The value of [tex]\sin(2\theta)[/tex] is [tex]-\frac {-4\sqrt 3 }9[/tex]
Sine ratio
The sine ratio is given as:
[tex]\sin(\theta) = \frac23[/tex]
Trigonometry
In trigonometry, we have the following ratio
[tex]\sin^2(\theta) + \cos^2(\theta) = 1[/tex]
Substitute [tex]\sin(\theta) = \frac23[/tex] in the above equation
[tex]\frac 23 + \cos^2(\theta) = 1[/tex]
Collect like terms
[tex]\cos^2(\theta) = 1 -\frac 23[/tex]
Evaluate
[tex]\cos^2(\theta) =\frac 13[/tex]
Take square roots
[tex]\cos(\theta) =\pm\frac 1{\sqrt 3 }[/tex]
Rationalize
[tex]\cos(\theta) =\pm\frac {\sqrt 3 }3[/tex]
From the question, [tex]\theta[/tex] is in the second quadrant, and cosine is negative.
So, we have:
[tex]\cos(\theta) =-\frac {\sqrt 3 }3[/tex]
In trigonometry, we have:
[tex]\sin(2\theta) = 2\sin(\theta)\cos(\theta)[/tex]
So, we have:
[tex]\sin(2\theta) = 2\times \frac 23 \times -\frac {\sqrt 3 }3[/tex]
[tex]\sin(2\theta) = -\frac {-4\sqrt 3 }9[/tex]
Hence, the value of [tex]\sin(2\theta)[/tex] is [tex]-\frac {-4\sqrt 3 }9[/tex]
Read more about trigonometry ratios at:
https://brainly.com/question/10417664
