Kalkulator Lingkaran Satuan – Nilai Trigonometri Tepat
Find exact sine, cosine, tangent, cosecant, secant, and cotangent values for any angle on the unit circle. Enter degrees or radians to get coordinates, trig ratios, and the reference angle — instantly.
Cara menggunakan kalkulator ini
- Masukkan Angle
- Masukkan Unit
- Klik tombol Hitung
- Baca hasil yang ditampilkan di bawah kalkulator
What Is the Unit Circle?
The unit circle is a circle with radius 1 centered at the origin of a coordinate plane. It's the foundation of trigonometry because every point on it has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis.
This means: for any angle θ, the x-coordinate of the point on the unit circle equals cos(θ), and the y-coordinate equals sin(θ). Since the radius is 1, the Pythagorean theorem gives us the fundamental identity: sin²θ + cos²θ = 1.
The unit circle connects geometry, algebra, and trigonometry into one elegant framework used throughout mathematics, physics, engineering, and computer science.
Key Angles and Their Exact Values
The unit circle is defined by specific "special angles" with exact, non-decimal values:
| Degrees | Radians | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 30° | π/6 | √3/2 | 1/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | 1/2 | √3/2 | √3 |
| 90° | π/2 | 0 | 1 | undefined |
| 120° | 2π/3 | −1/2 | √3/2 | −√3 |
| 135° | 3π/4 | −√2/2 | √2/2 | −1 |
| 150° | 5π/6 | −√3/2 | 1/2 | −1/√3 |
| 180° | π | −1 | 0 | 0 |
| 270° | 3π/2 | 0 | −1 | undefined |
| 360° | 2π | 1 | 0 | 0 |
Degrees vs Radians
Angles can be measured in degrees (0–360°) or radians (0–2π). Radians are the "natural" unit for mathematics — they directly relate angle to arc length on the unit circle.
Conversion formulas:
- Degrees to radians: multiply by π/180
- Radians to degrees: multiply by 180/π
Key equivalences: 180° = π rad, 360° = 2π rad, 90° = π/2 rad, 45° = π/4 rad.
In calculus and advanced mathematics, radians are always preferred. The derivative of sin(x) is cos(x) only when x is in radians.
Understanding Reference Angles
A reference angle is the acute angle (0°–90°) between the terminal side of an angle and the x-axis. Using reference angles, you can find trig values for any angle using values from Quadrant I.
Finding reference angles:
- Quadrant I (0°–90°): reference angle = θ
- Quadrant II (90°–180°): reference angle = 180° − θ
- Quadrant III (180°–270°): reference angle = θ − 180°
- Quadrant IV (270°–360°): reference angle = 360° − θ
Signs by quadrant (ASTC: All Students Take Calculus): Q1: all positive; Q2: sin positive; Q3: tan positive; Q4: cos positive.
All Six Trigonometric Functions
The six trig functions on the unit circle:
- sin θ = y-coordinate (opposite/hypotenuse in right triangle)
- cos θ = x-coordinate (adjacent/hypotenuse)
- tan θ = sin θ / cos θ = y/x (opposite/adjacent)
- csc θ = 1/sin θ (cosecant)
- sec θ = 1/cos θ (secant)
- cot θ = 1/tan θ = cos θ/sin θ (cotangent)
tan θ and sec θ are undefined when cos θ = 0 (at θ = 90°, 270°). csc θ and cot θ are undefined when sin θ = 0 (at θ = 0°, 180°, 360°).
Pythagorean Identities
The unit circle generates three fundamental Pythagorean identities:
- sin²θ + cos²θ = 1 (from the equation of the unit circle x² + y² = 1)
- 1 + tan²θ = sec²θ (divide the first identity by cos²θ)
- 1 + cot²θ = csc²θ (divide the first identity by sin²θ)
These identities are used throughout calculus, physics, and engineering to simplify expressions and solve equations involving trigonometric functions.
Real-World Applications of the Unit Circle
The unit circle isn't just abstract math — it underlies much of the physical world:
- Circular motion: Position of objects moving in circles (planets, rotating machinery, wheels)
- Wave mechanics: Sound waves, light waves, and electromagnetic radiation are described by sine and cosine functions
- Signal processing: Fourier analysis decomposes any signal into sine waves using unit circle principles
- Computer graphics: Rotation matrices use cos and sin to rotate 2D and 3D objects
- AC circuits: Alternating current is modeled as a sine wave; impedance calculations use complex numbers on the unit circle
Terakhir diperbarui: March 2026
Frequently Asked Questions
What is the unit circle used for?
The unit circle defines all trigonometric function values for any angle. It's used in algebra, precalculus, calculus, physics, engineering, and computer science to analyze periodic phenomena, rotations, and wave behavior.
How do you memorize the unit circle?
Use the pattern: for 0°, 30°, 45°, 60°, 90° — sin values are √0/2, √1/2, √2/2, √3/2, √4/2 (simplifying to 0, 1/2, √2/2, √3/2, 1). Cos is the reverse. Then apply ASTC signs for other quadrants.
What are the coordinates on the unit circle?
Every point on the unit circle has coordinates (cos θ, sin θ). For example: 0° = (1,0); 90° = (0,1); 180° = (-1,0); 270° = (0,-1); 45° = (√2/2, √2/2).
Why is pi important in the unit circle?
One complete revolution of the unit circle equals 2π radians (circumference = 2πr = 2π for r=1). Half revolution = π radians = 180°. Pi is the natural conversion between degrees and radians.
What is sin 30° exactly?
sin 30° = 1/2 = 0.5. This comes from the 30-60-90 triangle: the side opposite 30° is half the hypotenuse. On the unit circle, the y-coordinate at 30° is exactly 1/2.
What is the unit circle equation?
x² + y² = 1. Any point (x, y) on the unit circle satisfies this equation. Since x = cos θ and y = sin θ, this directly gives the fundamental identity: cos²θ + sin²θ = 1.
What is cos 45° in exact form?
cos 45° = √2/2 ≈ 0.7071. This comes from the 45-45-90 triangle, where both legs are equal. In exact form: √2/2 (sometimes written as 1/√2, which is equivalent after rationalization).
How do you find tan using the unit circle?
tan θ = sin θ / cos θ = y/x. It's also the length of the tangent line from the point on the unit circle to the x-axis intersection. tan θ is undefined where cos θ = 0 (at 90° and 270°).
What is the period of sine and cosine?
Both sine and cosine have a period of 2π radians (360°). This means they repeat their values every full revolution: sin(θ + 2π) = sin(θ) for all θ. Tangent has a period of π radians (180°).
Is the unit circle the same as a trigonometric chart?
They're related but different. The unit circle is a geometric representation showing coordinates at every angle. A trigonometric chart/table lists the numerical values. The unit circle is more fundamental — the chart is derived from it.