Trigonometry11 min read
Sine, Cosine & Tangent Explained: Visual Trigonometry Guide
Master sine, cosine, and tangent with visual explanations. Understand SOH-CAH-TOA, learn when to use each function, and see real-world applications.
Enter Angle
Quick reference
sin² θ + cos² θ = 1
tan θ = sin θ / cos θ
Results
Enter an angle to compute its trig values
Sine
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Cosine
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Tangent
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Cosecant
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Secant
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Cotangent
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Angle (degrees)
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Angle (radians)
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Sine, Cosine, and Tangent Explained
The three primary trigonometric functions — sine, cosine, and tangent — describe the relationship between the angle of a right triangle and the lengths of its sides. They're the building blocks of trigonometry and appear in every calculation that involves angles, from architectural drawings to GPS satellite positioning.
SOH-CAH-TOA: The Right Triangle Definitions
For an acute angle θ in a right triangle:
SOH — sin(θ) = Opposite ÷ Hypotenuse
CAH — cos(θ) = Adjacent ÷ Hypotenuse
TOA — tan(θ) = Opposite ÷ Adjacent
The "opposite" side is the leg across from the angle, the "adjacent" side is the leg next to the angle (not the hypotenuse), and the "hypotenuse" is the longest side, opposite the right angle.
The Reciprocal Functions
Cosecant, secant, and cotangent are the reciprocals of the primary functions:
- csc(θ) = 1 / sin(θ) = Hypotenuse ÷ Opposite
- sec(θ) = 1 / cos(θ) = Hypotenuse ÷ Adjacent
- cot(θ) = 1 / tan(θ) = Adjacent ÷ Opposite
Common Angle Values
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
| 180° | 0 | -1 | 0 |
Beyond Right Triangles: The Unit Circle
The unit circle generalizes sin, cos, and tan to any angle, including angles greater than 90° and negative angles. For any point on a circle of radius 1 centered at the origin, the x-coordinate equals cos(θ) and the y-coordinate equals sin(θ). Tangent is then sin(θ)/cos(θ), which becomes undefined wherever cos(θ) = 0. To explore this further, try our unit circle calculator.
Related Tools
- SOHCAHTOA Calculator — solve right triangles using the trig ratios
- Inverse Trig Calculator — find an angle from its sin, cos, or tan value
- Unit Circle Calculator — visualize trig functions on the unit circle
- Angle Converter — switch between degrees, radians, and gradians
Frequently Asked Questions
What does sin, cos, and tan mean?
Sine (sin), cosine (cos), and tangent (tan) are the three primary trigonometric functions. In a right triangle: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. They describe the ratio of side lengths for any given angle.
Should I use degrees or radians?
Use degrees for everyday problems, geometry, and construction. Use radians for calculus, physics, and most programming languages (JavaScript's Math.sin, Math.cos, Math.tan all expect radians). To convert: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
What are csc, sec, and cot?
Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocal trig functions: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). They appear less often than sin/cos/tan but are common in calculus and physics.
Why does the calculator return "undefined" for some values?
Trig functions become undefined when division by zero would occur. Tangent and secant are undefined whenever cos(θ) = 0 (at 90°, 270°, and so on). Cotangent and cosecant are undefined whenever sin(θ) = 0 (at 0°, 180°, 360°, etc.).
What are common sin, cos, tan values?
Memorize these key values: sin(0°)=0, sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1. Cos starts at 1 and decreases the same way. Tan(0°)=0, tan(30°)=√3/3, tan(45°)=1, tan(60°)=√3, tan(90°)=undefined.
How are sin, cos, and tan used in real life?
These functions describe anything that oscillates or relates to angles: building roof pitches, surveying distances, GPS positioning, AC electrical signals, sound and light waves, animation and game physics, and architecture. They convert between angles and lengths.
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