Question

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

Answer 1

Answer:

$\displaystyle \sin(2\theta)=\frac{2\sqrt{418}}{49}\approx0.8345$

Step-by-step explanation:

We are given that:

$\displaystyle \cos(\theta)=\frac{\sqrt{11}}{7}$

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:

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

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:

$\displaystyle =2\Big(\frac{\sqrt{38}}{7}\Big)\Big(\frac{\sqrt{11}}{7}\Big)$

Simplify. Therefore:

$\displaystyle \sin(2\theta)=\frac{2\sqrt{418}}{49}\approx0.8345$