Solving Inverse Trig Expressions

Aug 25, 2025

Overview

This lecture covers solving trigonometric expressions involving inverse trigonometric functions by constructing right triangles and using their side ratios, a foundational skill for Calculus II.

General Method for Solving Inverse Trig Problems

  • Start by identifying the inverse trig function (e.g., cotangent inverse, cosine inverse, etc.).
  • Let the argument of the inverse function equal theta: e.g., cot⁻¹(2x+1) = θ.
  • Restate using direct trig function: e.g., cotθ = 2x+1.
  • Rewrite as a fraction if needed (e.g., 2x+1/1) to identify triangle sides.
  • Sketch a right triangle with θ and assign sides according to the trig function's definition.
  • Use the Pythagorean theorem to find the missing side.
  • State the required trig ratio (e.g., sinθ = opposite/hypotenuse).

Example 1: sin(cot⁻¹(2x+1))

  • Let cot⁻¹(2x+1) = θ, so cotθ = 2x+1.
  • Cotangent is adjacent/opposite; set adjacent = 2x+1, opposite = 1.
  • Hypotenuse = √[1² + (2x+1)²] = √[4x² + 4x + 2].
  • sinθ = opposite/hypotenuse = 1/√[4x² + 4x + 2].

Example 2: cot(cos⁻¹(4x–2))

  • Let cos⁻¹(4x–2) = θ, so cosθ = 4x–2.
  • Cosine is adjacent/hypotenuse; adjacent = 4x–2, hypotenuse = 1.
  • Opposite = √[1 – (4x–2)²].
  • cotθ = adjacent/opposite = (4x–2)/√[1 – (4x–2)²].

Example 3: sin(sec⁻¹(5x+1))

  • Let sec⁻¹(5x+1) = θ, so secθ = 5x+1.
  • Secant is hypotenuse/adjacent; hypotenuse = 5x+1, adjacent = 1.
  • Opposite = √[(5x+1)² – 1].
  • sinθ = opposite/hypotenuse = √[(5x+1)² – 1]/(5x+1).

Example 4: cos(csc⁻¹(3x–3))

  • Let csc⁻¹(3x–3) = θ, so cscθ = 3x–3.
  • Cosecant is hypotenuse/opposite; hypotenuse = 3x–3, opposite = 1.
  • Adjacent = √[(3x–3)² – 1].
  • cosθ = adjacent/hypotenuse = √[(3x–3)² – 1]/(3x–3).

Key Terms & Definitions

  • Inverse trigonometric function — Function returning an angle whose trig ratio equals the input.
  • Cotangent (cot) — adjacent/opposite in a right triangle.
  • Cosine (cos) — adjacent/hypotenuse in a right triangle.
  • Secant (sec) — hypotenuse/adjacent in a right triangle.
  • Cosecant (csc) — hypotenuse/opposite in a right triangle.
  • Pythagorean theorem — a² + b² = c², relates sides of a right triangle.

Action Items / Next Steps

  • Watch the brief overview and sample videos as instructed.
  • Practice constructing triangles for each type of inverse trig problem.
  • Complete the assigned homework using the outlined six-step process.

Full transcript