Trigonometric Functions - All Six Trig Values From One Angle

A trigonometric functions calculator takes a single real angle and returns all six trig functions - sin, cos, tan, csc, sec, and cot - in one result panel. Pick the angle in degrees, radians, or multiples of pi, and the panel reports the six values with the unit-circle quadrant, the sign of sine and cosine, and the reference angle in degrees.

• • Six-function lookup at a reference angle: Read sin, cos, tan, csc, sec, and cot of 30, 45, 60, or any other angle at the same time.
• • Reciprocal consistency check: Confirm that csc equals 1/sin, sec equals 1/cos, and cot equals cos/sin from the same panel.
• • Undefined-case detection: Spot the angles where tan, sec, csc, or cot are undefined because their denominator is zero.
• • Quadrant sign read-out: Read the quadrant (I, II, III, IV, or on axis) and the sign of sin/cos at the reduced angle.

The six trig functions split into two natural groups. Sine, cosine, and tangent are the primary group, read from the y and x coordinates of a point on the unit circle. Cosecant, secant, and cotangent are the reciprocal group, defined as 1/sin, 1/cos, and cos/sin respectively.

Once you know the sine and cosine of the reduced angle, the other four functions are arithmetic: divide or invert. A single calculator returns all six trig values for any real angle in one panel.

When only one of the six trig functions is needed and the user wants to pick the function explicitly, Trigonometry Calculator returns a single value together with the quadrant and reference angle.

The trigonometric functions calculator reads the angle and the unit, converts the angle to radians, reduces it to the principal branch [0, 2*pi), and evaluates sine and cosine. Tangent, cosecant, secant, and cotangent are derived from those two values.

• angleValue: Numeric angle entered in the form, combined with angleUnit
• angleUnit: Unit of the input angle: degrees, radians, or multiples of pi
• theta (radians): Reduced angle in radians, kept on the principal branch [0, 2*pi)
• sin / cos: Primary outputs, computed from the reduced angle and rounded to 6 decimal places
• tan / csc / sec / cot: Derived outputs, reported as 'undefined' when their denominator is within 1e-12 of zero
• referenceAngle: Acute reference angle in degrees, always in [0, 90]

The principal-branch reduction is what makes the panel consistent across equivalent angles. A 30-degree input and a 390-degree input are the same angle on the unit circle, so after reduction both panels show the same six values.

The reciprocal functions come from the same sine and cosine, so csc equals 1/sin, sec equals 1/cos, and cot equals cos/sin whenever the denominator is not zero.

Wikipedia: Trigonometric functions defines the six trigonometric functions on the unit circle and in right triangles, with csc, sec, and cot stated explicitly as the reciprocals of sin, cos, and tan respectively.

When the problem only uses the three primary trig functions and does not need the reciprocals, Sin Cosine Tangent Calculator returns sin, cos, and tan together with a Pythagorean identity check.

Four ideas make all six trig function values come out right at any angle.

The five reference angles 0, 30, 45, 60, and 90 degrees come from the 30-60-90 and 45-45-90 right triangles, and the same values carry over to radians at 0, pi/6, pi/4, pi/3, and pi/2.

When the angle is one of the special reference angles like 30, 45, or 60 degrees and the user wants the exact form such as 1/2, sqrt(2)/2, or sqrt(3)/2, Exact Value of Trig Functions Calculator returns the closed-form values instead of rounded decimals.

Five short steps return all six trig functions of any real angle.

1. 1 Enter the angle value: Type the numeric angle. Negative angles, angles past 360 degrees, and decimals all work because the calculator reduces the input to the principal branch before evaluation.
2. 2 Pick the angle unit: Select degrees, radians, or multiples of pi. Use 'Multiples of pi' to enter values like 0.5 for pi/2 and 0.25 for pi/4 without typing the constant.
3. 3 Read all six trig functions: The result panel shows sin, cos, tan, csc, sec, and cot together. Any reciprocal function whose denominator is near zero is reported as 'undefined'.
4. 4 Check the quadrant and sign: Read the unit-circle quadrant (I, II, III, IV, or on axis) and the sign of sin and cos. The sign and quadrant come from the reduced sine and cosine, so a 30-degree input and a 390-degree input match.
5. 5 Use the reference angle for chart work: Read the acute reference angle in degrees. The reference-value chart lists values for angles in [0, 90], so the reference angle is what you compare to the chart entries.

Set the angle to 30, the unit to degrees: sin = 0.5, cos = 0.866025, tan = 0.577350, csc = 2, sec = 1.154700, cot = 1.732050, quadrant = I, reference angle = 30 degrees. Switch the unit to radians and enter 1.5707963 for pi/2, where tan and sec become 'undefined'.

If the input angle arrives in radians or pi multiples and the rest of the problem uses degrees, Trig Degree Calculator reformats the angle to a plain decimal degree value before the trig functions run.

A single six-function panel removes the need to switch tools when the same angle is needed in different forms.

• • All six trig functions from one input: sin, cos, tan, csc, sec, and cot return together, so the reciprocal relationships are visible in the same panel.
• • Three angle units on the same form: Degrees, radians, and multiples of pi are all accepted. The same panel works for classroom, engineering, and symbolic-form problems.
• • Built-in undefined handling: When the denominator of a reciprocal or ratio function is zero, the panel reports 'undefined' rather than a misleadingly large number.
• • Reciprocal and ratio consistency: Because csc, sec, and cot are derived from the same sin and cos, the panel always satisfies csc = 1/sin, sec = 1/cos, and cot = cos/sin.

When a downstream problem uses a reciprocal form - for example, the law of sines in 1/sin form - having csc, sec, and cot in the same panel saves a manual reciprocal step. A sign error is easy to spot: if sin reads negative in quadrant I, the input is mis-reduced.

Once all six trig functions are visible, identities like the Pythagorean identity sin^2 + cos^2 = 1 or the reciprocal identity csc * sin = 1 can be checked with Trig Identities Calculator for the same angle.

Four variables determine the value of the six trig functions, and two limitations tell you when the result is on the edge of validity.

• • The tool returns the principal real angle only. It does not evaluate the complex-valued trig functions, the hyperbolic functions (sinh, cosh, tanh), or the inverse trig functions.
• • Floating-point arithmetic means the six values are rounded to 6 decimal places, the reference angle to 4 decimal places, and the 'undefined' boundary is only detected when sine or cosine is within about 1e-12 of zero.

The reciprocal functions inherit the sign of their primary counterpart and the asymptotic behavior at the zeros of the denominator. tan is positive in quadrants I and III, negative in II and IV, undefined at 90 plus any integer multiple of 180 degrees. sec and csc follow cosine and sine.

The Pythagorean identity sin^2(theta) + cos^2(theta) = 1 holds for any real angle theta, and the remaining four trigonometric functions follow as reciprocals or ratios of sine and cosine, as documented in the Wolfram MathWorld: Sine reference entry.

When the workflow runs the trig functions in reverse and needs the principal angle from a known value, Inverse Trigonometric Calculator returns arcsin, arccos, arctan, arccsc, arcsec, and arccot in the same panel.