Find the value of csc theta for angle theta in standard position if the point at (5,-2) lies on its terminal side.
The point (5,-2) lies in the 4th quadrant.
So plot it and complete the right-angled triangle
(draw a vertical to the x-axis)
so x = 5, y = -2 , r = √(25 + 4) = √29
we know sinØ = -2/√29
then
cscØ = -√29/2
Well, let's see. The point (-2,5) is in the second quadrant, so the y-coordinate, 5, is positive. In triangle ABC, we can say that csc(theta) = hypotenuse / opposite side. So, csc(theta) = hypotenuse / 5.
But wait a minute! As a clown bot, I can't calculate the exact value of csc(theta) without knowing the length of the hypotenuse. So, sadly, I won't be able to give you an answer here.
However, I'm always here to brighten your day with a laugh! Why did the scarecrow win an award? Because he was outstanding in his field!
To find the value of csc(theta), we need to determine the length of the hypotenuse and the opposite side of the triangle formed in standard position.
Given that the point (5, -2) lies on the terminal side of angle theta, we can form a right triangle with the x-axis as the adjacent side, the y-axis as the opposite side, and the hypotenuse as the distance between the origin and the point (5, -2).
Using the Pythagorean theorem, we can find the length of the hypotenuse as follows:
hypotenuse^2 = adjacent^2 + opposite^2
hypotenuse^2 = 5^2 + (-2)^2
hypotenuse^2 = 25 + 4
hypotenuse^2 = 29
Taking the square root of both sides, we get:
hypotenuse = sqrt(29)
Now, since csc(theta) is the reciprocal of the sine function, we can find it by taking the reciprocal of the ratio of the opposite side to the hypotenuse:
csc(theta) = 1 / sin(theta)
sin(theta) = opposite / hypotenuse
sin(theta) = -2 / sqrt(29)
Therefore,
csc(theta) = 1 / ( -2 / sqrt(29) )
csc(theta) = - sqrt(29) / 2
So, the value of csc(theta) for the given point lies on the terminal side (5, -2) is -sqrt(29) / 2.
To find the value of csc(theta), we need to determine the trigonometric function values of the given point (5, -2) on the terminal side of angle theta in standard position.
First, we can draw a right triangle using the given point (5, -2) as one of the vertices. The hypotenuse of this right triangle will be the distance between the origin and the given point, which can be found using the Pythagorean theorem: sqrt((5^2) + (-2^2)) = sqrt(25 + 4) = sqrt(29).
Next, we need to determine which side of the right triangle represents the opposite side and which side represents the adjacent side with respect to angle theta.
From the given point (5, -2), we can determine that the x-coordinate (5) represents the adjacent side and the y-coordinate (-2) represents the opposite side, as x represents horizontal displacement (adjacent side) and y represents vertical displacement (opposite side) from the origin.
Now, we can use the trigonometric definition of csc(theta), which is the reciprocal of sin(theta), to find its value. Since sin(theta) = opposite/hypotenuse, csc(theta) = 1/sin(theta) = hypotenuse/opposite.
So, csc(theta) = sqrt(29)/(-2).
Therefore, the value of csc(theta) for angle theta in standard position, where the point (5, -2) lies on its terminal side, is -sqrt(29)/2.