Shoulders, Giants, and Cats🐈

We prove identities involving inverse trigonometric functions. Sum formulas from trigonometry are exploited. $\arctan{\frac{1}{2}}+\arctan{\frac{1}{3}} = \frac{\pi}{4}$ Suppose that $$\arctan{\frac{1}{2}}+\arctan{\frac{1}{3}} \neq \frac{\pi}{4}.$$ Then, $$\begin{align} \tan{\left(\arctan{\frac{1}{2}}+\arctan{\frac{1}{3}}\right)} &\neq \tan{\frac{\pi}{4}}\\ \frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\cdot\frac{1}{3}} &\neq 1 \\ \frac{5}{6-5} &\neq 1 \\ 1 &\neq 1, \end{align}$$ which is a contradiction. Thus, $$\arctan{\frac{1}{2}}+\arctan{\frac{1}{3}} = \frac{\pi}{4}.□$$ $\arcsin{\frac{16}{65}}+\arcsin{\frac{5}{13}} = \arcsin{\frac{3}{5}}$ Suppose that $$\arcsin{\frac{16}{65}}+\arcsin{\frac{5}{13}} \neq \arcsin{\frac{3}{5}}.$$ Then, $$\begin{align} \sin{\left(\arcsin{\frac{16}{65}}+\arcsin{\fra...

Equations involving inverse trigonometric functions are solved for the unknown. The following relation is assumed proven: $$\theta = \arcsin{\frac{a}{c}} = \arccos{\frac{\sqrt{c^2-a^2}}{c}},$$ which is clear from the figure below. $\arcsin{\frac{3}{5}}=\arctan{x}$ $$\begin{align} \arctan{x} &= \arcsin{\frac{3}{5}} = \arctan{\frac{3}{\sqrt{5^2-3^2}}} = \arctan{\frac{3}{4}}\\ &\Rightarrow x = \frac{3}{4}.□ \end{align}$$ $\arcsin{\frac{3}{5}}=2\arctan{x}$ $$\begin{align} 2\arctan{x} &= \arcsin{\frac{3}{5}} = \arctan{\frac{3}{\sqrt{5^2-3^2}}} = \arctan{\frac{3}{4}}\\ \frac{3}{4} &= \tan{\left(2\arctan{x}\right)} = \frac{2\tan{\left(\arctan{x}\right)}}{1-\tan^2{\left(\arctan{x}\right)}} \\ &= \frac{2x}{1-x^2}\\ &\Rightarrow 0 = 3x^2 +8x -3 = (3x-1)(3x+3) \\ &\Rightarrow x = \frac{1}{3} ( x = -1 \hbox{ is extraneous}).□ \end{align}$$ References

The following exercises involve the simplification of numerical expressions involving inverse trigonometric functions. We use the co-function relationships and reciprocal relationships among trigonometric functions. $\sin{\left(2\arccos{\frac{2}{5}}\right)}=?$ $$\begin{align} &=2\sin{\left(\arccos{\frac{2}{5}}\right)}\cos{\left(\arccos{\frac{2}{5}}\right)}% =2\cdot\frac{2}{5}\sin{\left(\arccos{\frac{2}{5}}\right)}\\% &=\frac{4}{5}\sin{\left(\arcsin{\frac{\sqrt{5^2-2^2}}{5}}\right)}\\ &=\frac{4}{5}\cdot\frac{\sqrt{21}}{5}\\ &=\frac{4}{25}\sqrt{21}.□ \end{align}$$ $\arcsin{a}+\arccos{a}=?$ We observe that because sine and cosine functions are co-functions of each other, for any $a$ in the domain of inverse cosine and inverse sine functions, $$y \equiv \sin(\theta) = \cos{\left(\frac{\pi}{2} - \theta\right)} \Rightarrow \arccos{y} = \frac{\pi}{2} -\arcsin{y}.$$ Thus, the gi...

In the following problems, we find the limits of sequences $\{a_n\}$ given the functional form of the sequences. We assume that some fundamental limits such as $$\lim_{n\rightarrow\infty}\frac{1}{n} = 0$$ and $$\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^n \equiv e$$ are known. $a_n = \left(1-\frac{1}{n}\right)^n$ $$\begin{align} a_n &= \left(\frac{n-1}{n}\right)^n% = \left(\frac{n}{n-1}\right)^{-n}\\ &=\left(\frac{n-1}{n-1}+\frac{1}{n-1}\right)^{-n}\\ &=\left(1+\frac{1}{n-1}\right)^{-n}\\ &=\left(1+\frac{1}{n-1}\right)^{-(n-1)}\left(1+\frac{1}{n-1}\right)^{-1}\\ &=\frac{\left(1+\frac{1}{n-1}\right)^{-1}}{\left(1+\frac{1}{n-1}\right)^{(n-1)}}\\ % \lim_{n\rightarrow\infty}a_n% &=\lim_{n\rightarrow\infty}\frac{\left(1+\frac{1}{n-1}\right)^{-1}}{\left(1+\frac{1}{n-1}\right)^{(n-1)}}\\ &=\frac{\lim_{n\rightarrow\infty}\left(1+\frac{1}{n-1}\right)^{-1}}{\lim_{{n-1}\rightarrow\infty}\left(1+\fra...