Calculator
Example Data Table
| Angle | Radians | sin θ | cos θ | tan θ | csc θ | sec θ | cot θ |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | Undefined | 1 | Undefined |
| 30° | π/6 | 0.5 | 0.866025 | 0.57735 | 2 | 1.1547 | 1.73205 |
| 45° | π/4 | 0.707107 | 0.707107 | 1 | 1.41421 | 1.41421 | 1 |
| 60° | π/3 | 0.866025 | 0.5 | 1.73205 | 1.1547 | 2 | 0.57735 |
| 90° | π/2 | 1 | 0 | Undefined | 1 | Undefined | 0 |
Formula Used
The calculator starts by converting the entered angle into radians. Trigonometric functions use radians internally. If the user enters degrees, the calculator uses radians = degrees × π / 180.
The six main formulas are:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = sin θ / cos θ
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = cos θ / sin θ
When a denominator is zero, the related function is undefined. This happens for tangent and secant when cosine equals zero. It also happens for cosecant and cotangent when sine equals zero.
How to Use This Calculator
- Enter an angle, such as 45, 180, pi/3, or 1.5708.
- Select degrees, radians, or multiples of pi.
- Choose the decimal precision.
- Press the calculate button.
- Review all six function values above the form.
- Use CSV or PDF buttons to save the result.
Advanced Guide to Six Trigonometric Functions
What This Calculator Does
Trigonometry connects angles with ratios. This calculator evaluates the six core trigonometric functions for one angle. It gives sine, cosine, tangent, cosecant, secant, and cotangent. It also shows radians, degrees, reference angle, and quadrant position. These details help students check signs and understand why a value appears positive, negative, zero, or undefined.
Why Six Functions Matter
Sine and cosine are the base functions. Tangent compares sine with cosine. The other three functions are reciprocals. Cosecant is the reciprocal of sine. Secant is the reciprocal of cosine. Cotangent is the reciprocal of tangent. Together, they describe wave motion, triangles, rotations, slopes, circles, and periodic models.
Degrees, Radians, and Pi Inputs
The calculator accepts common angle formats. You can enter degrees for school problems. You can enter radians for advanced math. You can also enter pi forms, such as pi/6, pi/4, pi/3, and 2pi. This makes unit circle work faster. The conversion step keeps all calculations consistent.
Quadrants and Signs
The quadrant matters because each function changes sign around the circle. In Quadrant I, all six functions are positive. In Quadrant II, sine and cosecant are positive. In Quadrant III, tangent and cotangent are positive. In Quadrant IV, cosine and secant are positive. This sign pattern is useful when checking answers.
Undefined Values
Some values cannot exist. Tangent and secant are undefined when cosine equals zero. Cosecant and cotangent are undefined when sine equals zero. The calculator detects these cases. It prevents misleading decimal answers near vertical asymptotes.
Practical Uses
This tool supports homework, graphing, navigation, engineering, and physics. It can verify unit circle values. It can test identities. It can support right triangle analysis. It can also help compare degree and radian answers. The export options make the results easy to store, print, or include in class notes.
FAQs
1. What are the six trigonometric functions?
They are sine, cosine, tangent, cosecant, secant, and cotangent. The first three are primary ratios. The last three are reciprocal ratios.
2. Can I enter angles in radians?
Yes. Select radians from the unit field. Then enter values such as 1.5708, 3.14159, or pi style values.
3. Can I enter pi values?
Yes. You can enter pi, pi/2, pi/3, pi/4, 2pi, or -pi. The calculator converts them internally.
4. Why is tangent sometimes undefined?
Tangent equals sine divided by cosine. When cosine is zero, division by zero occurs. Therefore, tangent becomes undefined.
5. Why is secant sometimes undefined?
Secant equals one divided by cosine. If cosine equals zero, secant has no valid finite value.
6. What is a reference angle?
A reference angle is the smallest positive angle made with the x-axis. It helps compare angles across different quadrants.
7. Are decimal answers exact?
Decimal answers are rounded approximations. Exact values may include radicals or fractions, especially for unit circle angles.
8. Can I download my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary.