TRIGONOMETRYTrigonometryMathematics Calculator
sin

The Sine Function

Sine relates an angle to the ratio of the opposite side over the hypotenuse. On the unit circle, sin(θ) is the y-coordinate — the vertical component of the point at angle θ.

Concept Fundamentals
[-1, 1]
Range
2π (360°)
Period
sin(-θ) = -sin(θ)
Odd Function
sin²θ + cos²θ = 1
Identity

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Sine is positive in quadrants 1 and 2, negative in 3 and 4. Use ASTC: All Students Take Calculus. sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2 — memorize these for quick problem-solving. The inverse is arcsin: arcsin(x) returns the angle whose sine is x, restricted to [-π/2, π/2].

Key quantities
[-1, 1]
Range
Key relation
2π (360°)
Period
Key relation
sin(-θ) = -sin(θ)
Odd Function
Key relation
sin²θ + cos²θ = 1
Identity
Key relation

Ready to run the numbers?

Why: Sine is fundamental to modeling waves, oscillations, and circular motion. Every periodic phenomenon — from sound waves to AC circuits — can be expressed using sine.

How: sin(θ) = opposite/hypotenuse in a right triangle. On the unit circle, it equals the y-coordinate of the point at angle θ. The function repeats every 2π radians.

Sine is positive in quadrants 1 and 2, negative in 3 and 4. Use ASTC: All Students Take Calculus.sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2 — memorize these for quick problem-solving.
Sources:Wolfram MathWorld — SineKhan Academy — Trigonometry

Run the calculator when you are ready.

Start CalculatingEnter an angle to compute sin(θ) and all related trig values

Examples — Click to Load

sine.sh
CALCULATED
$ sin --angle 45° --all-functions
sin(θ)
0.70710678
cos(θ)
0.70710678
tan(θ)
1
Quadrant
Q1
csc(θ)
1.41421356
sec(θ)
1.41421356
cot(θ)
1
Ref Angle
45°
Share:
Sine Calculator Result
sin(45°)
0.70710678
Q1ref 45°sin²+cos² = 1
numbervibe.com/calculators/mathematics/trigonometry/sine-calculator

Trig Value Breakdown

All 6 Trig Functions

sin² vs cos² (Pythagorean Identity)

Calculation Breakdown

CONVERSION
Input Angle
45°
Convert to Radians
0.78539816 rad
45° × π/180
Normalized Angle
45°
\text{theta} mod 360^{circ}
Quadrant
Q1
45.0° is in quadrant 1
Reference Angle
45°
ext{Acute} ext{angle} ext{to} x- ext{axis}
PRIMARY RESULT
SINE VALUE
0.70710678
sin(45°)
RELATED VALUES
Cosine
0.70710678
cos(45°)
Tangent
1
tan(45°) = sin/cos
Cosecant
1.41421356
1/\text{sin}(\text{theta} )
Secant
1.41421356
1/\text{cos}(\text{theta} )
Cotangent
1
\text{cos}(\text{theta} )/\text{sin}(\text{theta} )
PYTHAGOREAN IDENTITY
sin²(θ)
0.5
cos²(θ)
0.5
sin²(θ) + cos²(θ)
1
ext{Always} ext{equals} 1

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🌊

Sound waves, ocean waves, and electromagnetic radiation all follow sinusoidal patterns.

— MIT OpenCourseWare

📜

The sine function was first developed by Indian mathematician Aryabhata around 500 AD.

— Wolfram MathWorld

Key Takeaways

  • • sin(θ) = opposite / hypotenuse in a right triangle; on the unit circle it is the y-coordinate
  • • Range is always [-1, 1]. Key values: sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2, sin(90°) = 1
  • • Sine is an odd function: sin(-θ) = -sin(θ) with period 2π (360°)
  • • The Pythagorean identity sin²(θ) + cos²(θ) = 1 always holds
  • • Sine is positive in Q1 and Q2 (0°–180°), negative in Q3 and Q4 (180°–360°)

Did You Know?

