Calculus II ep11: Inverse trig functions (Oct 1, 2026)

Overview

Inverse trigonometric functions return angles, but defining them as functions requires restricting sine and cosine to one-to-one intervals: arcsin uses sine on [-π/2, π/2], while arccos uses cosine on [0, π]. Prof Staecker uses right-triangle constructions to simplify compositions such as cos(arcsin x), then derives the arcsin and arccos derivatives by implicit differentiation and applies the chain and product rules to examples.

Key takeaways

An inverse function exists only after restricting a non-one-to-one function: arcsin uses sine on [-π/2, π/2], and arccos uses cosine on [0, π].
Arcsin and arccos both accept inputs in [-1, 1], but their output ranges differ: [-π/2, π/2] for arcsin and [0, π] for arccos.
Right triangles derive compositions efficiently: cos(arcsin(x)) = √(1 - x²) and sin(arccos(x)) = √(1 - x²).
Implicit differentiation gives d/dx arcsin(x) = 1/√(1 - x²) and d/dx arccos(x) = -1/√(1 - x²).
When an inverse-trig derivative has a composite input, apply the chain rule separately; for arcsin(2x² + 7), the inner derivative contributes a factor of 4x.

Chapters

0:00 Inverse Trig Notation and the Angle-Solving Purpose

Arcsin, arccos, and arctan are alternate names for inverse sine, cosine, and tangent.
The notation sin⁻¹(x) means inverse sine, not 1/sin(x); the reciprocal is cosecant.
For a right triangle with opposite side 1 and adjacent side 3, tan(θ) = 1/3, so θ = arctan(1/3).

5:00 Why Sine Must Be Restricted Before It Has an Inverse

Reflecting the full sine graph across y = x produces a curve that fails the vertical line test.
Sine is not one-to-one: for example, sin(π) and sin(2π) both equal 0.
Restricting sine to a one-to-one interval makes its inverse a function.

10:00 Arcsin’s Principal Interval, Domain, and Range

The principal arcsin branch uses sine on [-π/2, π/2], where sine is one-to-one.
Arcsin has domain [-1, 1], since sine never takes values outside that interval.
Its range is [-π/2, π/2]; for instance, arcsin(2) is undefined over the reals.

15:00 Calculator Modes and Standard Arcsin Values

Calculator output depends on whether the angle mode is radians or degrees: arcsin(1) is π/2 radians or 90°.
Unit-circle values give arcsin(1/2) = π/6 and arccos(1/2) = π/3.
Other standard values include arcsin(1) = π/2 and arcsin(-1/2) = -π/6.

20:00 Right-Triangle Method for Trig Functions of Arcsin

Set θ = arcsin(x), so sin(θ) = x; represent this as opposite side x and hypotenuse 1.
The Pythagorean theorem gives the adjacent side √(1 - x²), so cos(arcsin(x)) = √(1 - x²).
The same triangle yields tan(arcsin(x)) = x/√(1 - x²), without memorizing a separate identity.

25:00 Arccos’s Principal Branch and Its Different Range

Arccos uses the one-to-one cosine interval [0, π], beginning at cos(0) = 1 and ending at cos(π) = -1.
Its domain is [-1, 1], like arcsin, but its range is [0, π].
Reflecting this restricted cosine graph across y = x gives the arccos graph, which differs in orientation and range from arcsin.

30:00 Triangle Identities for Sine and Tangent of Arccos

Set θ = arccos(x), so cos(θ) = x; use adjacent side x and hypotenuse 1.
The remaining side is √(1 - x²), giving sin(arccos(x)) = √(1 - x²).
Taking opposite over adjacent gives tan(arccos(x)) = √(1 - x²)/x wherever the expression is defined.

35:00 Implicit Differentiation Derives the Arcsin Rule

Starting with y = arcsin(x), rewrite the inverse relationship as x = sin(y).
Implicit differentiation gives 1 = cos(y)·dy/dx, so dy/dx = 1/cos(y).
Substituting y = arcsin(x) and using cos(arcsin(x)) = √(1 - x²) gives d/dx arcsin(x) = 1/√(1 - x²).

40:00 Arccos Derivative and the Chain-Rule Example

The corresponding derivative is d/dx arccos(x) = -1/√(1 - x²); the negative sign comes from the derivative of cosine.
For arcsin(2x² + 7), apply the arcsin derivative formula to the inside expression, then multiply by its derivative 4x.
The chain-rule factor stays outside the inverse-trig derivative expression; it is not inserted under the radical.

45:00 Practice with Nested Inverse Trig and Product Rules

For arcsin(√x), the chain rule combines 1/√(1 - x) with the inner derivative 1/(2√x).
For a product involving arccos(-x) and a polynomial factor, use the product rule and the arccos derivative.
Differentiating arccos(-x) also requires the chain-rule factor -1 from the inner function; the next class will introduce arctan.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Prof Staecker.