the terminal side of an angle in standard position coincides with the line x-2y=0 in quadrant 3, find cosine theta to the nearest ten-thousandth

since the slope of the line is 1/2, tanθ = 1/2 = y/x.

Now draw that triangle in QIII and you can see that cosθ = x/r = -2/√5

To find the cosine of theta, we need to determine the value of x and y.

Given that the terminal side of the angle coincides with the line x-2y=0, we can use this equation to find the corresponding values of x and y.

First, solve the equation for x:
x = 2y

Since the line lies in quadrant 3, both x and y will be negative.

Now, let's substitute x into the cosine formula:
cos(theta) = x / r

We need to find r (the distance from the origin to the point (x, y)) to calculate the cosine of theta.

From the equation of the line, we can rewrite x in terms of y:
x = 2y

Using the Pythagorean theorem, we can find r:
r^2 = x^2 + y^2

Substituting x = 2y:
r^2 = (2y)^2 + y^2
r^2 = 4y^2 + y^2
r^2 = 5y^2

Taking the square root of both sides:
r = √(5y^2)
r = √(5 * y^2)
r = √5 * √y^2
r = √5y

Now substitute x = 2y and r = √5y into the cosine formula:
cos(theta) = x / r
cos(theta) = (2y) / (√5y)
cos(theta) = 2 / √5

To get the value of cosine theta to the nearest ten-thousandth, we can approximate √5 to four decimal places (1.4142) and compute:
cos(theta) = 2 / √5
cos(theta) = 2 / 1.4142
cos(theta) ≈ 1.4142

Therefore, the cosine of theta to the nearest ten-thousandth is approximately 1.4142.

To find the cosine of an angle, we need the coordinates of a point on the terminal side of the angle in standard position.

Given that the terminal side of the angle coincides with the line x - 2y = 0 in quadrant 3, we can determine the coordinates by setting one of the variables to a known value.

Let's set y equal to 1, and solve for x:

x - 2(1) = 0
x - 2 = 0
x = 2

So, we have the point (2, 1) on the terminal side of the angle.

To find the cosine of the angle, we need the length of the adjacent side and the hypotenuse of the right triangle formed by the terminal side.

The length of the adjacent side is the x-coordinate (2), and the length of the hypotenuse can be found using the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let's calculate the length of the hypotenuse:

h^2 = x^2 + y^2
h^2 = 2^2 + 1^2
h^2 = 4 + 1
h^2 = 5
h = √5

Now, we have the lengths of the adjacent side (2) and the hypotenuse (√5). We can use these values to calculate the cosine of the angle.

cos(theta) = adjacent side / hypotenuse
cos(theta) = 2 / √5

To find the value of cos(theta) to the nearest ten-thousandth, we need to divide 2 by the square root of 5 and round the result.

cos(theta) ≈ 2 / √5 ≈ 0.89443 (rounded to the nearest ten-thousandth)

Therefore, the cosine of theta to the nearest ten-thousandth is approximately 0.8944.