Determine the value of $x$.

A) $\frac { 34 \sqrt { 3 } } { 3 }$
B) $\frac { 1 } { 34 }$
C) 34
D) $\frac { \sqrt { 3 } } { 34 }$
E) $\frac { 17 } { 60 }$

Question

Determine the value of $x$.

A) $\frac { 34 \sqrt { 3 } } { 3 }$ 
B) $\frac { 1 } { 34 }$ 
C) 34 
D) $\frac { \sqrt { 3 } } { 34 }$ 
E) $\frac { 17 } { 60 }$

Quizplus · Accepted Answer

This is a problem involving complex numbers. The given number can be written in rectangular form as $x+yi$, where $y=\frac{1}{x}$ and $x>0$. We can then convert to trigonometric form:
$$x+yi=|x+yi|\text{cis}\theta=\sqrt{x^2+\frac{1}{x^2}}\text{cis}\theta$$
where $\theta$ is the argument of $x+yi$. Since $x>0$, we know that $\theta$ is in the first or fourth quadrant. Furthermore, we know that $\text{cis}\theta$ must have a positive imaginary part, so $\theta$ is in the first quadrant. Therefore, we have
$$x+yi=\sqrt{x^2+\frac{1}{x^2}}\left(\cos\theta+i\sin\theta\right).$$
Equating real and imaginary parts with the given information, we have
$$x=\sqrt{x^2+\frac{1}{x^2}}\cos\theta,\qquad y=\sqrt{x^2+\frac{1}{x^2}}\sin\theta.$$
Squaring both equations and adding them yields
$$x^2+y^2=x^2+\frac{1}{x^2}\implies y^2=\frac{1}{x^2}.$$
Substituting into the equation for $y$ gives
$$x=\sqrt{x^2+y^2}\sin\theta\implies \sin\theta=\frac{x}{\sqrt{x^2+y^2}}.$$
Recall that $y=\frac{1}{x}$, so we have
$$\cos\theta=\sqrt{1-\sin^2\theta}=\frac{1}{\sqrt{x^2+y^2}}=\frac{\sqrt{x^2}}{\sqrt{x^4+1}}=\frac{1}{\sqrt{x^2+\frac{1}{x^2}}}$$
Squaring both equations and adding them yields
$$\sin^2\theta+\cos^2\theta=\frac{x^2}{x^2+\frac{1}{x^2}}+ \frac{1}{x^2+\frac{1}{x^2}}=1,$$
so $\text{cis}\theta=\cos\theta+i\sin\theta$ lies on the unit circle. We're given that the imaginary part of $\text{cis}\theta$ is $\frac{1}{34}$. Therefore,
$$\sin\theta=\frac{1}{34}\implies\cos\theta=\frac{\sqrt{3}}{34}.$$
Substituting back into the expression for $x$ yields
$$x=\sqrt{x^2+\frac{1}{x^2}}\cos\theta=\frac{\sqrt{3}}{34}\cdot\frac{\sqrt{34^2+1}}{\sqrt{3}}=\boxed{\frac{34\sqrt{3}}{3}}.$$

Determine the Value Of xxx A) 3433\frac { 34 \sqrt { 3 } } { 3 }3343​​

Determine the Value Of $x$

A) $\frac { 34 \sqrt { 3 } } { 3 }$