find the exact value of cos(theta) if the terminal side of angle(theta) contains the point (-3,5)
hypotenuse = sqrt (-3^2 + 5^2) = sqrt (34)
cos T = -3/sqrt 34 in quadrant 2
To find the exact value of cos(theta) when the terminal side of angle theta contains the point (-3,5), we need to use the coordinates of the point to determine the values of the trigonometric ratios.
We can use the Pythagorean theorem to find the length of the hypotenuse of the right triangle created by the point (-3,5). The hypotenuse is the distance between the origin (0,0) and the point (-3,5).
The formula for the hypotenuse, or the distance, between two points (x1,y1) and (x2,y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of the point are (-3,5), and the origin is (0,0). We substitute these values into the formula:
d = sqrt((-3 - 0)^2 + (5 - 0)^2)
= sqrt((-3)^2 + 5^2)
= sqrt(9 + 25)
= sqrt(34)
So, the length of the hypotenuse is sqrt(34).
To find the cosine of theta, we use the definition of cosine in a right triangle. In this case, theta is the angle between the x-axis and the hypotenuse.
cos(theta) = adjacent / hypotenuse
The x-coordinate of the point (-3,5) is -3, which represents the adjacent side of the triangle.
cos(theta) = -3 / sqrt(34)
Therefore, the exact value of cos(theta) is -3 / sqrt(34).