Inverse Trigonometric Functions: Properties, Derivatives & Related Rates

Posted on Oct 21, 2024 in Mathematics

Inverse Trigonometric Functions

Here’s a concise summary of each inverse trigonometric function, including their notations, domains, ranges, and key properties:

1. Inverse Sine Function

  • Notation: sin-1(x) or arcsin(x)
  • Domain: -1 ≤ x ≤ 1
  • Range: -π/2 ≤ y ≤ π/2
  • Key Property: sin(y) = x

2. Inverse Cosine Function

  • Notation: cos-1(x) or arccos(x)
  • Domain: -1 ≤ x ≤ 1
  • Range: 0 ≤ y ≤ π
  • Key Property: cos(y) = x

3. Inverse Tangent Function

  • Notation: tan-1(x) or arctan(x)
  • Domain: -∞ < x < ∞
  • Range: -π/2 < y < π/2
  • Key Property: tan(y) = x

4. Inverse Cosecant Function

  • Notation: csc-1(x) or arccsc(x)
  • Domain: -∞ < x ≤ -1 or 1 ≤ x < ∞
  • Range: -π/2 ≤ y ≤ 0 or 0 < y ≤ π/2
  • Key Property: csc(y) = x

5. Inverse Secant Function

  • Notation: sec-1(x) or arcsec(x)
  • Domain: -∞ < x ≤ -1 or 1 ≤ x < ∞
  • Range: 0 ≤ y < π/2 or π/2 < y ≤ π
  • Key Property: sec(y) = x

6. Inverse Cotangent Function

  • Notation: cot-1(x) or arccot(x)
  • Domain: -∞ < x < ∞
  • Range: 0 < y < π
  • Key Property: cot(y) = x

Derivatives of Inverse Trigonometric Functions

Here are the derivatives for quick reference:

  • d/dx [sin-1(x)] = 1 / √(1 – x2)
  • d/dx [cos-1(x)] = -1 / √(1 – x2)
  • d/dx [tan-1(x)] = 1 / (1 + x2)
  • d/dx [csc-1(x)] = -1 / (|x|√(x2 – 1))
  • d/dx [sec-1(x)] = 1 / (|x|√(x2 – 1))
  • d/dx [cot-1(x)] = -1 / (1 + x2)

Geometric Formulas for Related Rates

For related rates problems in calculus, it’s helpful to know these geometric formulas:

1. Circle

  • Area: A = πr2
  • Circumference: C = 2πr
  • Volume of a Sphere: V = (4/3)πr3
  • Surface Area of a Sphere: SA = 4πr2

2. Triangle

  • Area: A = (1/2)bh (where b is the base and h is the height)
  • Pythagorean Theorem: a2 + b2 = c2 (for right triangles, where c is the hypotenuse)

3. Rectangle

  • Area: A = l × w (where l is the length and w is the width)
  • Perimeter: P = 2l + 2w

4. Cylinder

  • Volume: V = πr2h
  • Surface Area: SA = 2πr2 + 2πrh

5. Cone

  • Volume: V = (1/3)πr2h
  • Surface Area: SA = πr2 + πrl (where l is the slant height, calculated as l = √(r2 + h2))

6. Rectangular Prism

  • Volume: V = l × w × h
  • Surface Area: SA = 2(lw + lh + wh)

7. Square

  • Area: A = s2
  • Perimeter: P = 4s (where s is the side length)

General Tips for Related Rates Problems

  • Identify Variables: Start by identifying what quantities are changing and what you need to find.
  • Differentiate: Use implicit differentiation with respect to time t for the relevant formulas.
  • Relate Rates: Express the relationship between the rates of change of the different quantities involved.
  • Units: Be mindful of the units when dealing with rates (e.g., meters per second, square meters, etc.).

Example Problem Structure

  1. Identify the relationship between the changing quantities.
  2. Write down the formula for the relevant geometric shape.
  3. Differentiate with respect to time t.
  4. Substitute known values into the differentiated equation to find the desired rate.

Understanding these formulas and how to apply them will help you solve related rates problems effectively!