Calculate cotangent precisely
The default is π/2. Use the advanced controls to inspect other angles, output precision, undefined points, and graph details.
Cotangent graph
Vertical breaks appear where sine becomes zero. The highlighted point uses your submitted angle when a defined cotangent value exists.
Formula used
When θ = π/2: cos(π/2) = 0 and sin(π/2) = 1.
Therefore: cot(π/2) = 0 / 1 = 0.
Cotangent is undefined only when sine equals zero. The calculator tests that condition with your chosen tolerance before dividing.
How to use this calculator
- Choose a preset or enter a custom angle.
- Select π multiplier mode for fractions of π.
- Select numeric mode for decimal radians or degrees.
- Set output precision and the near-zero tolerance.
- Press Calculate Cotangent to view the result above.
- Use the graph, CSV, or PDF export for review.
Example data table
These unit-circle angles help you check common cotangent values.
| Angle | Degrees | sin(θ) | cos(θ) | cot(θ) | Result state |
|---|---|---|---|---|---|
| 0 | 0° | 0 | 1 | Undefined | Sine is zero |
| π/6 | 30° | 1/2 | √3/2 | √3 | Defined |
| π/4 | 45° | √2/2 | √2/2 | 1 | Defined |
| π/3 | 60° | √3/2 | 1/2 | √3/3 | Defined |
| π/2 | 90° | 1 | 0 | 0 | Defined |
| π | 180° | 0 | −1 | Undefined | Sine is zero |
Understanding cotangent at π/2
Cotangent compares cosine with sine. The standard rule is cot θ = cos θ ÷ sin θ. At π/2 radians, cosine equals zero. Sine equals one. Dividing zero by one gives zero. Therefore, cot(π/2) equals zero. This result is defined. It is not an asymptote or an indeterminate value.
Why radians matter
Pi describes a half turn around a circle. The angle π/2 marks one quarter turn. It also equals 90 degrees. Unit-circle coordinates at this position are (0, 1). The x-coordinate represents cosine. The y-coordinate represents sine. Cotangent uses x divided by y. The quotient is 0 ÷ 1. The calculator shows the same result after converting your selected input.
Checking the reciprocal relationship
Cotangent is the reciprocal of tangent where tangent exists. Tangent at π/2 is undefined because cosine is zero. That reciprocal statement does not mean cotangent is undefined there. Use the primary ratio instead. Cotangent equals cosine divided by sine. Since sine is one at π/2, the ratio remains valid. This distinction prevents a common identity mistake.
Reading the graph
The cotangent graph repeats every π radians. It has vertical breaks wherever sine equals zero. Those breaks occur at 0, π, and their integer multiples. At π/2, the curve crosses the horizontal axis. The plotted point appears at y = 0. Adjust the graph range to compare nearby values. Near a break, cotangent becomes very large in magnitude.
Using tolerance and precision
Computers store decimal approximations. A value intended as zero may appear as a tiny number. The tolerance setting treats sufficiently small sine or cotangent values as zero when appropriate. The precision setting controls displayed decimals. It does not change the underlying formula. Keep a small tolerance for normal classroom work. Use more precision when comparing numerical methods or engineering data.
Practical study tips
Begin by choosing the correct input format. Use a π multiplier for angles such as π/2 or 3π/4. Use numeric mode for values written in degrees or decimal radians. Review sine and cosine in the result panel. Then check the cotangent ratio. Download the CSV for a worksheet record. Download the PDF when you need a compact summary. The example table gives useful benchmark angles for quick verification. Use each result to connect formulas, coordinates, graphs, units, and exact values.
Frequently asked questions
1. What is cot(π/2)?
cot(π/2) equals 0. Cosine is 0 at π/2, while sine is 1. The ratio cos(θ) ÷ sin(θ) is therefore 0 ÷ 1.
2. Why is cot(π/2) defined?
Cotangent divides by sine. Sine at π/2 is 1, not 0. The division is valid, so cotangent is defined and equals zero.
3. Why is tan(π/2) undefined instead?
Tangent divides sine by cosine. Cosine at π/2 is 0. Division by zero is undefined, so tangent has a vertical asymptote at that angle.
4. How do I enter π/2?
Choose π multiplier mode and enter 1/2. The calculator multiplies your entry by π and evaluates the resulting angle in radians.
5. Can I use degrees?
Yes. Choose numeric angle mode, select Degrees, and enter a value such as 90. The calculator converts it to radians internally before evaluating functions.
6. What does zero tolerance do?
Zero tolerance handles tiny rounding values. It helps the calculator recognize values intended to be zero and prevents misleading huge quotients near sine-zero locations.
7. When is cotangent undefined?
Cotangent is undefined whenever sin(θ) equals zero. This occurs at 0, π, 2π, and every integer multiple of π.
8. What is the period of cotangent?
The cotangent function repeats every π radians, or 180 degrees. Its graph has one full repeating pattern between consecutive vertical breaks.
9. Why does the graph have gaps?
The graph has gaps where sine is zero. Cotangent would require division by zero there, so those x-values are excluded from the function.
10. Does decimal precision change the answer?
No. Precision changes only how many decimal places are shown. The calculator still evaluates the same trigonometric ratio using its internal numeric value.
11. Can I download my calculation?
Yes. After a valid calculation, use Download CSV for tabular data or Download PDF for a concise result summary.