The sine of an angle θ is positive in quadrant II, and the value of sine θ is [tex]\mathbf{ \frac{\sqrt{117}}{11}}[/tex]
The given parameters are:
[tex]\mathbf{cos(\theta_1)=-\frac{2}{11}}[/tex]
Using trigonometry ratio, we have:
[tex]\mathbf{sin^2(\theta) + cos^2(\theta) = 1}[/tex]
Substitute [tex]\mathbf{cos(\theta_1)=-\frac{2}{11}}[/tex]
[tex]\mathbf{sin^2(\theta) + (-\frac{2}{11})^2 = 1}[/tex]
Evaluate the square
[tex]\mathbf{sin^2(\theta) + \frac{4}{121} = 1}[/tex]
Subtract 4/121 from both sides
[tex]\mathbf{sin^2(\theta) = 1 - \frac{4}{121}}[/tex]
Take LCM
[tex]\mathbf{sin^2(\theta) = \frac{121 -4}{121}}[/tex]
Simplify the numerator
[tex]\mathbf{sin^2(\theta) = \frac{117}{121}}[/tex]
Take square roots of both sides
[tex]\mathbf{sin(\theta) = \pm \sqrt{\frac{117}{121}}}[/tex]
This gives
[tex]\mathbf{sin(\theta) = \pm \frac{\sqrt{117}}{11}}[/tex]
From the question, we understand that angle θ is located in quadrant II.
The sine of an angle θ is positive in quadrant II
So, we have:
[tex]\mathbf{sin(\theta) = \frac{\sqrt{117}}{11}}[/tex]
Hence, the value of sine θ is [tex]\mathbf{ \frac{\sqrt{117}}{11}}[/tex]
Read more about quadrants at:
https://brainly.com/question/7196312
