Given the point A(2,-1) on the terminal side of angle 0, what is the value of the cosecant of angle 0?
x = 2
y = -1
sketch graph
in Quadrant 4
if theta is angle below x axis
sin theta = -1/2
csc theta = 1/sin theta = -2/1
To find the value of the cosecant of angle θ, we need to determine the value of the sine of angle θ first.
Since we are given the point A(2,-1) on the terminal side of angle θ, we can use the coordinates (x, y) of the point A to find the sine of angle θ.
We know that in a right triangle, the sine of an angle θ is equal to the ratio of the length of the opposite side to the length of the hypotenuse. However, since we don't have a right triangle, we need to find the values of the opposite, adjacent, and hypotenuse sides.
We can calculate the values of the opposite side and the hypotenuse using the coordinates of point A:
- The length of the opposite side is given by the y-coordinate of point A, which is -1.
- The length of the hypotenuse is the distance from the origin (0, 0) to the point A, which can be calculated using the distance formula: √(x^2 + y^2) = √(2^2 + (-1)^2) = √(4 + 1) = √5.
Now, we can calculate the sine of angle θ by dividing the length of the opposite side (-1) by the length of the hypotenuse (√5):
sin(θ) = opposite/hypotenuse = -1/√5
Finally, we can find the value of the cosecant of angle θ by taking the reciprocal of the sine of angle θ:
csc(θ) = 1/sin(θ) = 1/(-1/√5) = -√5.
Therefore, the value of the cosecant of angle θ is -√5.