Sine Cosine Tangent Calculator

How to use the sine, cosine and tangent calculator

What this calculator does

This calculator evaluates the three principal trigonometric functions for an angle and can also solve a right triangle from two independent measurements. In Angle mode, it converts the entered angle to radians internally and returns sine, cosine and tangent. In Right triangle mode, it treats c as the hypotenuse, a as the leg opposite angle α and b as the leg adjacent to α. It then solves the missing sides and acute angles and reports the ratios for α. The tool performs mathematical identities and geometric calculations; it does not determine whether a real-world survey, construction layout or measurement procedure is appropriate.

When to use it

Use Angle mode when checking homework, converting an angle into a direction ratio, or verifying values before using them in engineering or physics formulas. Use Right triangle mode when you know two sides, or one side and one acute angle, and need the remaining dimensions. It is also useful for checking a 3 – 4 – 5 layout, estimating slope from rise and run, or confirming that two acute angles are complementary.

How to calculate

1. The calculator opens with a complete 3 – 4 – 5 right-triangle demonstration. Its results are already calculated, and the example Excel workbook is immediately available.
2. Select Angle or Right triangle under Calculation mode. Choose Degrees or Radians in Angle unit; changing the unit converts any angle values currently entered.
3. For Angle mode, replace the demonstration Angle value. For Right triangle mode, choose a pair under Given, then enter the two active measurements.
4. Read the primary result, Sine, Cosine, Tangent and the supporting values in Calculation details. Results update as you type.
5. Select Download Excel to export the current validated inputs and full-precision results. Reset clears the demonstration and every data field; export is then disabled until a complete valid state is entered again.

Input guide

Calculation mode is required and selects either a single-angle calculation or a right-triangle solution. Angle unit is required and accepts Degrees or Radians. A value such as 30 degrees converts to about 0.523599 radians; a common mistake is entering a degree value while Radians is selected. The Angle field is required in Angle mode, accepts a finite decimal with a period as the decimal separator, and may be positive, negative or greater than one revolution. For example, 390 degrees has the same trigonometric values as 30 degrees. Scientific notation and ambiguous decimal-comma input are rejected.

Given is required in Right triangle mode and identifies which two measurements are independent. Side length a, Side length b and Side length c accept positive lengths up to one trillion in any consistent unit; 3, 4 and 5 are valid examples. Side c must be longer than either selected leg. Angle α and Angle β accept acute angles only: greater than 0 and less than 90 degrees, or the equivalent range in radians. Increasing a while b is fixed increases α and tan(α); increasing b while a is fixed decreases α. Do not mix units among side lengths, and do not enter a right or obtuse angle for α or β.

Output guide

Sine is opposite divided by hypotenuse, Cosine is adjacent divided by hypotenuse, and Tangent is opposite divided by adjacent. In Angle mode these are exact mathematical-function evaluations subject to floating-point rounding. Tangent is shown as “Not defined” where cosine is effectively zero. Normalized angle expresses the same direction within one complete turn. In triangle mode, Hypotenuse c, Angle α, Angle β, Triangle area and Perimeter are solved geometric values. The live summary pills labeled Mode, Angle, sin and cos provide a compact snapshot of the same canonical result. In Calculation details, the Quantity column names the measurement or ratio, Value shows its rounded result, and Interpretation identifies the geometric role or formula. Sine and cosine always remain between – 1 and 1; unusually large tangent values indicate an angle close to 90 degrees plus a half-turn.

Worked example

The startup example uses Right triangle mode with side a = 3 and side b = 4. The Pythagorean theorem gives c = √(3² + 4²) = 5. Therefore sin(α) = 3 ÷ 5 = 0.600000, cos(α) = 4 ÷ 5 = 0.800000 and tan(α) = 3 ÷ 4 = 0.750000. The angle is α = arctan(3 ÷ 4) = 36.869898°, while β = 90° – α = 53.130102°. The area is (3 × 4) ÷ 2 = 6, and the perimeter is 3 + 4 + 5 = 12. These values match the first-open cards, table and workbook checkpoints.

Learn more

NASA's educational explanation of sine, cosine and tangent as right-triangle ratios shows why the three functions connect an angle to side lengths. For a more formal treatment of definitions, zeros and periodicity, consult the NIST Digital Library of Mathematical Functions.