Unit Circle
Compute sine, cosine, tangent, and radian measure for any angle on the unit circle with this free online tool. Essential for trigonometry, calculus, and precalculus students.
About This Calculator
The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. It is the foundation of trigonometry because it provides a geometric way to define sine, cosine, and tangent for any angle. For a given angle theta measured from the positive x-axis, the coordinates of the intersection point on the circle give sin(theta) as the y-coordinate and cos(theta) as the x-coordinate.
This calculator accepts any angle in degrees and returns sin(theta), cos(theta), tan(theta), and the equivalent angle in radians. The tangent result displays "Undefined" when cos(theta) = 0 (at 90 deg and 270 deg). All trigonometric values are rounded to six decimal places, while the radian measure is shown to four decimal places. This tool is ideal for students learning trigonometry, teachers preparing lessons, and professionals needing quick trigonometric lookups.
How to Use
Enter an angle in degrees in the input field and click Calculate. The calculator outputs sin(theta), cos(theta), tan(theta), and the radian equivalent. Use the URL permalink feature to save and share specific calculations.
Regional Notes
Angle measurement in degrees is used universally, though radians are standard in higher mathematics and physics (US, UK, India, and worldwide). The unit circle is consistent across all educational systems -- the definitions of sine, cosine, and tangent via the unit circle are identical in CBSE (India), GCSE/A-Level (UK), and Common Core (US) curricula. The key difference is grade level at which it is introduced: India (Class 11), US (typically 10th-11th grade), and UK (GCSE foundation for basics, A-Level for depth).
Frequently Asked Questions
What is a unit circle?
A unit circle is a circle with a radius of 1 centered at the origin (0,0) of a coordinate plane. It is fundamental in trigonometry because any angle's sine and cosine are simply the y- and x-coordinates, respectively, of the point where the terminal side of the angle intersects the circle.
How do you find sin, cos, and tan on the unit circle?
For an angle theta in standard position, the intersection point on the unit circle has coordinates (cos theta, sin theta). Sine is the y-coordinate, cosine is the x-coordinate, and tangent equals sin theta / cos theta. Our calculator computes all three instantly for any angle in degrees.
What are the common angles on the unit circle?
The most commonly used angles on the unit circle are 0 deg, 30 deg (pi/6), 45 deg (pi/4), 60 deg (pi/3), 90 deg (pi/2), 180 deg (pi), 270 deg (3pi/2), and 360 deg (2pi). Their sine, cosine, and tangent values are frequently used in trigonometry, calculus, and physics.
What is the formula for converting degrees to radians?
To convert degrees to radians, multiply the angle in degrees by pi / 180. For example, 180 deg x pi/180 = pi radians, and 45 deg x pi/180 = pi/4 radians. Our calculator automatically shows the radian measure alongside sine, cosine, and tangent values.
What does tan(90 deg) equal on the unit circle?
Tan(90 deg) is undefined because cos(90 deg) = 0, and division by zero is mathematically undefined. On the unit circle, this occurs because the vertical line at x = 0 never intersects the tangent line at (1,0). Our calculator correctly displays 'Undefined' for such angles.
Is the unit circle the same for degrees and radians?
Yes, the unit circle is the same geometric construct regardless of angle units. Degrees and radians are simply two ways to measure angles on the same circle. The sine, cosine, and tangent values are identical; only the angle measure changes. Our calculator accepts degrees and outputs both trigonometric values and the radian equivalent.
How is the unit circle used in real life?
The unit circle underpins all periodic phenomena modeled by sine and cosine waves -- sound waves, alternating current, tides, pendulum motion, and planetary orbits. Engineers use it for signal processing, physicists for wave mechanics, and computer scientists for graphics rotations. Understanding the unit circle is essential for STEM fields worldwide.