Don't Get Confused By The Text

Geometry Level 3

Which of these trigonometric equations is/are impossible in $\mathbb{R}$ ?

A. $\displaystyle \tan x=\frac{\pi}{2}$

B. $\displaystyle \tan \frac{\pi}{2}=x$

C. $\displaystyle \sin x=\frac{\pi}{2}$

D. $\displaystyle \sin \frac{\pi}{2}=x$

E. $\displaystyle \arcsin x=x$

Only B B, C and E B and C A, B and D All of these A and D

1 solution

Ralph James
Mar 8, 2016

A. $\tan x = \frac{\pi}{2}$ has real solutions because the trigonometric function $\tan x$ has range $(-\infty, \infty)$ .

B. $\tan \frac{\pi}{2} = x$ has no real solutions because the tangent of $\frac{\pi}{2}$ is undefined.

Short Proof: $\tan x = \frac{\sin x}{\cos x} \rightarrow \tan \frac{\pi}{2} = \frac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}} = \frac{1}{0}$ which is undefined. Division by 0 could blow up the universe!

C. $\sin x = \frac{\pi}{2}$ has no real solutions because the trigonometric function $\sin x$ has range $[1,-1]$ , and $\frac{\pi}{2} ≈ 1.57$ .

D. $\sin \frac{π}{2} = 1$ .

E. $\arcsin x = x$ has solution $x = 0$ , which is real.

Therefore, the answers are $\boxed{B}$ and $\boxed{C}$ .

Exactly. In particular, we can see that equation A. has $\displaystyle x=\arctan\frac{\pi}{2}+k\pi$ (approximately 1.004), with $k\in\mathbb{Z}$ . And in D. the solution is obviously $x=1$ .

Ervin Tronati - 5 years, 3 months ago

Don't Get Confused By The Text

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