), csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), and cot(theta) = cos(theta)/sin(theta). These functions describe the relationships between angles and side lengths in right triangles and are fundamental to mathematics, physics, and engineering.
How do I convert degrees to radians manually?
To convert degrees to radians, multiply the angle in degrees by pi/180. For example, 45 deg x pi/180 = pi/4 ≈ 0.7854 radians, and 90 deg x pi/180 = pi/2 ≈ 1.5708 radians. Our calculator performs this conversion automatically -- enter any angle in degrees and it displays both the trigonometric values and the equivalent radian measure as part of the results.
What are the common values of trig functions for standard angles?
Common exact values: sin(0 deg) = 0, sin(30 deg) = 1/2, sin(45 deg) = sqrt2/2 ≈ 0.7071, sin(60 deg) = sqrt3/2 ≈ 0.8660, sin(90 deg) = 1; cos(0 deg) = 1, cos(30 deg) = sqrt3/2 ≈ 0.8660, cos(45 deg) = sqrt2/2 ≈ 0.7071, cos(60 deg) = 1/2, cos(90 deg) = 0; tan(0 deg) = 0, tan(30 deg) = 1/sqrt3 ≈ 0.5774, tan(45 deg) = 1, tan(60 deg) = sqrt3 ≈ 1.7321, tan(90 deg) is undefined. Our calculator provides decimal approximations with up to 6 decimal places.
Why are some trig functions undefined for certain angles?
A trigonometric function is undefined when its denominator equals zero. tan(theta) = sin(theta)/cos(theta) is undefined at theta = 90 deg + 180 degk because cos(theta) = 0 at those angles. sec(theta) = 1/cos(theta) is also undefined at those same angles. csc(theta) = 1/sin(theta) is undefined at theta = 0 deg + 180 degk (multiples of 180 deg) because sin(theta) = 0. The calculator displays undefined for these cases.
Can I enter negative angles and angles over 360 degrees?
Yes, the calculator accepts any real number input including negative angles and angles greater than 360 deg. Negative angles represent clockwise rotation on the unit circle. For negative angles: sin(-theta) = -sin(theta), cos(-theta) = cos(theta), and tan(-theta) = -tan(theta). Angles beyond 360 deg wrap around due to periodicity -- sin(theta + 360 deg) = sin(theta) and cos(theta + 360 deg) = cos(theta).
How are trig functions used in real-world applications?
Trigonometric functions are used everywhere: in physics (projectile motion, wave mechanics, optics, and pendulum analysis), engineering (structural load analysis, signal processing, AC circuit design, and robotics kinematics), navigation (GPS triangulation, bearing calculations, and celestial navigation), computer graphics (3D rotation matrices, game physics, and animation), architecture and construction (roof slopes, stair angles, and truss design), and music theory (sound wave analysis). They are fundamental to any field that involves angles, oscillations, or periodic phenomena.
What is the relationship between sine and cosine?
Sine and cosine are related by the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, which holds for any angle theta. They are phase-shifted by 90 deg: sin(theta) = cos(90 deg - theta) and cos(theta) = sin(90 deg - theta). On the unit circle, sin(theta) is the y-coordinate and cos(theta) is the x-coordinate of the point at angle theta measured counterclockwise from the positive x-axis. Both functions have a range of [-1, 1] and a period of 360 deg (2pi radians).
Is this trig calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration or download required. You can also share your calculation results by copying the URL -- the input values are preserved in the link, making it easy to save and share specific angle calculations with students, colleagues, or teammates.