FAC1004 Tutorial 6 — Inverse Trigonometric Functions
Centre for Foundation Studies in Science
University of Malaya
FAC1004 Advanced Mathematics II, 2025/2026
Question 1
Consider two inverse trigonometric functions:
$$f_1(x) = \tan^{-1} x \quad \text{and} \quad f_2(x) = \cos^{-1} 3x$$
State the domain for each of $f_1(x)$ and $f_2(x)$. Hence, state the domain of the function $g(x) = f_1(x) - f_2(x)$.
Question 2
Given that $\theta = \tan^{-1}\left(\frac{3}{4}\right)$, find the exact values of $\sin \theta$, $\cos \theta$, $\cot \theta$, $\sec \theta$ and $\csc \theta$.
Question 3
Given that $\theta = \sec^{-1}(2\sqrt{6})$, find the exact values of $\sin \theta$, $\cos \theta$, $\cot \theta$, $\sec \theta$ and $\csc \theta$.
Question 4
Evaluate the exact value for following expressions.
[Hint: Refer to the list of trigonometric identities]
(a) $\sin\left(2\tan^{-1} 3\right)$
(b) $\cos\left(\sin^{-1} \frac{3}{5} + \sec^{-1} 2\right)$
(c) $\tan\left(\frac{\pi}{4} + \cot^{-1} 5\right)$
(d) $\sin\left(\tan^{-1} \sqrt{3} + \cos^{-1} \frac{1}{\sqrt{2}}\right)$
(e) $\sin\left(\sec^{-1}\left(\frac{2}{\sqrt{3}}\right)\right)$
(f) $\cos\left(2\sin^{-1}\left(\frac{3}{5}\right) + \frac{\pi}{2}\right)$
(g) $\cos\left(\tan^{-1}(3) - \csc^{-1}\left(\frac{3}{\sqrt{5}}\right)\right)$
(h) $\csc(\sec^{-1} 3) + \tan\left(\cos^{-1} \frac{1}{2}\right)$
Question 5
Simplify the following and show the valid interval for $x$ as follows:
(a) $\cos\left(\sin^{-1}\left(\frac{x-1}{x}\right)\right)$ valid for $x \geq 2$
(b) $\sin\left(\cos^{-1}\left(\sqrt{\frac{x+1}{x^2}}\right)\right)$ valid for $x \leq -1$
Question 6
A camera is positioned $x$ feet from the base of a missile launching pad. If a missile of length $a$ feet is launched vertically, show that when the base of missile is $b$ feet above the camera lens, the angle $\theta$ subtended at the lens by the missile is:
$$\theta = \cot^{-1}\left(\frac{x}{a+b}\right) - \cot^{-1}\left(\frac{x}{b}\right)$$
Question 7
The law of cosine states that $a^2 = b^2 + c^2 - 2bc\cos\theta$ where $a$, $b$ and $c$ are lengths of the sides of a triangle and $\theta$ is the angle formed by sides $b$ and $c$.
Find $\theta$, to the nearest degree, for the triangle with $a = 4$, $b = 2$ and $c = 3$.
Key Concepts Covered
- Inverse Trigonometric Functions — arcsin, arccos, arctan, arcsec, arccsc, arccot
- Domain and Range — Restrictions on inverse trig functions
- Right Triangle Method — Finding exact values
- Trigonometric Identities — Compound angles, double angles
- Law of Cosines — Applications in geometry
- Angle Subtended — Physical applications