Inverse Trigonometric Functions: Formulas and Graphs

The sine function cannot be reversed directly because it repeats: sin 30° and sin 150° are both 1/2, so "which angle has sine 1/2?" has no single answer. To build a proper inverse, mathematicians cut the sine function down to one interval where it never repeats, and the inverse of that cut-down piece is called arcsine, written sin⁻¹x.

This matters everywhere angles must be recovered from measurements. A robot arm needs the actual joint angle behind a length ratio, and a navigator needs the bearing behind a measured slope. Inverse trigonometric functions turn ratios back into angles, and their carefully chosen domains and ranges keep answers single-valued.

Key Concepts

• Inverse trigonometric function: the inverse of a trigonometric function restricted to an interval where it is one-to-one, written sin⁻¹, cos⁻¹, tan⁻¹, and so on.
• Principal value: the single angle returned, always inside the function's fixed range.
• Domain: the set of inputs the inverse function accepts. For sin⁻¹ and cos⁻¹ it is [−1, 1], because sine and cosine never go outside that interval.
• Range: the set of output angles. For sin⁻¹ it is [−π/2, π/2]; for cos⁻¹ it is [0, π]; for tan⁻¹ it is (−π/2, π/2).
• All angles in this article are in radians.

Formulas

• y = sin⁻¹x means x = sin y with −1 ≤ x ≤ 1 and −π/2 ≤ y ≤ π/2.
• y = cos⁻¹x means x = cos y with −1 ≤ x ≤ 1 and 0 ≤ y ≤ π.
• y = tan⁻¹x means x = tan y with x any real number and −π/2 < y < π/2.
• y = cot⁻¹x means x = cot y with x any real number and 0 < y < π.
• y = sec⁻¹x means x = sec y with |x| ≥ 1 and 0 ≤ y ≤ π, y ≠ π/2.
• y = cosec⁻¹x means x = cosec y with |x| ≥ 1 and −π/2 ≤ y ≤ π/2, y ≠ 0.
• Graph shapes: arcsine rises through the origin from (−1, −π/2) to (1, π/2); arccosine falls from (−1, π) to (1, 0); arctangent is an S-curve through the origin flattening toward y = ±π/2.

Worked Example

Evaluate sin⁻¹(−1/2) + cos⁻¹(1/2), giving the exact answer.

1. sin⁻¹(−1/2) = −π/6, the angle in [−π/2, π/2] whose sine is −1/2.
2. cos⁻¹(1/2) = π/3, the angle in [0, π] whose cosine is 1/2.
3. Add them: −π/6 + π/3 = π/6.

The exact value is π/6.

Common Misconceptions

• "sin⁻¹(sin x) = x always." Only when x is inside [−π/2, π/2]; e.g. sin⁻¹(sin(5π/6)) = π/6.
• "sin⁻¹(2) is some angle." It is undefined over the reals, because no angle has a sine of 2. Inputs outside [−1, 1] are rejected.
• "sin⁻¹x means 1/sin x." It does not. The ⁻1 denotes an inverse function, not a reciprocal. The reciprocal of sin x is written cosec x.

Real-World Applications

• Robotics: robot arms compute joint angles from target positions with arctangent (inverse kinematics).
• Navigation: bearings and slopes are converted from measured ratios into degrees with arcsine and arctangent.
• Physics: the launch angle for a projectile to land at a given distance is found with inverse trig functions.
• Computer graphics: rotating objects and aiming cameras use arctangent to turn direction vectors into angles.

Inverse trigonometric functions are the undo buttons of trigonometry: sin⁻¹ and cos⁻¹ accept inputs from [−1, 1] and return angles in [−π/2, π/2] and [0, π] respectively, while tan⁻¹ accepts any real number and returns an angle strictly between −π/2 and π/2. Remember that the output is always a single principal value, so sin⁻¹(sin x) only simplifies to x inside the principal range.