Trig Identities
Free online trig identities calculator evaluates all six trigonometric functions for any angle in degrees. Includes formula reference, reciprocal identities, and interactive function graph.
About This Calculator
This trig identities calculator evaluates all six trigonometric functions -- sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) -- for any angle entered in degrees. Whether you are solving right triangle problems, graphing periodic functions, or verifying trigonometric identities, this tool provides instant access to all six function values.
The calculator uses standard JavaScript Math functions: sin(theta), cos(theta), tan(theta), and their reciprocals csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), and cot(theta) = cos(theta)/sin(theta). Values that are undefined (due to division by zero) are displayed as 'undefined'. The interactive chart plots the selected function across 0 deg to 360 deg with proper handling of asymptotes.
How to use
Enter any angle in degrees (e.g. 30, 45, 90, 180) and click Calculate. All six function values appear at once. Use the chart tabs to switch between viewing sin(theta), cos(theta), or tan(theta) on the interactive graph. The URL updates automatically so you can share or bookmark specific angle calculations.
Frequently Asked Questions
What are the six trigonometric functions?
The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc = 1/sin), secant (sec = 1/cos), and cotangent (cot = 1/tan = cos/sin). Together they form the complete set of circular functions relating angles to side ratios of a right triangle.
How are reciprocal trig identities defined?
Reciprocal identities relate the three primary functions to their reciprocals: csc(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), and cot(theta) = 1/tan(theta) = cos(theta)/sin(theta). These are undefined wherever the denominator equals zero, such as csc(0 deg) and cot(0 deg) where sin(0 deg) = 0.
What is the Pythagorean identity?
The fundamental Pythagorean identity is sin^2(theta) + cos^2(theta) = 1, which holds for all angles. It is derived from the unit circle and the Pythagorean theorem. Two other forms are 1 + tan^2(theta) = sec^2(theta) and 1 + cot^2(theta) = csc^2(theta), obtained by dividing the original identity by cos^2(theta) and sin^2(theta) respectively.
When are trigonometric functions undefined?
tan(theta) and sec(theta) are undefined when cos(theta) = 0, which occurs at theta = 90 deg + nx180 deg. cot(theta) and csc(theta) are undefined when sin(theta) = 0, which occurs at theta = 0 deg + nx180 deg. sin(theta) and cos(theta) are defined for all real angles.
How do you find exact values of trig functions?
Exact values can be found using the unit circle for standard angles (0 deg, 30 deg, 45 deg, 60 deg, 90 deg). For example, sin(30 deg) = 1/2, cos(45 deg) = sqrt2/2, and tan(60 deg) = sqrt3. For non-standard angles, use sum/difference or double-angle identities combined with known exact values.
What is the range and domain of each trig function?
sin(theta) and cos(theta) have domain all real numbers and range [-1, 1]. tan(theta) has domain all reals except theta = 90 deg + nx180 deg and range all real numbers. csc(theta) and sec(theta) have range (-∞, -1] union [1, ∞). cot(theta) has domain all reals except theta = nx180 deg and range all real numbers.