simplify the expression: csc⁡(arctan⁡2x)

Question

Jayden-James Duffy · Accepted Answer

To solve this, drawing an atriangle is the simplest method. Create a right triangle with an acute angle that is $α$ , the adjacent side(of the angle) is A, the opposite side is B and the hypotenuse isC.
Now let $α = \arctan (2 x)$ . Since $\tan (\arctan (2 x)) = 2 x$ and tan is calculated bythe opposite side/adjacent side from the angle let B= 2x and A=1. To calculate C use pythagorean theorem. We get, $C = \sqrt{1 + 4 x^{2}}$
$\csc (\arctan (2 x)) = \frac{1}{\sin (\arctan (2 x))} = \frac{1}{\sin α} = \frac{\frac{1}{B}}{C} = \frac{C}{B} = \frac{\sqrt{1 + 4 x^{2}}}{2 x}$
(Remember that $\sin α = o p p o s i t e / h y p o t e n u s e = B / C$ )
We can check if this answer works. For example if we let $x = \frac{1}{2}$ , we get $\csc = \csc (\arctan (1)) = \csc (\frac{π}{4}) = \frac{1}{\sin (\frac{π}{4}}) = \frac{\frac{1}{1}}{\sqrt{2}} = \sqrt{2}$
Use the formula derived, if we sub in x=1/2 we get
$\csc (\arctan (1)) = \frac{\sqrt{1 + 4 {(\frac{1}{2})}^{2}}}{2 (\frac{1}{2})} = \sqrt{2}$
I hope this helps! Please remember to rate.

simplify the expression: csc(arctan 2x)

Answered question

Answer & Explanation

New Questions in Other