A point on the terminal side of an angel theta is given. Find the exact value of the indicated trigometric function

(-1/5,1/2 ) find cos theta

your given point is in quad II, so the cosine of the angle will have to negative.

x = -1/5 , y = 1/2
r^2 = 1/25 + 1/4 = 29/100
r = √29/10

cosØ = x/r = (-1/5)/(√29/10)
= (-1/5)(10/√29)
= -2/√29

they might expect it to be rationalized and want the answer as -2√29/29

cuz you know that the angle is in the top left

you can use tan(x) = y/x to calculate the angle(it must be bigger than 90)
than use cos(x) on your cal to find cos(x)

or
you can use x^2 + y^2 = hyp^2
to find the hyp and do (-1/5)/(hyp length)
to get cos(x)

To find the exact value of cos(theta), we can use the Pythagorean Identity:

cos^2(theta) + sin^2(theta) = 1

Given that the point on the terminal side of the angle theta is located at (-1/5, 1/2), we can determine the values of cos(theta) and sin(theta) using the coordinates.

Since cos(theta) is the x-coordinate of the point on the unit circle, we have:

cos(theta) = -1/5

Therefore, the exact value of cos(theta) for this point is -1/5.

To find the exact value of cosine (cos) theta, given a point on the terminal side of angle theta, we can use the coordinates of the point and the Pythagorean theorem to determine the length of the hypotenuse.

1. Start by plotting the given point (-1/5, 1/2) on the coordinate plane. The x-coordinate represents the horizontal distance from the origin (a negative value indicates a point in the left half plane), and the y-coordinate represents the vertical distance (positive vertical distance means the point is above the x-axis).

2. Looking at the point on the coordinate plane, draw a right triangle with one leg along the x-axis, another leg going vertically up to the point, and the hypotenuse connecting the origin to the point.

3. Using the Pythagorean theorem, calculate the length of the hypotenuse. In this case, we have:

hypotenuse^2 = (horizontal distance)^2 + (vertical distance)^2
hypotenuse^2 = (-1/5)^2 + (1/2)^2
hypotenuse^2 = (1/25) + (1/4)

To get a common denominator, we have:
hypotenuse^2 = (1/25) + (5/25)
hypotenuse^2 = 6/25

Taking the square root of both sides gives:
hypotenuse = sqrt(6)/5

4. Now that we know the length of the hypotenuse, we can determine the cosine (cos) theta value by dividing the horizontal distance (x-coordinate) by the hypotenuse. In this case, we have:

cos(theta) = (horizontal distance) / hypotenuse
cos(theta) = (-1/5) / (sqrt(6)/5)
cos(theta) = -1/sqrt(6)

Since sqrt(6) is not a perfect square, this is the exact value of cos(theta) using the given coordinates.