The point (-4, 6) is on the terminal side of theta in standard position. Which is the cosine of theta?
1. - 2sqrt13/13
2. -2/13
3. 2/13
4. 2sqrt13/13
I think it is one of the negative ones, but I'm not sure of which.
you are correct, since (-4,6) is in QII.
Recall that cosine is x/r.
r^2 = 4^2+6^2 = 42
r = √42 = 2√13
so, your cos is -4/2√13 = -2/√13
Or, as they clumsily put it, -2√13/13
Ah, the point (-4, 6) is a real party animal on the coordinate plane, hanging out on the terminal side of theta! Now let's figure out which cosine it is grooving to.
To find the cosine of theta, we need to calculate the x-coordinate of the point on the unit circle that corresponds to theta. In this case, the x-coordinate is -4.
Since the point is on the terminal side, which is in the second quadrant, it means that the x-coordinate is negative.
Among the answer options, we see that only option 1, -2sqrt13/13, and option 2, -2/13, are negative. So let me show you which one it is:
Let's simplify option 1:
- 2sqrt(13)/13 = -2 * sqrt(13)/13
And option 2:
-2/13 = -2/13 (already simplified)
So, the cosine of theta is -2/13, option 2. Now that we've got that settled, theta can continue to boogie on the coordinate plane!
To determine the cosine of theta, we need to find the x-coordinate of the point on the unit circle that corresponds to the angle theta.
Since the point (-4, 6) is on the terminal side of theta, we can determine the length of the hypotenuse and the ratios by using the Pythagorean theorem.
The length of the hypotenuse can be found using the formula:
c^2 = x^2 + y^2,
where x = -4 and y = 6.
Substituting the given values, we have:
c^2 = (-4)^2 + 6^2
c^2 = 16 + 36
c^2 = 52
c = sqrt(52)
c = 2 * sqrt(13)
Now, the cosine of theta is defined as the ratio of the adjacent side to the hypotenuse. The adjacent side is the x-coordinate of the point, which is -4.
Therefore, the cosine of theta is:
cos(theta) = -4 / (2 * sqrt(13))
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of sqrt(13), which is -sqrt(13):
cos(theta) = (-4 / (2 * sqrt(13))) * (-sqrt(13) / -sqrt(13))
cos(theta) = (4 sqrt(13) / 2 sqrt(13) * -sqrt(13))
cos(theta) = 4 sqrt(13) / -26
cos(theta) = -2 sqrt(13) / 13
Thus, the correct answer is option 1: -2 sqrt(13) / 13.
To find the cosine of theta, we can use the coordinates (-4, 6) of the point that lies on the terminal side of theta. Recall that the cosine of an angle is the ratio of the adjacent side to the hypotenuse in a right triangle.
In this case, we can construct a right triangle with one side being the x-coordinate of the given point (-4) and the other side being the y-coordinate of the point (6). The hypotenuse can be found using the Pythagorean theorem.
Let's calculate it step by step:
1. Determine the length of the hypotenuse using the Pythagorean theorem:
hypotenuse = square root of (x^2 + y^2)
= square root of ((-4)^2 + 6^2)
= square root of (16 + 36)
= square root of 52
= 2 * square root of 13
2. Now, we can find the cosine of theta by dividing the adjacent side (-4) by the hypotenuse (2 * square root of 13):
cosine(theta) = adjacent / hypotenuse
= -4 / (2 * square root of 13)
= -2 / (square root of 13)
= -2 * square root of 13 / 13
Comparing this with the given options, we find that the cosine of theta is:
Option 1: -2 * square root of 13 / 13
Therefore, the correct answer is 1. -2sqrt13/13.