Section_4.7_Notes

Chapter 4

Section 4.7 - Inverse Trigonometric Functions

A. Inverse Sine Function

• A function must be one-to-one to have an inverse.

• The graph of the sine function, y = sin(x), does not pass the Horizontal Line Test, hence it is not one-to-one on its entire domain.

• To establish an inverse, we restrict the sine function:

  • Commonly restricted to the interval [-E, B]

  • Thus, y = sin(x) within the interval [-B, E] is one-to-one.

• The inverse function is denoted as:

  • y = sin^(-1)(x) or y = arcsin(x)

• The domain of arcsin(x) is [-1, 1], and its range is [-E, B].

B. Properties of the Inverse Sine Function

• Recall:

  • sin(y) = x is valid for -B ≤ y ≤ B.

• When evaluating arcsin(x):

  • Only x-values between -1 and 1 can be used for evaluation.

• Example 1:

  • If x = -1, then y = sin^(-1)(-1) = -B

• Example 2:

  • If x = 0, then sin^(-1)(0) = 0 (valid as y = 0).

C. Inverse Cosine Function

• The cosine function, y = cos(x), can be made one-to-one by restricting its domain:

  • Vale of x limited to [0, π].

• The output of cos^(-1)(x) will yield angles in the range of [0, π].

• Example:

  • If x = -1, then y = cos^(-1)(-1) = π.

• Verify the relationship:

  • cos(y) = x, where -1 ≤ x ≤ 1.

D. Inverse Tangent Function

• The tangent function y = tan(x) can be one-to-one if we restrict its domain to (-π/2, π/2).

• Inverse tangent or arctan is defined for all real numbers.

• Example evaluations:

  • If x = √3, then y = tan^(-1)(√3) = π/3 (valid angle).

• If x = -1, y = tan^(-1)(-1) = -π/4, but since angles must be in (-π/2, π/2):

• Identify angles carefully using the unit circle.

E. Using Calculators for Inverse Trig Functions

• Ensure calculator settings are correct (Degree or Radian Mode).

• Calculate inverse trigonometric values:

  • Example: tan^(-1)(5.69)

  • Example: cos^(-1)(-√3)

• Use reference angles to determine the exact quadrants for angle values.

F. Applications of Inverse Trigonometric Functions

• Example 1: Angle of elevation based on a shadow length.

  • Given a shadow of 4ft, height of 3ft: tan(θ) = 3/4 leads to θ = tan^(-1)(3/4).

• Example 2: Hiker's incline down a path:

  • alt = 480 yd, path = 1250 yd: sin(θ) = 480/1250, find θ.

G. Additional Calculator Practice

• Example 1: Find Csc(√2).

• Example 2: Find csc^(-1)(-7) and sec(7).

H. Composition of Inverse Trig Functions

• Recall the cancellation property:

  • sin(sin^(-1)(x)) = x for -1 ≤ x ≤ 1.

  • cos(cos^(-1)(x)) = x for 0 ≤ x ≤ π.

  • tan(tan^(-1)(x)) = x for all x in (-π/2, π/2).

• Be careful with angles outside the standard range when composing functions.