Question

Given the unit circle what is the value of x

Answer 1

Hello there. To solve this question, we'll have to remember some properties about the unit circle.

Given the unit circle:

Notice we have a right triangle with opening angle theta and hypotenuse equals 1.

Therefore all the points in this unit circle have coordinates (cos(Θ), sin(Θ)).

To find the value of x, we first have to know that it is a point from the third quadrant, so the angle is between pi and 3pi/2.

In this case, cos is negative and so do sin.

We just have to make:

`\sin (\theta)=-\frac{\sqrt[]{2}}{2}`

For an angle theta between pi and 3pi/2.

We know that sin(pi/4) = sqrt2/2. Also, from the identity sin(x + pi) = -sin(x), we have that:

sin(pi/4 + pi) = sin(5pi/4) = -sin(pi/4) = -sqrt2/2.

Therefore we say the angle theta is 5pi/4.

Taking its cosine, we have:

`x=\cos \mleft((5\pi)/(4)\mright)=-\cos \mleft((\pi)/(4)\mright)=-\frac{\sqrt[]{2}}{2}`

Thus we have that the expressions between brackets are: