The terminal side of an angle theta in standard position coincides with the line 2x – y = 0 in Quadrant III. Find cos theta to the nearest ten-thousandth.

-.4472

did you get the answer to this?

To find the value of cos(theta), we need to determine the x and y coordinates of the point where the terminal side of the angle intersects the unit circle.

Given that the terminal side lies on the line 2x - y = 0, we can substitute y in terms of x as follows:
2x - y = 0
y = 2x

Since the terminal side is in the third quadrant, both x and y coordinates are negative.

To find the x-coordinate, we need to solve the equation:
2x = -1 (since y = 1 on the unit circle)
x = -1/2

Now let's find the y-coordinate:
y = 2x
y = 2(-1/2) = -1

So, the point of intersection is (-1/2, -1).

Next, we find the length of the radius of the unit circle, which is 1.

Using the Pythagorean theorem, we can find the hypotenuse of the right triangle formed by the given coordinates:
h^2 = (-1/2)^2 + (-1)^2
h^2 = 1/4 + 1
h^2 = 5/4

Taking the square root of both sides, we have:
h = √(5/4) = √5 / √4 = √5 / 2

Finally, we can calculate cos(theta) as:
cos(theta) = x / h = (-1/2) / (√5 / 2)
cos(theta) = -1 / √5
cos(theta) ≈ -0.4472 (rounded to the nearest ten-thousandth)

Therefore, cos(theta) is approximately -0.4472.

To find the cosine of an angle theta, we can use the coordinates of a point on the terminal side of the angle. Since the terminal side of theta coincides with the line 2x - y = 0 in Quadrant III, we can find a point on this line.

First, let's rearrange the equation of the line to solve for y:
2x - y = 0
y = 2x

Since the line doesn't have a specific value for x or y, we can assign a value to x and find the corresponding y-coordinate. Let's choose x = -1 as an example:
y = 2(-1) = -2

Now we have the coordinates (-1, -2) of a point on the terminal side of theta.

Next, we can find the length of the hypotenuse by using the distance formula, which is given by:
r = sqrt(x^2 + y^2)

Substituting the values of x and y:
r = sqrt((-1)^2 + (-2)^2) = sqrt(1 + 4) = sqrt(5)

Finally, we can calculate the cosine of theta using the x-coordinate and the length of the hypotenuse:
cos(theta) = x / r = -1 / sqrt(5)

To get the value to the nearest ten-thousandth, we can use a calculator and get approximately:
cos(theta) ≈ -0.4472