Hong Life CalculatorsUnderstanding the trigonometric functions calculator — concept and principle
This calculator returns all six trigonometric functions of an angle: sine, cosine, tangent and their reciprocals cosecant, secant and cotangent. You can enter the angle in degrees or radians. It is useful for checking homework, solving right triangles, resolving forces into components, and any calculation involving waves, rotation or slopes.
How it is calculated
- sin θ, cos θ and tan θ = sin θ ÷ cos θ are computed directly.
- csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ = cos θ ÷ sin θ.
- In degree mode the angle is first converted: θ(rad) = θ(°) × π ÷ 180.
- tan is reported as "undefined" when cos θ is zero, as at 90° and 270°.
Results are shown to 10 significant digits.
The principle
Place the angle at the centre of a circle of radius 1, starting from the positive x-axis. The point where the angle's arm meets the circle has coordinates (cos θ, sin θ). In a right triangle with hypotenuse 1, cos is the adjacent side and sin is the opposite side, which is why, for any right triangle, sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse and tan = opposite ÷ adjacent. The unit-circle view extends these ratios to every angle, including obtuse and negative ones, and gives the identity sin²θ + cos²θ = 1.
Worked example
- Default input: angle = 30, unit = degrees.
- θ = 30 × π ÷ 180 = π/6 ≈ 0.5235987756 rad.
- sin 30° = 1/2 = 0.5; cos 30° = √3/2 = 0.8660254038.
- tan 30° = 0.5 ÷ 0.8660254 = 1/√3 = 0.5773502692.
- csc 30° = 1 ÷ 0.5 = 2; sec 30° = 2/√3 = 1.154700538; cot 30° = √3 = 1.732050808.
| Angle | sin |
|---|---|
| 7.5 | 0.1305261922 |
| 15 | 0.2588190451 |
| 22.5 | 0.3826834324 |
| 30 | 0.5 |
| 37.5 | 0.608761429 |
| 45 | 0.7071067812 |
| 60 | 0.8660254038 |
How to read the result
sin and cos always lie between −1 and 1; csc and sec are therefore never between −1 and 1. Signs follow the quadrant: in quadrant I all are positive, in II only sin (and csc) is positive, in III only tan (and cot), in IV only cos (and sec). Values worth memorising: sin 0° = 0, sin 30° = 0.5, sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660, sin 90° = 1, with cos running in the reverse order.
Common mistakes
- Leaving the unit on degrees while entering radians (or the reverse). sin 30 in radians is −0.988, not 0.5.
- Confusing csc θ = 1/sin θ with the inverse function arcsin θ; sin⁻¹ usually means arcsin, not the reciprocal.
- Expecting an exact 0 or infinity near 0°, 90°, 180° or 270°: floating-point results there may be tiny or huge numbers instead.
Frequently asked questions
Why is tan 90° undefined?
tan 90° = sin 90° ÷ cos 90° = 1 ÷ 0. As the angle approaches 90° from below, tan grows without bound.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, or radians by 180/π to get degrees. 180° = π rad.
What does sin² θ mean?
It means (sin θ)², the square of the sine, not sin(sin θ).
Sources: NIST Digital Library of Mathematical Functions, Chapter 4, §4.14 (trigonometric functions: definitions) · Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 4