AHL 3.9 (HL)—Reciprocal and inverse trigonometric functions

Reciprocal and inverse trigonometric functions undo different operations.

Reciprocal functions invert a value, while inverse trigonometric functions return an angle; the notation and domain restrictions are different.

Example

For sin θ=0.6 with θ acute, θ=sin⁻¹(0.6)≈36.9°. By contrast, csc θ=1/sin θ, so csc θ≈1.67.

Choose inverse trig when the unknown is an angle; choose a reciprocal when the operation is division by the trig value.

sin⁻¹x is not 1/sin x, and inverse-trig answers must be checked against the stated interval.

Reciprocal definitions: $\sec\theta=1/\cos\theta$, $\cosec\theta=1/\sin\theta$, and $\cot\theta=1/\tan\theta$, wherever the denominator is non-zero. Hence $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\cosec^2\theta$. Principal inverse ranges are $\arcsin x\in[-\pi/2,\pi/2]$ for $x\in[-1,1]$, $\arccos x\in[0,\pi]$ for $x\in[-1,1]$, and $\arctan x\in(-\pi/2,\pi/2)$ for all real $x$.