Question

Given tanx=1/2 and sinx < 0, find sin2x,cos2x, and tan2x. I found tan2x but i need help with sin2x and cos2x

Answer 1

Clearly, we have
\[ \sin(x)=\frac{\cos(x)}{2} \]
Thus,
\[ \sin^2(x)+\cos^2(x)=\frac{\cos^2(x)}{4}+\cos^2(x) \]
\[ 1= \frac{5}{4}\cos^2(x) \]
Since \[ \sin(x) <0\]therefore \[ \cos(x) <0 \]
From the equation above, we derive at
\[ \cos(x)=\frac{-2}{\sqrt{5}} \]
and
\[ \sin(x)=\frac{-1}{\sqrt{5}} \]
Moreover, using double angle formula, we have
\[ \sin(2x)=2\sin(x)\cos(x) \\
\cos(2x)=\cos^2(x)-\sin^2(x) \]

Answer 2

can u show me how to draw it on a quadrant plane? i know its in the 3rd one but i'm confused on how to draw it

Answer 3

you are right, it is in the third quadrant. The moral lesson is that you should be more comfortable with your answer, believe in your instinct.

Answer 4

i understand. thx ^_^