Find two values of angle A between 2pi where sin A = ( - √(2 ) )/(2)

Question

Find two values of angle A between 2pi where sin A =
`( - √(2 ) )/(2)`

Answer 1

Explanation:

First of all, we know that

`\sin (\pi)/(4)=\frac{\sqrt[]{2}}{2}`

And knowing the unit circle, we know that the sine takes negative values in 3rd and 4th quadrants. Therefore, from the above value of the angle, If we go π radians counterclockwise, we encounter negative values of sine; hence,

`\sin \lbrack(\pi)/(4)+\pi\rbrack=-\frac{\sqrt[]{2}}{2}`
`\rightarrow\sin (5\pi)/(4)=\frac{-\sqrt[]{2}}{2}`

The second value of the angle that yields the above value for sine is found by adding π/2 radians to the angle above (we are now in the 4th quadrant)

`\sin (5\pi)/(4)+(\pi)/(2)=-\frac{\sqrt[]{2}}{2}`

`\rightarrow\sin (7\pi)/(4)=-\frac{\sqrt[]{2}}{2}`

Hence, the two values of angles between 0 and 2π are 5π/4 and 7π/4.