Select Function
Sine
Ratio of opposite side to hypotenuse
sin(x) = opposite / hypotenuse
Domain: All real numbers
Range: [-1, 1]
Input Value
Enter angle in degrees
Unit Circle
All Trigonometric Values
sin
0.500000
cos
0.866025
tan
0.577350
csc (1/sin)
2.000000
sec (1/cos)
1.154701
cot (1/tan)
1.732051
Sine Wave Graph
Amplitude
1
Period
2pi
Phase Shift
0
Vertical Shift
0
Equation:
y = sin((x))
Trigonometric Identities
Pythagorean Identities
sin^2(x) + cos^2(x) = 1
The fundamental trigonometric identity
1 + tan^2(x) = sec^2(x)
Derived by dividing the basic identity by cos^2(x)
1 + cot^2(x) = csc^2(x)
Derived by dividing the basic identity by sin^2(x)
Reciprocal Identities
csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x) = cos(x) / sin(x)
tan(x) = sin(x) / cos(x)
Double Angle Formulas
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x)
tan(2x) = 2tan(x) / (1 - tan^2(x))
Quadrant Signs (ASTC)
Quadrant 1
All positive (0 to 90)
Quadrant 2
Sin positive (90 to 180)
Quadrant 3
Tan positive (180 to 270)
Quadrant 4
Cos positive (270 to 360)
Remember: All Students Take Calculus - All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4.
Special Angles Reference
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 |
| 30 | pi/6 | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| 45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 | pi/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90 | pi/2 | 1 | 0 | undefined |
| 120 | 2pi/3 | sqrt(3)/2 | -1/2 | -sqrt(3) |
| 135 | 3pi/4 | sqrt(2)/2 | -sqrt(2)/2 | -1 |
| 150 | 5pi/6 | 1/2 | -sqrt(3)/2 | -sqrt(3)/3 |
| 180 | pi | 0 | -1 | 0 |
Result
Sine
0.500000
Exact: 1/2
Try These Examples
Quick-start with common scenarios
Practice Problems
Test your skills with practice problems
Practice with 3 problems to test your understanding.
?What Are Trigonometric Functions?
Trigonometric functions relate angles to ratios of sides in a right triangle. The six basic functions are: sin (opposite/hypotenuse), cos (adjacent/hypotenuse), tan (opposite/adjacent), and their reciprocals csc, sec, cot. Use inverse functions (arcsin, arccos, arctan) to find angles from ratios. The unit circle shows all values for angles 0-360 degrees.
About Trigonometry
Trigonometry is the branch of mathematics dealing with the relationships between angles and sides of triangles. The six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are fundamental to geometry, physics, engineering, and many other fields. This calculator evaluates any trig function in degrees, radians, or gradians.
Key Facts
- sin(x) = opposite/hypotenuse, range [-1, 1]
- cos(x) = adjacent/hypotenuse, range [-1, 1]
- tan(x) = sin(x)/cos(x) = opposite/adjacent
- Pythagorean identity: sin^2(x) + cos^2(x) = 1
- 360 degrees = 2pi radians = 400 gradians
- Special angles: 30 (sin=1/2), 45 (sin=cos=sqrt(2)/2), 60 (sin=sqrt(3)/2)
- Quadrant I: all positive, II: sin+, III: tan+, IV: cos+
- Inverse functions: arcsin returns angle in [-90, 90]
Frequently Asked Questions
What is the difference between sin, cos, and tan?
In a right triangle: sin(angle) = opposite/hypotenuse, cos(angle) = adjacent/hypotenuse, tan(angle) = opposite/adjacent. Sine measures vertical displacement on the unit circle, cosine measures horizontal displacement, and tangent is their ratio.
How do I convert between degrees and radians?
To convert degrees to radians: multiply by pi/180. To convert radians to degrees: multiply by 180/pi. For example, 90 = 90 x (pi/180) = pi/2 radians. Remember: 360 = 2pi radians.
What are the inverse trigonometric functions?
Inverse trig functions (arcsin, arccos, arctan) return the angle when given a ratio. For example, arcsin(0.5) = 30 because sin(30) = 0.5. They have restricted ranges to ensure unique outputs.
What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin. For any angle, the x-coordinate of the point on the circle equals cos(angle) and the y-coordinate equals sin(angle). It visualizes all trig values.
Why is tan undefined at 90 and 270 degrees?
tan = sin/cos. At 90 and 270 degrees, cos equals 0, making tan undefined (division by zero). On a graph, these appear as vertical asymptotes where the function approaches positive or negative infinity.
Last updated: 2026-01-15
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How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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