Trigonometric Function Graphs
Plot and explore sin, cos, tan, cot, sec, and csc with full parameter control
Trigonometric Functions Reference
| Function | Formula | Period | Domain | Range |
|---|---|---|---|---|
| Sine (sin) | opposite / hypotenuse | 2π | All reals | [−1, 1] |
| Cosine (cos) | adjacent / hypotenuse | 2π | All reals | [−1, 1] |
| Tangent (tan) | sin / cos | π | x ≠ π/2 + nπ | All reals |
| Cotangent (cot) | cos / sin | π | x ≠ nπ | All reals |
| Secant (sec) | 1 / cos | 2π | x ≠ π/2 + nπ | (−∞,−1]∪[1,∞) |
| Cosecant (csc) | 1 / sin | 2π | x ≠ nπ | (−∞,−1]∪[1,∞) |
Key Angle Values
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −1/√3 |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Understanding Trigonometric Function Graphs
Trigonometric function graphs visually display how sin, cos, tan, cot, sec, and csc values change as the angle x increases. These graphs are foundational in mathematics, physics, engineering, and signal processing.
The General Form: f(x) = A · func(Bx + C) + D
Every trigonometric function can be written in a general form where four parameters control its appearance. Understanding each parameter lets you immediately read any trig graph.
- A (Amplitude): stretches or compresses the graph vertically. For sin and cos, amplitude = |A|.
- B (Frequency factor): compresses or stretches horizontally. Period = 2π / |B| for sin, cos, sec, csc. Period = π / |B| for tan and cot.
- C (Phase shift): shifts the graph horizontally. The shift is −C/B units.
- D (Vertical shift): moves the midline up or down by D units.
Graphs of Sine and Cosine
Sine and cosine graphs share the same smooth S-shaped wave pattern. Both oscillate between −1 and 1 with a period of 2π radians (360 degrees) in their natural form.
Sine Graph (sin x)
The sine function starts at 0 when x = 0, rises to a maximum of 1 at x = π/2, returns to 0 at x = π, falls to −1 at x = 3π/2, and completes one full cycle at x = 2π.
Cosine Graph (cos x)
The cosine function starts at its maximum value of 1 when x = 0. It is the sine function shifted left by π/2 radians. Cosine and sine are called cofunctions because cos(x) = sin(π/2 − x).
Graphs of Tangent and Cotangent
Tangent and cotangent graphs have vertical asymptotes where the denominator function equals zero. They have a period of π, half the period of sine and cosine.
Where Asymptotes Occur
- Tangent: undefined at x = π/2 + nπ, where n is any integer
- Cotangent: undefined at x = nπ, where n is any integer
Between each pair of asymptotes, the graph increases (for tan) or decreases (for cot) continuously through all real values.
Graphs of Secant and Cosecant
Secant and cosecant graphs are the reciprocals of cosine and sine respectively. They produce U-shaped curves that open upward and downward alternately, with vertical asymptotes at each zero of the parent function.
Neither sec nor csc has a defined amplitude since both extend to positive and negative infinity. Both have a period of 2π. The secant graph has its minimum values where cosine is at maximum, and vice versa.
Frequently Asked Questions
The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Tangent equals sine divided by cosine. Cotangent is the reciprocal of tangent. Secant is the reciprocal of cosine. Cosecant is the reciprocal of sine.
The general form is f(x) = A sin(Bx + C) + D, where A is the amplitude, B determines the period (period = 2π/B), C is the phase shift, and D is the vertical shift. The same structure applies to cosine and the other trig functions.
The natural period of sine and cosine is 2π radians (approximately 6.283). The natural period of tangent and cotangent is π radians (approximately 3.14159). When using the form A sin(Bx), the period becomes 2π divided by B.
Amplitude is the maximum vertical distance from the midline to the peak of the graph. For A sin(Bx + C) + D, the amplitude equals the absolute value of A. Tangent, cotangent, secant, and cosecant do not have a defined amplitude because they extend to positive and negative infinity.
A phase shift is a horizontal translation of a trigonometric graph. In f(x) = A sin(Bx + C) + D, the phase shift equals negative C divided by B. A positive C value shifts the graph to the left; a negative C value shifts it to the right.
Both sine and cosine graphs have the same shape, amplitude, and period. The key difference is a horizontal shift: cosine is equivalent to sine shifted left by π/2 radians (90 degrees). The sine graph starts at 0 when x = 0, while the cosine graph starts at its maximum of 1.