What is the exact value of the trigonometric expression in simplest form? 2 cos(3π/4)−4 sin(7π/6)
[tex] 2 cos(\frac{3\pi}{4})-4sin(\frac{7\pi}{6}) [/tex]
Let us find the value of cos([tex] \frac{3\pi}{4} [/tex]) and [tex] sin(\frac{7\pi}{6} ) [/tex]
[tex] cos(\frac{3\pi}{4} )=cos(\pi -\frac{\pi}{4} ) [/tex]
The angle [tex] \pi -\frac{\pi}{4} [/tex] lies in second quadrant.
So, [tex] cos(\frac{3\pi}{4} )=-cos(\frac{\pi}{4} ) =-\frac{1}{\sqrt{2}} [/tex]
[tex] So, cos(\frac{3\pi}{4} )=-\frac{1}{\sqrt{2}} [/tex]
Now, Let us find the value of [tex] sin(\frac{7\pi}{6} ) [/tex]
[tex] sin(\frac{7\pi}{6} ) =sin(\pi +\frac{\pi}{6} )=-sin(\frac{\pi}{6} ) =\frac{-1}{2} [/tex]
[tex] sin(\frac{7\pi}{6} ) =\frac{-1}{2} [/tex]
[tex] 2 cos(\frac{3\pi}{4})-4sin(\frac{7\pi}{6}) [/tex]=2*[tex] \frac{-1}{\sqrt{2}} [/tex]-4*[tex] \frac{-1}{2} [/tex]
=[tex] \frac{-2}{\sqrt{2}} +\frac{4}{2} [/tex]
=[tex] \frac{-2\sqrt{2}}{2} +\frac{4}{2} [/tex]
=[tex] \frac{-1\sqrt{2}}{1} +\frac{2}{1} [/tex]
=[tex] -\sqrt{2} +2 [/tex]
=[tex] 2-\sqrt{2} [/tex]
[tex] 2 cos(\frac{3\pi}{4})-4sin(\frac{7\pi}{6})=2-\sqrt{2} [/tex]
