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Answer:
[tex]\displaystyle \sin(2\theta)=\frac{2\sqrt{418}}{49}\approx0.8345[/tex]
Step-by-step explanation:
We are given that:
[tex]\displaystyle \cos(\theta)=\frac{\sqrt{11}}{7}[/tex]
Where θ is in QI.
And we want to determine sin(2θ).
First, note that since θ is in QI, all trig ratios will be positive.
Next, recall that cosine is the ratio of the adjacent side to the hypotenuse. Therefore, the adjacent side a = √(11) and the hypotenuse c = 7.
Then by the Pythagorean Theorem, the opposite side to θ is:
[tex]b=\sqrt{(7)^2-(\sqrt{11})^2}=\sqrt{49-11}=\sqrt{38}[/tex]
So, with respect to θ, the adjacent side is √(11), the opposite side is √(38), and the hypotenuse is 7.
We can rewrite as expression as:
[tex]\sin(2\theta)=2\sin(\theta)\cos(\theta)[/tex]
Using the above information, substitute. Remember that all ratios will be positive:
[tex]\displaystyle =2\Big(\frac{\sqrt{38}}{7}\Big)\Big(\frac{\sqrt{11}}{7}\Big)[/tex]
Simplify. Therefore:
[tex]\displaystyle \sin(2\theta)=\frac{2\sqrt{418}}{49}\approx0.8345[/tex]