Inverse trig calculator
arcsin(3/5) is 36.87° in the inverse trig calculator: the angle of a 3-4-5 triangle opposite the side of 3.
arcsin(3/5)
- arcsin(3/5), principal value arcsin answers in -90° to 90° (-π/2 to π/2).
- 36.86989765°
- Every angle in one full turn Sine is the y-coordinate, and two points on the unit circle share each height: θ and 180° - θ.
- 36.86989765° and 143.1301024°
- Every solution
- θ = 36.86989765° + 360°k or θ = 143.1301024° + 360°k, for any integer k
- In radians
- 0.6435011088 rad and 2.498091545 rad
- Check
- sin(36.86989765°) = 0.6
- Other inverses of 3/5 Principal values only.
- arccos 53.13010235°, arctan 30.96375653°
- The sin⁻¹ notation means the inverse function, not 1/sin. The reciprocal of sin is csc, which is a different thing.
How the Trigonometry Calculator works
An inverse trig function runs a trig function backward: given the ratio, it returns the angle. If sin θ = 3/5, then θ = arcsin(3/5) = 36.87°. Arccos does the same for a cosine and arctan for a tangent, and sin⁻¹, cos⁻¹ and tan⁻¹ are the same functions written another way.
In a right triangle with sides 3, 4 and 5, the angle opposite the 3 has a sine of 3/5, a cosine of 4/5 and a tangent of 3/4, so arcsin(0.6), arccos(0.8) and arctan(0.75) all give 36.87°.
Each inverse returns one principal value: arcsin between -90° and 90°, arccos between 0° and 180°, and arctan between -90° and 90°. A sine of 3/5 also belongs to 180° - 36.87° = 143.13°, so there are two angles in each turn, and adding any multiple of 360° to either gives another.
Sine and cosine never go above 1 or below -1, so arcsin and arccos of 1.5 have no real angle. Arctan takes any number.
In radians, arcsin(3/5) is 0.6435. The standard ratios give exact angles: arcsin(1/2) = 30° = π/6, arccos(√2/2) = 45° = π/4 and arctan(√3) = 60° = π/3.
Trigonometry Calculator FAQ
What is arctan(1)?
45°, or π/4 radians: the angle at which the opposite and adjacent sides are equal.
How do I find an angle from two sides of a right triangle?
Divide the sides and take the matching inverse. Opposite ÷ adjacent is the tangent, so sides of 5 and 12 give arctan(5/12) = 22.62°.
Why does my calculator give a different answer?
Check whether it is set to degrees or radians. arcsin(0.5) is 30 in degrees and 0.5236 in radians.