QuickCalculators evaluates the six trigonometric functions for an angle in degrees, radians, or pi-radians and returns exact radical values for special angles when those values exist. Enter the angle and the function, then read sine, cosine, tangent, or a reciprocal with any undefined case explained.
Trigonometric functions relate an angle to ratios on the unit circle or in a right triangle. Sine, cosine, and tangent are the primary ratios; cotangent, secant, and cosecant are their reciprocals. This is the first trig tool on the math hub, built so later trig pages can extend the same engine.
Calculate the six trigonometric functions
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. QuickCalculators computes each for a supported angle and unit, showing a decimal and, for special angles, an exact form such as one half or the square root of 2 over 2. Undefined inputs receive an explanation instead of a blank or NaN result.
Primary functions come from side ratios or unit-circle coordinates. Reciprocal functions invert those values when the primary value is nonzero. Choosing degrees or radians changes how the same numeric angle token is interpreted, so the unit control must match the problem statement before any comparison to a key.
Calculate sine, cosine and tangent
Sine, cosine, and tangent are the core ratios most courses introduce first. On the unit circle, cosine is the x-coordinate and sine is the y-coordinate of the point at the given angle; tangent is sine divided by cosine when cosine is not zero. QuickCalculators reports those three values together when a full primary set is requested.
In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Both views agree for acute angles. For 30 degrees, sine equals one half, cosine equals the square root of 3 over 2, and tangent equals 1 over the square root of 3, which rationalizes to the square root of 3 over 3.
Calculate cotangent, secant and cosecant
Cotangent, secant, and cosecant are the reciprocals of tangent, cosine, and sine in that order. Cotangent is cosine over sine, secant is 1 over cosine, and cosecant is 1 over sine, when those denominators are nonzero. QuickCalculators computes the reciprocal set from the same angle so a prompt that asks for sec or csc does not need a second tool.
Undefined cases appear when the primary value in the denominator is zero. Secant of 90 degrees is undefined because cosine of 90 degrees is 0. Cosecant of 0 degrees is undefined because sine of 0 is 0. The page names the zero denominator instead of inventing a finite stand-in.
Work in degrees or radians
Angles may be entered in degrees, radians, or multiples of pi radians. A full turn is 360 degrees or 2 pi radians. QuickCalculators converts according to the selected unit so that 30 degrees and pi over 6 radians map to the same geometric angle. Mixing units without converting is a frequent source of wrong calculator output.
Pi-radian mode accepts coefficients of pi so special angles stay exact: pi over 4 matches 45 degrees, and pi over 3 matches 60 degrees. Decimal radian entries still compute, but exact radical outputs appear when the angle matches a known special value. Confirm the unit label on the problem before comparing decimals to a key that used the other system.
Find exact values for special angles
Special angles 0, 30, 45, 60, and 90 degrees, with matching radian forms, have exact sine and cosine values built from 0, 1/2, the square root of 2 over 2, the square root of 3 over 2, and 1. QuickCalculators returns those exact forms rather than only a rounded decimal when the angle matches. Exact values appear on many exams.
Tangent, cotangent, secant, and cosecant inherit exact forms from those sine and cosine values when defined. Memorizing the 30-45-60 set covers a large share of early trig drills. When the angle is not special, a decimal approximation appears with a note that no simpler exact classroom form was produced.
Calculate the sine of 30 degrees
The worked case finds the sine of 30 degrees on QuickCalculators. In a 30-60-90 triangle the side opposite 30 degrees is half the hypotenuse, so sine of 30 degrees equals one half. The exact value 1/2 is the reference fixture for this page.
- Set the angle to 30 and the unit to degrees.
- Select sine.
- Read the exact value 1/2 and confirm with the 30-60-90 ratio.
The same angle in radians is pi over 6 and yields the same sine.
Avoid this common mistake
The misconception named "trig buttons always use degrees" ignores calculator mode. Many devices default to radians, so entering 30 while in radian mode computes sine of 30 radians, not 30 degrees. Always match the unit control to the problem. This page makes the unit an explicit choice so that mode mismatch is visible before the answer is trusted.
Frequently asked questions
What are the six trigonometric functions?
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. The first three are primary ratios; the last three are reciprocals. QuickCalculators evaluates all six for a supported angle and unit.
What is the sine of 30 degrees?
The sine of 30 degrees equals one half. The matching radian measure is pi over 6. Entering that angle in degree mode returns the exact value 1/2.
How do degrees and radians differ?
Degrees divide a full turn into 360 equal parts; radians measure arc length on the unit circle, with a full turn equal to 2 pi. The same geometric angle has different numeric labels in each system. Select the unit that matches the problem.
When is tangent undefined?
Tangent is undefined when cosine is zero, such as at 90 degrees or pi over 2 radians. Division by zero has no real value. The calculator explains that case instead of returning a silent error code.
What is a reciprocal trig function?
A reciprocal trig function inverts a primary value: cosecant inverts sine, secant inverts cosine, and cotangent inverts tangent when those values are nonzero. Exact special-angle values invert to exact reciprocals when defined.
Why return exact values for special angles?
Exact values such as the square root of 2 over 2 match exam keys that reject rounded decimals for 30, 45, and 60 degree problems. Decimals remain available for non-special angles. The engine prefers exact radical forms when the angle is special.
Summary
QuickCalculators evaluates sine, cosine, tangent, and their reciprocals in degrees, radians, or pi-radians, with exact special-angle values when available. The fixture sine of 30 degrees equals one half. Undefined cases such as tangent of 90 degrees are explained rather than hidden. Match the unit to the problem so a degree prompt is never evaluated as radians by accident.