Question

Let O be a quadrant I angle with cos(0) ✓11/7 Find sin(20).​

Answer 1

`\displaystyle \sin(2\theta)=(2√(418))/(49)\approx0.8345`

Explanation:

We are given that:

`\displaystyle \cos(\theta)=(√(11))/(7)`

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:

`b=\sqrt{(7)^2-(√(11))^2}=√(49-11)=√(38)`

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:

`\sin(2\theta)=2\sin(\theta)\cos(\theta)`

Using the above information, substitute. Remember that all ratios will be positive:

`\displaystyle =2\Big((√(38))/(7)\Big)\Big((√(11))/(7)\Big)`

Simplify. Therefore:

`\displaystyle \sin(2\theta)=(2√(418))/(49)\approx0.8345`