FAC1004 Advanced Mathematics II Tutorial 6

Inverse Trigonometric Functions (Tutorial 6)

Domain of Inverse Trig Functions

• tan⁻¹(x): Domain = ℝ (all real numbers), Range = (−π/2, π/2)
• cos⁻¹(x): Domain = [−1, 1], Range = [0, π]
• sec⁻¹(x): Domain = (−∞, −1] ∪ [1, ∞), Range = [0, π/2) ∪ (π/2, π]

Example: Domain of Composite Function

For 𝑔(𝑥) = tan⁻¹(𝑥) − cos⁻¹(3𝑥):

• tan⁻¹(𝑥): Domain ℝ
• cos⁻¹(3𝑥): Requires −1 ≤ 3𝑥 ≤ 1 ⇒ −1/3 ≤ 𝑥 ≤ 1/3
• Domain of 𝑔(𝑥) = intersection: 𝑥 ∈ [−1/3, 1/3]

Evaluating Exact Values from Inverse Trig

Given θ	Method
θ = tan⁻¹(4/3)	Draw right triangle: opposite = 4, adjacent = 3, hypotenuse = 5
θ = sec⁻¹(2.6) = sec⁻¹(13/5)	sec θ = 13/5 ⇒ adjacent = 5, hypotenuse = 13, opposite = 12

Results for θ = tan⁻¹(4/3):

• sin θ = 4/5, cos θ = 3/5, cot θ = 3/4, sec θ = 5/3, csc θ = 5/4

Key Trigonometric Identities Used

• Double angle: sin(2θ) = 2 sin θ cos θ
• Sum/difference: sin(A+B), cos(A+B), tan(A+B) formulas
• Reciprocal: sec θ = 1/cos θ, csc θ = 1/sin θ

Simplifying Expressions with Inverse Trig

Example: cos(sin⁻¹((x−1)/x)) valid for x ≥ 1/2

• Let θ = sin⁻¹((x−1)/x)
• Then sin θ = (x−1)/x, cos θ = √(1 − ((x−1)/x)²) = √( (2x−1)/x² )

Valid interval: x ≥ 1/2 ensures radicand ≥ 0

Law of Cosines

Formula: a² = b² + c² − 2bc cos θ

Given a = 4, b = 2, c = 3:

• cos θ = (b² + c² − a²)/(2bc) = (4 + 9 − 16)/(12) = −3/12 = −1/4
• θ = cos⁻¹(−1/4) ≈ 104° (to nearest degree)