Enter Angle Details
Function Graph
The chart shows the selected function across -360° to 360°. Large asymptote values are hidden for readability.
Example Data Table
| Angle | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0.000000 | 1.000000 | 0.000000 | Undefined | 1.000000 | Undefined |
| 30° | 0.500000 | 0.866025 | 0.577350 | 2.000000 | 1.154701 | 1.732051 |
| 45° | 0.707107 | 0.707107 | 1.000000 | 1.414214 | 1.414214 | 1.000000 |
| 60° | 0.866025 | 0.500000 | 1.732051 | 1.154701 | 2.000000 | 0.577350 |
| 90° | 1.000000 | 0.000000 | Undefined | 1.000000 | Undefined | 0.000000 |
| 180° | 0.000000 | -1.000000 | 0.000000 | Undefined | -1.000000 | Undefined |
Formula Used
sinθ = opposite / hypotenuse
cosθ = adjacent / hypotenuse
tanθ = sinθ / cosθ
cscθ = 1 / sinθ
secθ = 1 / cosθ
cotθ = cosθ / sinθ
Identity: sin²θ + cos²θ = 1
Angle conversion: radians = degrees × π / 180, and radians = gradians × π / 200.
How to Use This Calculator
- Enter the angle value in the first field.
- Select degrees, radians, or gradians.
- Choose the decimal precision for the final table.
- Select signed values for direction, or absolute values for size only.
- Pick a graph function for visual comparison.
- Click the calculate button.
- Review the results above the form.
- Download the CSV or PDF file when needed.
Understanding All Six Functions
Trigonometry connects an angle with ratios inside a right triangle and with coordinates on the unit circle. This calculator brings sine, cosine, tangent, cosecant, secant, and cotangent into one clear view. You enter one angle. The tool converts it into common units. It then reports signs, reciprocal values, reference angles, and quadrant details.
Why these ratios matter
Sine and cosine describe vertical and horizontal movement on the unit circle. Tangent compares sine with cosine. Cosecant, secant, and cotangent are reciprocal ratios. These values appear in wave motion, surveying, navigation, robotics, sound, lighting, and classroom problems. A single wrong unit can change the answer, so the calculator lets you choose degrees, radians, or gradians before solving.
Using angles with care
Angles repeat after a full revolution. For degrees, that cycle is 360. For radians, it is two pi. The calculator normalizes the angle to show where it lands in the standard cycle. This makes signs easier to understand. The reference angle also helps because many identities use it for quick checking.
Interpreting undefined values
Some ratios are not defined at special angles. Tangent and secant are undefined when cosine equals zero. Cosecant and cotangent are undefined when sine equals zero. Instead of forcing a huge number, the calculator labels those results as undefined. This prevents confusing output near vertical asymptotes.
Better checking and reporting
The results table gives values at the chosen precision. The identity panel checks common relationships, such as sine squared plus cosine squared equals one. The graph helps you see periodic behavior. CSV and PDF buttons make it easier to keep records, share work, or attach results to reports. Use the example table to compare popular angles before entering your own values.
Practical workflow
Start with the correct angle unit. Set the precision to match your task. Choose absolute output when you only need size, or keep signed output for direction. Review the quadrant before trusting any manual answer. Then compare the plotted curve with the numeric table. This simple routine catches most entry mistakes and supports clean explanations in study notes, lab sheets, or design calculations for reliable future reviews and project records.
FAQs
1. What does this calculator solve?
It calculates sine, cosine, tangent, cosecant, secant, and cotangent for one angle. It also shows the quadrant, reference angle, normalized angle, identities, and a graph.
2. Can I enter radians?
Yes. Select radians from the unit menu. The calculator converts the value internally, then reports equivalent degree and normalized angle information.
3. Why are some answers undefined?
A function is undefined when its denominator becomes zero. Tangent and secant fail when cosine is zero. Cosecant and cotangent fail when sine is zero.
4. What is a reference angle?
A reference angle is the acute angle between the terminal side and the x-axis. It helps compare signs and exact values across quadrants.
5. What does signed output mean?
Signed output keeps positive and negative values. This is useful for direction, coordinates, wave behavior, and quadrant-based trigonometry problems.
6. When should I use absolute output?
Use absolute output when you only need magnitude. It is helpful for size comparisons, quick checks, or cases where direction is not important.
7. Does the graph show asymptotes?
The graph hides extreme values near asymptotes to keep the chart readable. Undefined regions still appear as gaps or sharp breaks.
8. Can I export my results?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a clean report-style summary.