Loona Calculators
Enter an angle and get all six trig ratios — sin, cos, tan, csc, sec and cot — in degrees or radians. Or work backwards: give a ratio to the inverse functions and find the angle.
They are ratios between the sides of a right-angled triangle. For an angle θ: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. The other three ratios are their flips: csc = 1 ÷ sin, sec = 1 ÷ cos, cot = 1 ÷ tan.
Because tan θ = sin θ ÷ cos θ, and cos 90° = 0. Dividing by zero has no answer, so tan 90° is genuinely undefined — the same goes for csc 0° and sec 90°. The calculator says so honestly instead of showing a huge wrong-looking number.
Degrees are the everyday unit (a right angle is 90°) and what most school work uses. Radians are the science unit (a right angle is π/2 ≈ 1.5708). The calculator converts for you and shows the angle in both units.
For an angle θ in a right-angled triangle, with hypotenuse H, opposite side O and adjacent side A:
sin θ = O ÷ H cos θ = A ÷ H tan θ = O ÷ A
csc θ = 1 ÷ sin θ sec θ = 1 ÷ cos θ cot θ = 1 ÷ tan θ
The inverse functions run the other way: sin⁻¹ takes a ratio and returns the angle whose sine it is. To convert, degrees = radians × 180 ÷ π, and radians = degrees × π ÷ 180.
| Angle | sin | cos | tan |
| 0° (0 rad) | 0 | 1 | 0 |
| 30° (π/6) | 1/2 = 0.5 | √3/2 ≈ 0.8660 | 1/√3 ≈ 0.5774 |
| 45° (π/4) | √2/2 ≈ 0.7071 | √2/2 ≈ 0.7071 | 1 |
| 60° (π/3) | √3/2 ≈ 0.8660 | 1/2 = 0.5 | √3 ≈ 1.7321 |
| 90° (π/2) | 1 | 0 | Undefined |
Use SOH-CAH-TOA: Sin = Opposite ÷ Hypotenuse, Cos = Adjacent ÷ Hypotenuse, Tan = Opposite ÷ Adjacent. It is the memory trick used in classrooms worldwide.
Sine repeats: 150° also has sine 0.5, as does 390°. Calculators return the principal value — the standard single answer in the range −90° to 90° for sin⁻¹. If your problem needs another angle, use the symmetry: for sine, 180° − the principal angle has the same value.
A sine or cosine can never be bigger than 1 or smaller than −1, because a triangle's opposite or adjacent side can never be longer than its hypotenuse. So there is no angle with sin θ = 2, and the calculator will ask you for a value between −1 and 1. Tan⁻¹ accepts any value.
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