Question

What is the value of the x-coordinate of point A?

Answer 1

-0.75 is x coordinate
A coordinate : (-0.75, -0.5)

Answer 2

The x-coordinate of point A, situated in the third quadrant on the unit circle with an angle of
`\((4\pi)/(5)\)`, is
`\(-(1 + √(5))/(4)\)`.

The x-coordinate of a point on the unit circle in the third quadrant can be found using the cosine function. In this case, since the angle
`\((\pi)/(5)\)` is measured counterclockwise from the positive x-axis, the x-coordinate is given by:

`\[ \text{x-coordinate of A} = \cos\left(\pi - (\pi)/(5)\right) \]`

Simplifying the expression inside the cosine function:

`\[ \text{x-coordinate of A} = \cos\left((4\pi)/(5)\right) \]`

Now, evaluate the cosine of
`\((4\pi)/(5)\)`:

`\[ \text{x-coordinate of A} = \cos\left((4\pi)/(5)\right) \]`

Without numerical calculation, we know that
`\(\cos\left((4\pi)/(5)\right)\)` is negative because
`\((4\pi)/(5)\)` is in the second quadrant. Therefore, the x-coordinate of point A is negative.

Let's evaluate the cosine of
`\((4\pi)/(5)\)`:

`\[ \text{x-coordinate of A} = \cos\left((4\pi)/(5)\right) \]`

Cosine values for special angles can be determined without a calculator. The angle
`\((4\pi)/(5)\)` corresponds to a
`\(72^\circ\)` angle in degrees. In the unit circle, cosine is negative in the second quadrant.

So,
`\(\cos\left((4\pi)/(5)\right) = -\cos(72^\circ)\)`.

Using known values:

`\[ \cos(72^\circ) = (1 + √(5))/(4) \]`

Therefore,

`\[ \text{x-coordinate of A} = -(1 + √(5))/(4) \]`

This is the exact value of the x-coordinate of point A.