Question

Given sine of theta is equal to radical 3 over 2 comma determine three possible angles θ on the domain [0,∞).

Answer 1

Let's check the given function:

`\sin \theta=\frac{\sqrt[]{3}}{2}`

Where sinθ is positive in quadrants 1 and 2.

Also, sinθ= opposite side/ hypotenuse:

Then:

Now, the first angle can be found solving theta:

`\theta=\arcsin (\frac{\sqrt[]{3}}{2})`
`\theta=60\text{ or }\theta=(\pi)/(3)`

To find the second angle we use:

`\theta_2=\pi-\theta`

Then:

`\theta_2=\pi-(\pi)/(3)`
`\theta_2=(2)/(3)\pi\text{ or }\theta_2=120`

Now, sinθ will be again positive when we complete a whole circle.

Then, we use

If a circle has 2π, then 2π+1/2π =5/2π is when sinθ is positive again.

There, we use:

120* 4= 480

Then:

`\sin (480)=\frac{\sqrt[]{3}}{2}`

Hence:

`\theta_3=(8)/(3)\pi\text{ or }\theta_3=480`