Inverse Trigonometric Functions Practice Flashcards

These flashcards cover complex identity proofs, evaluations, and series summations for Inverse Trigonometric Functions as found in JEE Advanced level lecture notes.

What is the solution to the equation $\cos^{-1}(2x) - 2\text{cos}^{-1}(\sqrt{1-x^2}) = \pi$ in the interval $x \in [-1, 1]$?

The equation reduces to $2x = -(1-2x^2)$, and the valid root is $x = \frac{1-\sqrt{3}}{2}$.

If $\text{cos}^{-1}(\alpha) = \text{tan}^{-1}(\alpha)$, what is the value of $\text{sin}(\text{cos}^{-1}\alpha)$?

By setting $\text{cos}^{-1}(\alpha) = \theta$, we get $\text{cos}(\theta) = \text{tan}(\theta)$, which leads to $\text{sin}(\theta) = \frac{\sqrt{5}-1}{2}$.

Evaluate $\frac{d}{d(\text{tan}(\theta))} f(\theta)$ if $f(\theta) = \text{sin}\left(\text{tan}^{-1}\left(\frac{\text{sin}(\theta)}{\sqrt{\text{cos}(2\theta)}}\right)\right)$.

The expression simplifies to $f(\theta) = \text{tan}(\theta)$, so its derivative with respect to $\text{tan}(\theta)$ is $1$.

For the series $S_n(x) = \sum_{k=1}^{n} \text{cot}^{-1}(1 + k(k+1)x^2)$, identify the value of $\text{cot}(S_n(x))$.

$\text{cot}(S_n(x)) = \frac{1 + n(n+1)x^2}{nx}$.

If $a_1 = 1, a_2, a_3, \dots, a_n$ are consecutive natural numbers, what is the value of the sum $\sum_{k=1}^{2021} \text{tan}^{-1}\left(\frac{1}{1 + a_k a_{k+1}}\right)$?

The sum simplifies to $\text{tan}^{-1}(2022) - \text{tan}^{-1}(1) = \text{tan}^{-1}(2022) - \frac{\pi}{4}$.

In the context of the JEE problem on Page 12, what are the values of $x$ and $y$ given $3\text{sin}^{-1}(\text{log}_2 x) + \text{cos}^{-1}(\text{log}_2 y) = \frac{\pi}{2}$ and $\text{sin}^{-1}(\text{log}_2 x) + 2\text{cos}^{-1}(\text{log}_2 y) = \frac{11\pi}{6}$?

The specific values are $x = 2^{-1/2} = \frac{1}{\sqrt{2}}$ and $y = 2^{-1} = \frac{1}{2}$.

What is the minimum value of $x^2 + y^2 + 2xy\text{sin}(\alpha)$ if $\text{cos}^{-1}(x) + \text{sin}^{-1}(y) = \alpha$ where $\frac{\pi}{2} \le \alpha \le \pi$?

$$\text{cos}^2(\alpha)$$

Find the value of $\theta + \phi$ if $\theta = \text{tan}^{-1}(\text{tan}(\frac{3\pi}{4}))$ and $\phi = \text{tan}^{-1}(-\text{tan}(\frac{3\pi}{4}))$.

$0$ (since $\theta = -\frac{\pi}{4}$ and $\phi = \frac{\pi}{4}$ within the principal values).

According to Page 19, what is the result of the sum $\sum_{k=0}^{10} \text{sec}\left(\frac{7\pi}{12} + \frac{k\pi}{2}\right) \text{sec}\left(\frac{7\pi}{12} + \frac{(k+1)\pi}{2}\right)$?

$$-8\text{sin}\left(\frac{7\pi}{6}\right) = 4$$

What is the value of $\text{cot}^{-1}(\text{cot}(-11)) + 10\text{sin}(2\text{cos}^{-1}(\frac{1}{\sqrt{2}})) + 10\text{sin}(2\text{tan}^{-1}(2))$?

$$45\pi - 11$$

Given $S = \{x \in \mathbb{R} : \text{sin}^{-1}\left(\frac{2(x+1)}{x^2+2x+2}\right) - \text{cos}^{-1}\left(\frac{x^2+2x}{x^2+2x+2}\right) = \frac{\pi}{2}\}$, what are the elements of set $S$?

$S = \{0, -1\}$ (Based on calculations derived on Page 2).

If $\text{cos}(x) + \text{cos}(y) + \text{cos}(z) = 0$ and $\text{sin}(x) + \text{sin}(y) + \text{sin}(z) = 0$, what is the possible value of $\text{cos}(x-y)$?

$$-\frac{1}{2}$$