Question

If θ is a first quadrant angle in standard position with P(u,v) = (3,4), evaluate sin 2θ

Answer 1

So,

We could draw the situation here below:

Remember that sin(θ) is a relation between the opposite side of the angle θ, and the hypotenuse of the right triangle.

In this case, we could write that:

`\sin (\theta)=(4)/(r)`

But, using the pythagorean theorem, we could find the value of r:

`\begin{gathered} r=\sqrt[]{3^2+4^2} \\ r=\sqrt[]{25} \\ r=5 \end{gathered}`

Then,

`\sin (\theta)=(4)/(5)`

Now, we're asked to find the value of sin(2θ). To do this, we could use the fact that:

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

So, we would need to know the value of cos(θ), which is the ratio between the adjacent side of the angle θ and the hypotenuse r. This is,

`\cos (\theta)=(3)/(5)`

Now, we could replace these values in the expression given, to obtain:

`\begin{gathered} \sin (2\theta)=2((4)/(5))((3)/(5)) \\ \sin (2\theta)=(24)/(25) \end{gathered}`

Therefore, sin(2θ) = 24/25