Theta
Calculate sin, cos, tan, csc, sec, cot for any angle in degrees instantly. Free theta calculator with degree-to-radian conversion and 6-decimal precision.
About This Calculator
The Theta Calculator computes all six fundamental trigonometric functions for any angle theta entered in degrees. The trigonometric functions--sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)--are essential tools in mathematics, physics, engineering, and countless other fields. Whether you are a student learning trigonometry, a physicist analyzing wave motion, an engineer designing structures, or a professional in navigation and surveying, this calculator provides instant access to all trigonometric values for any angle.
The calculator converts the input angle from degrees to radians (multiply by pi/180) before evaluating the trigonometric functions. The primary functions are defined on the unit circle: sin(theta) is the y-coordinate and cos(theta) is the x-coordinate of the point at angle theta. From these, 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.
The sine and cosine functions have a range of [-1, 1] and are periodic with period 360 deg (2pi radians), while tangent and cotangent have period 180 deg (pi radians) and can take any real value. At angles where a function is undefined (division by zero), the calculator displays "undefined" -- for example, tan(90 deg) is undefined because cos(90 deg) = 0. Understanding when each function is undefined is a key part of mastering trigonometry and is important for avoiding division errors in real-world calculations.
Frequently Asked Questions
What is theta in trigonometry?
Theta (theta) is a Greek letter commonly used to represent an unknown angle in mathematics and physics. In trigonometry, theta is the angle variable around which the trigonometric functions--sine, cosine, tangent, and their reciprocals--are defined. When you see theta in a formula, it represents an angle measured in degrees or radians.
How are the six trigonometric functions related to theta?
Given an angle theta in a right triangle or on the unit circle: 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 are periodic with sin, cos, sec, csc having period 360 deg (2pi rad) and tan, cot having period 180 deg (pi rad).
How do you convert degrees to radians for theta?
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. The theta calculator automatically performs this conversion internally -- enter any angle in degrees and it returns both the trigonometric values and the equivalent radian measure.
What are the common trigonometric values for standard angles?
Common values: sin(0 deg) = 0, sin(30 deg) = 0.5, sin(45 deg) ≈ 0.7071, sin(60 deg) ≈ 0.8660, sin(90 deg) = 1, sin(180 deg) = 0, sin(270 deg) = -1; cos(0 deg) = 1, cos(30 deg) ≈ 0.8660, cos(45 deg) ≈ 0.7071, cos(60 deg) = 0.5, cos(90 deg) = 0, cos(180 deg) = -1, cos(270 deg) = 0; tan(0 deg) = 0, tan(30 deg) ≈ 0.5774, tan(45 deg) = 1, tan(60 deg) ≈ 1.7321.
What does it mean when tan or sec is undefined?
Tan(theta) is undefined when cos(theta) = 0, which occurs at theta = 90 deg + 180 degk (odd multiples of 90 deg). Sec(theta) = 1/cos(theta) is also undefined at these angles. Similarly, csc(theta) = 1/sin(theta) is undefined at theta = 0 deg + 180 degk (multiples of 180 deg) and cot(theta) is undefined when tan(theta) = 0. The calculator displays undefined for these cases.
How is the angle theta used in real-world applications?
The angle theta is used extensively in physics (projectile motion, wave mechanics, optics), engineering (structural analysis, signal processing, robotics), navigation (GPS triangulation, bearing calculations), computer graphics (rotation matrices, 3D rendering), and architecture (roof slopes, stair angles, truss design). Trigonometric functions of theta are fundamental to any field involving angles and periodic phenomena.
Can I use this calculator for negative angles?
Yes, you can enter negative angles. Negative angles represent clockwise rotation on the unit circle. The trigonometric functions handle negative inputs consistently: sin(-theta) = -sin(theta), cos(-theta) = cos(theta), and tan(-theta) = -tan(theta). For example, sin(-30 deg) = -0.5 and cos(-30 deg) ≈ 0.8660. The calculator accepts any real number input.
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 also 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. Both functions are periodic with period 360 deg and range [-1, 1].