To find the exact value of the cosecant (csc) of an angle in standard position with a given point on its terminal side, we can use the coordinates of the point to determine the values of the adjacent, opposite, and hypotenuse sides of a right triangle.
In this case, the given point is (-7, 6). To determine the values of the sides, we can use the Pythagorean theorem.
The adjacent side (x-coordinate) is -7.
The opposite side (y-coordinate) is 6.
Using the Pythagorean theorem:
hypotenuse^2 = adjacent^2 + opposite^2
hypotenuse^2 = (-7)^2 + 6^2
hypotenuse^2 = 49 + 36
hypotenuse^2 = 85
Taking the square root of both sides, we get:
hypotenuse = √85
The cosecant (csc) of an angle is the reciprocal of the sine (sin) of that angle. The sine can be determined using the opposite side and the hypotenuse:
sin(theta) = opposite / hypotenuse
sin(theta) = 6 / √85
To find the csc(theta), we take the reciprocal of the sine:
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (6 / √85)
csc(theta) = √85 / 6
Therefore, the exact value of csc(theta) with a rational denominator is √85 / 6.