🌊Sound waves, ocean waves, and electromagnetic radiation all follow sinusoidal patterns — sine is the foundation of wave physicsSource: MIT OpenCourseWare
📜The sine function was first developed by Indian mathematician Aryabhata around 500 AD, centuries before European adoptionSource: Wolfram MathWorld
🎵Fourier's theorem proves that ANY periodic function can be decomposed into a sum of sine and cosine wavesSource: Khan Academy
🔭NASA uses sine functions to model satellite orbits, signal processing, and trajectory calculationsSource: NASA Technical Reports
📱Every time your phone processes audio or displays graphics, sine calculations happen millions of times per secondSource: IEEE Signal Processing
🏗️Civil engineers use sine to calculate forces in bridge cables, building loads, and structural stress analysisSource: Engineering Toolbox

How the Sine Function Works

The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. On the unit circle, it is simply the y-coordinate of the point at angle θ.

Unit Circle Definition

For any angle θ measured counterclockwise from the positive x-axis, the terminal point on a circle of radius 1 has coordinates (cos θ, sin θ). This geometric definition extends sine to all real numbers, not just acute angles.

Quadrant Sign Rules (ASTC)

All positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4. Memorize "All Students Take Calculus" to remember which functions are positive in each quadrant.

Reference Angles

The reference angle is the acute angle formed between the terminal side and the x-axis. sin(θ) = ±sin(reference angle), with the sign determined by the quadrant. This lets you evaluate sine for any angle using only values from 0° to 90°.

Expert Tips

Memorize the Special Angles

sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2. The pattern is √1/2, √2/2, √3/2 for 30°, 45°, 60°. Use the Unit Circle Calculator to visualize.

Use Identities to Simplify

sin(A+B) = sinA·cosB + cosA·sinB lets you find exact values for non-standard angles. Try the Sum & Difference Calculator.

Degrees vs Radians

Always check your calculator mode! π radians = 180°. In calculus, radians are required for derivatives: d/dx[sin(x)] = cos(x) only when x is in radians.

The Pythagorean Identity

sin²θ + cos²θ = 1 always. If you know sin(θ), then cos(θ) = ±√(1 - sin²θ), with the sign from the quadrant. See the Trig Identities Calculator.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorScientific CalculatorManual Computation
All 6 trig functions at once✅❌ One at a time❌
Quadrant & reference angle✅❌✅ Slow
Visual charts & breakdown✅❌❌
Step-by-step explanation✅❌✅
Pythagorean identity check✅❌⚠️ Manual
Copy & share results✅❌❌
Degrees and radians✅✅✅
Preset examples✅❌❌

Frequently Asked Questions

What is the range of the sine function?

The sine function always outputs values between -1 and 1, inclusive. sin(90°) = 1 is the maximum and sin(270°) = -1 is the minimum. This bounded range makes sine ideal for modeling oscillations.

What is the difference between degrees and radians?

Degrees divide a full rotation into 360 parts, while radians use the radius as the measuring unit (2π radians = 360°). To convert: radians = degrees × π/180. Radians are preferred in calculus because derivative formulas are simpler.

Why is sine called an odd function?

A function is odd when f(-x) = -f(x). For sine: sin(-θ) = -sin(θ). Graphically, this means the sine curve has rotational symmetry about the origin — rotating it 180° produces the same graph.

How do I find sine without a calculator?

Memorize the special angles: sin(0°)=0, sin(30°)=1/2, sin(45°)=√2/2≈0.707, sin(60°)=√3/2≈0.866, sin(90°)=1. For other angles, use reference angles and quadrant signs, or the Taylor series: sin(x) ≈ x - x³/6 + x⁵/120.

What is sin²(θ) + cos²(θ)?

Always exactly 1, for every angle. This is the Pythagorean identity, derived from the Pythagorean theorem on the unit circle. It is one of the most important identities in trigonometry and is used to derive many other formulas.

Where is sine used in real life?

Sine appears in physics (wave motion, pendulums, AC circuits), engineering (signal processing, structural analysis), music (sound waves, synthesizers), navigation (GPS calculations), and computer graphics (rotations, animations).

What is the period of sine?

The period of sin(θ) is 2π radians (360°), meaning sin(θ + 2π) = sin(θ). For sin(Bθ), the period becomes 2π/B. This is fundamental to understanding frequency in wave analysis.

How is sine related to cosine?

Sine and cosine are co-functions: sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). They are also phase-shifted versions of each other: sin(θ) = cos(θ - 90°). Together they form the basis of all trigonometric functions.

Sine Function by the Numbers

[-1, 1]
Output Range
360°
Period
4
Special Angles
∞
Applications

Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices), always verify with certified computational tools. Not a substitute for professional engineering analysis.

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