Trigonometric Functions
Free online trigonometric functions calculator evaluates sin, cos, tan, csc, sec, and cot for any angle in degrees. Interactive graph for visualizing sine, cosine, and tangent functions.
About This Calculator
This trigonometric functions calculator evaluates all six fundamental trigonometric functions -- sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) -- for any angle entered in degrees. These are the core circular functions that form the foundation of trigonometry, calculus, and their applications in science and engineering. Students, teachers, engineers, and professionals across physics, astronomy, architecture, and electronics rely on these functions daily for solving angle-based problems.
Each function is computed using standard mathematical formulas: sin and cos from the JavaScript Math library, tan = sin/cos, csc = 1/sin, sec = 1/cos, and cot = cos/sin. Values that are undefined (division by zero) display as 'undefined'. The interactive chart plots sin(theta), cos(theta), or tan(theta) across 0 deg to 360 deg, with proper handling of asymptotic behavior at vertical asymptotes where tangent approaches infinity.
The six functions relate directly to the unit circle: cos(theta) gives the x-coordinate and sin(theta) gives the y-coordinate of a point on the circle at angle theta. Tangent represents the slope of the radius line, while cosecant, secant, and cotangent are the reciprocal functions. Understanding these relationships is key to mastering trigonometry, solving triangles, and modeling periodic phenomena such as sound waves, alternating current, and seasonal cycles.
How to use
Enter any angle in degrees into the input field, then click Calculate. The result panel displays all six function values simultaneously. Use the chart tabs to switch between visualizing individual functions on the interactive graph. The calculator URL updates automatically, allowing you to share or bookmark specific calculations.
Regional notes
Trigonometry is a universal branch of mathematics taught worldwide. Degrees are the most common unit for angle measurement in secondary education across all regions (IN, US, UK). While this calculator accepts degrees and converts internally to radians for computation, professionals in higher mathematics and physics often work in radians. The conversion formula radians = degrees x pi/180 is standard internationally.
Frequently Asked Questions
What are the six trigonometric functions?
The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). They describe the ratios of sides in a right triangle relative to an angle and define the coordinates of points on the unit circle.
How do you evaluate trig functions for any angle?
For any angle theta, trig functions are evaluated using the unit circle. The angle is measured counterclockwise from the positive x-axis. The x-coordinate of the intersection point equals cos(theta), and the y-coordinate equals sin(theta). Other functions are derived as reciprocals or ratios of these two.
What is the difference between degrees and radians?
Degrees and radians are two units for measuring angles. A full circle is 360 deg or 2pi radians. To convert: radians = degrees x pi/180, degrees = radians x 180/pi. This calculator accepts degrees and internally converts to radians for computation.
What are the signs of trig functions in each quadrant?
In quadrant I (0 deg-90 deg): all six functions are positive. In quadrant II (90 deg-180 deg): only sine and cosecant are positive. In quadrant III (180 deg-270 deg): only tangent and cotangent are positive. In quadrant IV (270 deg-360 deg): only cosine and secant are positive. The mnemonic is 'All Students Take Calculus'.
How are trig functions used in real-world applications?
Trigonometric functions are used in countless real-world applications including: GPS navigation (triangulation), architecture (roof angles and load calculations), music theory (sound wave analysis), medical imaging (CT scan reconstruction), astronomy (planetary orbit calculations), and electrical engineering (AC circuit analysis).
What is the period of each trigonometric function?
Sine, cosine, cosecant, and secant have period 360 deg (2pi radians). Tangent and cotangent have period 180 deg (pi radians). This means sin(theta + 360 deg) = sin(theta) for all theta, while tan(theta + 180 deg) = tan(theta). The period is the smallest positive angle after which the function repeats.
Why does tan(90 deg) show as undefined?
Tan(theta) equals sin(theta)/cos(theta). At 90 deg, cos(90 deg) = 0, so the ratio sin(90 deg)/cos(90 deg) = 1/0, which is mathematically undefined. On the unit circle, this corresponds to a vertical line with infinite slope. The graph of tan(theta) has a vertical asymptote at 90 deg and 270 deg where the function approaches positive or negative infinity.
How do you remember which trig functions are positive in each quadrant?
The mnemonic 'All Students Take Calculus' helps you remember which functions are positive in each quadrant starting from quadrant I: All six functions positive (0 deg-90 deg), only Sine and cosecant positive (90 deg-180 deg), only Tangent and cotangent positive (180 deg-270 deg), only Cosine and secant positive (270 deg-360 deg). Some textbooks use 'ASTC' or the phrase 'Add Sugar To Coffee' as alternatives.