Inverse trig

What it shows

sin x takes the value 1/2 at π/6, 5π/6, 13π/6 and infinitely many other angles, so “the angle whose sine is 1/2” has no single answer. To make an inverse function you keep one piece of the graph, the principal branch, where the function takes each of its values exactly once. Reflect that piece in the line y = x and you get the graph of the inverse function, sin⁻¹ x.

This page shows that reflection as a slide you can play, for all six ratios. It also works out principal values (exact, like 2π/3, when the answer is a standard angle), lists every angle from −2π to 2π with the same ratio, checks the properties that link the inverse functions, and shows the common trap: sin⁻¹(sin θ) is not always θ.

It follows inverse trigonometric functions in Class 12 maths, after the trigonometric functions of Class 11. The idea of restricting a domain to make a function one-one is in the one-one and onto tester; the unit circle shows where the ratios come from; and the trig ratio explorer tool covers the standard angles.

How to use it

1. Pick sin, cos, tan, cot, sec or cosec. The page gives the domain of the inverse and its principal values.
2. In the first picture, grey is the whole graph, blue is the principal branch and green is that branch mirrored in y = x. Drag the Reflect slider back to 0 and press Play the reflection to watch it turn. Tick Reflect the whole graph to see why the full mirror image is not a function.
3. Under Principal value, type x: a decimal, a fraction such as -1/2 or an expression such as sqrt(3)/2. You get the principal value in radians and degrees, a check, the value at −x, the identity with its partner function and the list of all angles with the same ratio.
4. In The trap, slide θ (in steps of 5°) and compare sin⁻¹(sin θ) (the blue zig-zag) with θ itself (dashed).

Worked example

With sin picked and x = 1/2, the page shows sin⁻¹(0.5) = π/6, that is 0.523599 radians or 30°. The list says the angles from −2π to 2π with sin θ = 0.5 are −11π/6, −7π/6, π/6, 5π/6; only π/6 lies in [−π/2, π/2], so it is the principal value. It also shows sin⁻¹(−0.5) = −π/6 and sin⁻¹x + cos⁻¹x = π/2.

Now press cos: x changes to −1/2 and the page shows cos⁻¹(−0.5) = 2π/3 (120°), which matches π − cos⁻¹(1/2) = π − π/3.

In the trap with sin and θ = 150° (that is 5π/6), the page says sin⁻¹(sin 5π/6) = π/6, not 5π/6, because 5π/6 is outside [−π/2, π/2].

Properties shown

• sin⁻¹(−x) = −sin⁻¹x, tan⁻¹(−x) = −tan⁻¹x, cosec⁻¹(−x) = −cosec⁻¹x.
• cos⁻¹(−x) = π − cos⁻¹x, cot⁻¹(−x) = π − cot⁻¹x, sec⁻¹(−x) = π − sec⁻¹x.
• sin⁻¹x + cos⁻¹x = tan⁻¹x + cot⁻¹x = sec⁻¹x + cosec⁻¹x = π/2.
• cosec⁻¹(1/x) = sin⁻¹x and sec⁻¹(1/x) = cos⁻¹x; cot⁻¹(1/x) = tan⁻¹x for x > 0.

Check yourself

Five fresh questions on principal values of standard angles, the sin⁻¹(sin θ) trap and sin⁻¹x + cos⁻¹x. Answers such as pi/6 or 2pi/3 are accepted. Nothing is saved.

FAQ

Why is cot⁻¹(−1) = 3π/4 and not −π/4? The school convention takes cot⁻¹ in (0, π). Some calculators and computer libraries use (−π/2, π/2] instead, so their answers differ for negative x.

Is sin⁻¹x the same as 1/sin x? No. sin⁻¹x is an angle; 1/sin x is cosec x, written (sin x)⁻¹.

Is this from my textbook? It covers the standard school topic with its own examples. It is an independent study aid and is not part of NCERT.

Last updated 2026-11-09.