Enter angle details
Results
Trigonometric values
Coterminal angles
Example data table
| Input | Unit | Normalized (0–360°) | Quadrant/Axis | sin | cos | tan |
|---|---|---|---|---|---|---|
| 45 | Degrees | 45° | Quadrant I | 0.7071 | 0.7071 | 1 |
| 120 | Degrees | 120° | Quadrant II | 0.8660 | -0.5000 | -1.7321 |
| -30 | Degrees | 330° | Quadrant IV | -0.5000 | 0.8660 | -0.5774 |
| 0.523599 | Radians | 30° | Quadrant I | 0.5000 | 0.8660 | 0.5774 |
| 50 | Gradians | 45° | Quadrant I | 0.7071 | 0.7071 | 1 |
Values are rounded to four decimals in this example table.
Formula used
- Radians = Degrees × π / 180
- Degrees = Radians × 180 / π
- Degrees = Gradians × 0.9 and Gradians = Degrees / 0.9
- sin(θ), cos(θ), tan(θ) are evaluated using θ in radians.
- sec(θ)=1/cos(θ), csc(θ)=1/sin(θ), cot(θ)=1/tan(θ) when defined.
- Normalization: θ_mod = θ − 360°×floor(θ/360°), keeping 0 ≤ θ_mod < 360.
- Reference angle depends on quadrant: α is the acute angle to the x‑axis.
How to use this calculator
- Enter your angle value, including negatives if needed.
- Select the unit: degrees, radians, or gradians.
- Choose decimal places and a coterminal k range.
- Enable exact special-angle results for cleaner outputs.
- Click Evaluate angle to view conversions and trig values.
- Use the download buttons to export your latest result.
Evaluating angles in real problems
1) Why evaluation matters
Angle evaluation turns a raw number into usable geometry. This tool converts units, identifies position, and returns trigonometric ratios. That helps with triangles, circles, rotations, waves, and periodic signals, where the same direction repeats every full turn. A correct evaluation prevents sign mistakes and wrong quadrants.
2) Units and conversion data
Degrees are common in drawings, while radians are standard in formulas. The conversion uses π ≈ 3.1415926536 with 180° = π rad. Gradians are used in some surveying workflows, with 400 grad per turn and 1 grad = 0.9°. This calculator shows all three forms at once.
3) Normalization and coterminals
Normalization maps any angle to the range 0–360° for easy interpretation. For example, −30° becomes 330°, and 765° becomes 45°. Coterminal angles follow θ + 360°×k, where k is any integer. Listing several k values helps you verify periodic motion and repeated orientations.
4) Quadrants and reference angle
Quadrants define signs: in Quadrant II, sine is positive and cosine is negative. The reference angle is the acute angle to the x-axis, always between 0° and 90°. For 120°, the reference angle is 60°. For 210°, it is 30°. These relationships guide quick mental checks.
5) Trig values and domain limits
The calculator evaluates sin, cos, and tan from the input angle in radians. When cos is near zero, tan and sec become undefined, such as at 90° and 270°. When sin is near zero, csc becomes undefined, such as at 0° and 180°. The display flags these cases clearly.
6) Special angles and exact forms
Common angles often have clean exact values. Examples include sin 30° = 1/2, cos 60° = 1/2, and sin 45° = √2/2. When you enable exact mode, the tool shows these forms and also provides rounded decimals. This is useful for homework, proofs, and quick simplification.
7) Practical checks with numbers
Try 50 grad: it equals 45°, so sin and cos should match at about 0.7071. Try 0.523599 rad: it is about 30°, so tan should be about 0.5774. Try 300°: cosine should be 0.5 and sine negative. Use these benchmarks to validate results fast.
For accuracy, increase decimal places when angles are extreme, and compare coterminals to ensure consistent sign patterns always carefully.
FAQs
1) What does “normalized 0–360°” mean?
Normalization rewrites any angle to an equivalent direction between 0° and 360°. It helps you read quadrants quickly. For example, −30° becomes 330°, and 765° becomes 45°, with the same final direction.
2) Why are tan and sec sometimes undefined?
tan(θ) = sin(θ)/cos(θ) and sec(θ) = 1/cos(θ). When cos(θ) is zero or extremely close to zero, division fails. This happens at angles like 90° and 270°, so the calculator marks them as undefined.
3) What is a reference angle?
A reference angle is the acute angle between the terminal side and the x-axis. It is always between 0° and 90°. It helps you reuse known values. Example: 210° has a 30° reference angle, with signs set by its quadrant.
4) How does exact mode work?
Exact mode detects common special angles such as 30°, 45°, 60°, and related quadrants. When matched, it prints exact forms like √2/2 or √3/3 and also shows a rounded decimal approximation beside them for verification.
5) What are coterminal angles used for?
Coterminal angles point in the same direction and differ by whole turns. They follow θ + 360°×k in degrees. They are useful in rotations, periodic signals, and checking that sine and cosine patterns repeat consistently across multiple revolutions.
6) How do the download buttons decide what to export?
The CSV and PDF exports use the most recent evaluated result from your session. Calculate once, then download. If you change inputs, press Evaluate again to refresh stored values. This keeps exports consistent with the results you see on screen.
Note: When sin or cos is near zero, some ratios are undefined.