FAC1004 Tutorial 6 — Inverse Trigonometric Functions

Practice problems on domains, evaluation, and properties of inverse trigonometric functions.

Topics Covered

• Domain and range of inverse trigonometric functions
• Evaluating inverse trig expressions
• Trigonometric identities with inverse functions
• Applications in geometry

Problem Set

1. Domain Analysis: For $f_1(x) = \tan^{-1} x$ and $f_2(x) = \cos^{-1}(3x)$, state the domain of each and of $g(x) = f_1(x) - f_2(x)$

2. Exact Values from Tangent: Given $\theta = \tan^{-1}(3)$, find exact values of $\sin\theta$, $\cos\theta$, $\cot\theta$, $\sec\theta$, $\csc\theta$

3. Exact Values from Secant: Given $\theta = \sec^{-1}(2.6)$, find exact values of all trig functions

4. Evaluate Expressions:

   • $\sin(2\tan^{-1} 3)$
   • $\cos(\sin^{-1}\frac{3}{5} + \sec^{-1} 2)$
   • $\tan(\frac{\pi}{4} + \cot^{-1} 5)$
   • $\sin(\tan^{-1}\sqrt{3} + \cos^{-1}\frac{1}{\sqrt{2}})$
   • And more complex expressions

5. Simplification: Simplify expressions and show valid intervals:

   • $\cos(\sin^{-1}(\frac{x-1}{x}))$ for $x \geq 2$
   • $\sin(\cos^{-1}(\sqrt{\frac{x+1}{x^2}}))$ for $x \leq -1$

6. Geometric Application: Camera angle problem involving inverse cotangent

7. Law of Cosines: Find angle given triangle side lengths

Related

• FAC1004 - Advanced Mathematics II (Computing) — main course page
• Inverse Trigonometric Functions — concept page
• FAC1004 L13 — Inverse Trigonometric Functions — related lecture
• FAC1004 L14 — Properties of Inverse Trig Functions — related lecture

Source File

TUTORIALS_SET_2526/FAC1004 Tutorial 6 25-26.pdf