Question

What is the exact value of tan 2 theta in simplest radical form? Picture gives more context.

Answer 1

now, we know the angle is in the I Quadrant, where sine and cosine or opposite and adjacent sides are both positive, so

`\sin(\theta )=\cfrac{\stackrel{opposite}{2}}{\underset{hypotenuse}{√(7)}}\hspace{5em}\textit{let's find the \underline{adjacent side}} \\\\\\ \begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=√(c^2 - o^2) \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{√(7)}\\ a=adjacent\\ o=\stackrel{opposite}{2} \end{cases}`

`a=\pm\sqrt{ √(7^2) - 2^2}\implies a=\pm√( 7 - 4 ) \implies a=\pm√( 3 )\implies \stackrel{ I~Quadrant }{a=+√(3)} \\\\[-0.35em] ~\dotfill\\\\ \tan(\theta )=\cfrac{\stackrel{opposite}{2}}{\underset{adjacent}{√(3)}}\hspace{9em} \stackrel{\textit{Double Angle Identities}}{\tan(2\theta)=\cfrac{2\tan(\theta)}{1-\tan^2(\theta)}}`

`\tan(2\theta)\implies \cfrac{2\cdot (2)/(√(3))}{1-\left( (2)/(√(3)) \right)^2}\implies \cfrac{~~ (4 )/(√(3) ) ~~}{1-(4)/(3)}\implies \cfrac{~~ (4 )/(√(3) ) ~~}{(3-4)/(3)}\implies \cfrac{~~ (4 )/(√(3) ) ~~}{(-1)/(3)} \\\\\\ \cfrac{4 }{√(3) }\cdot \cfrac{3}{-1}\implies -\cfrac{4\cdot 3}{√(3)}\implies \boxed{-4√(3)}`