Inverse Trigonometric Functions

How does a phone know which direction the user is facing? How does a 3D engine compute the tilt angle of an object? How does GPS determine heading? All of it uses atan2 - the smart version of inverse tangent.

• Game dev: atan2(y, x) to rotate a character toward a target
• Navigation: computing heading between GPS coordinates via haversine with arctan
• Computer graphics: computing Euler angles from a rotation matrix
• Physics: angle of reflection of a ray from a surface

Предварительные знания

• Inverse Trigonometric Functions
• Graphs of Trigonometric Functions

Inverse functions: restricting the domain

sin(x) is not a one-to-one function - sin(0) = sin(π) = 0. To define arcsin as an inverse, we must restrict the range of sin to [-π/2, π/2], where it is strictly increasing.

**Inverse trigonometric functions:** - **arcsin x**: domain [-1, 1], range [-π/2, π/2] - **arccos x**: domain [-1, 1], range [0, π] - **arctan x**: domain (-∞, +∞), range (-π/2, π/2) - **arccot x**: domain (-∞, +∞), range (0, π) **Key identity:** sin(arcsin x) = x, but arcsin(sin x) = x ONLY when x ∈ [-π/2, π/2]

**In programming**: always use `Math.atan2(y, x)`, never `Math.atan(y/x)`. The latter loses information about the signs of x and y, leading to the wrong quadrant.

Derivatives of inverse trigonometric functions

The derivatives of inverse trig functions involve a square root - they are derived via the inverse function theorem: if y = arcsin x, then sin y = x; differentiate both sides.

**Derivatives of inverse trig functions:** - (arcsin x)' = 1/√(1-x²), x ∈ (-1, 1) - (arccos x)' = -1/√(1-x²), x ∈ (-1, 1) - (arctan x)' = 1/(1+x²), x ∈ ℝ - (arccot x)' = -1/(1+x²), x ∈ ℝ **Note:** (arcsin x)' + (arccos x)' = 0 ⟺ arcsin x + arccos x = const = π/2

Applications: navigation, geometry, integrals

Inverse trig functions appear in problems that require finding an angle given known sides or coordinates.

Key ideas

• arcsin: [-1,1] → [-π/2, π/2]; arccos: [-1,1] → [0,π]; arctan: ℝ → (-π/2, π/2)
• arcsin x + arccos x = π/2 for all x ∈ [-1,1]
• (arcsin x)' = 1/√(1-x²), (arctan x)' = 1/(1+x²)
• atan2(y, x) is the correct way to find an angle in 2D
• ∫ dx/(1+x²) = arctan x + C - fundamental integration formula

Related topics

Inverse functions complete the trigonometric toolkit:

• Core trigonometric functions — Inverse functions reverse the originals
• Applications of trigonometry — atan2 is used in navigation and geometry problems

Вопросы для размышления

• Why is arctan(y/x) insufficient compared to atan2(y, x)? What information is lost by dividing?
• Compute arcsin(sin(7π/6)) = ? Draw the unit circle to solve it.
• Leibniz's formula π/4 = 1 - 1/3 + 1/5 - ... converges slowly. How can arctan(1/5) and arctan(1/239) be used to compute π much faster?

Связанные уроки

• calc-06-derivative-intro