Question

What is the exact value of the trigonometric expression in simplest form? 2 cos(3π/4)−4 sin(7π/6)

Answer 1

2−√2 This is the answer I found. I am doing this for something else as well.

Answer 2

What is the exact value of the trigonometric expression in simplest form? 2 cos(3π/4)−4 sin(7π/6)

`2 cos((3\pi)/(4))-4sin((7\pi)/(6))`

Let us find the value of cos(
`(3\pi)/(4)`) and
`sin((7\pi)/(6) )`

`cos((3\pi)/(4) )=cos(\pi -(\pi)/(4) )`

The angle
`\pi -(\pi)/(4)` lies in second quadrant.

So,
`cos((3\pi)/(4) )=-cos((\pi)/(4) ) =-(1)/(√(2))`

`So, cos((3\pi)/(4) )=-(1)/(√(2))`

Now, Let us find the value of
`sin((7\pi)/(6) )`

`sin((7\pi)/(6) ) =sin(\pi +(\pi)/(6) )=-sin((\pi)/(6) ) =(-1)/(2)`

`sin((7\pi)/(6) ) =(-1)/(2)`

`2 cos((3\pi)/(4))-4sin((7\pi)/(6))`=2*
`(-1)/(√(2))`-4*
`(-1)/(2)`

=
`(-2)/(√(2)) +(4)/(2)`

=
`(-2√(2))/(2) +(4)/(2)`

=
`(-1√(2))/(1) +(2)/(1)`

=
`-√(2) +2`

=
`2-√(2)`

`2 cos((3\pi)/(4))-4sin((7\pi)/(6))=2-√(2)`