If the point (-3,0) lies on the terminal side of angle theta in standard position find:
a) csc theta
b) cot theta
a)1/sin = 1/0 = undefined
b) cos/sin = 1/0 = undefined
To find the values of csc theta and cot theta given that the point (-3,0) lies on the terminal side of angle theta in standard position, we can use the Pythagorean Theorem and the definitions of csc and cot.
a) csc theta:
In standard position, the point (-3,0) lies on the x-axis, which forms a right triangle with a hypotenuse of length 3 (the distance from the origin to (-3,0)) and a leg adjacent to theta with length 3 (the x-coordinate of (-3,0)).
Using the Pythagorean Theorem, we can find the length of the other leg (opposite to theta):
(opposite)^2 + (adjacent)^2 = (hypotenuse)^2
(opposite)^2 + 3^2 = 3^2
(opposite)^2 = 3^2 - 3^2
(opposite)^2 = 0
opposite = 0
The length of the opposite side is 0, which means that the sine of theta is 0. Since csc theta is the reciprocal of sine theta, csc theta is undefined for this angle.
b) cot theta:
The cotangent of theta is defined as the ratio of the adjacent side to the opposite side in a right triangle.
In this case, the adjacent side is 3 (the x-coordinate of (-3,0)) and the opposite side is 0 (the y-coordinate of (-3,0)).
cot theta = adjacent/opposite
cot theta = 3/0
Since division by zero is undefined, the value of cot theta is undefined for this angle.
To find the values of csc(theta) and cot(theta), we need to determine the position of the point (-3, 0) on the unit circle.
In standard position, the initial side of the angle is always along the positive x-axis, and the terminal side is where the point lies. In this case, the point (-3, 0) lies on the left side of the x-axis, in the third quadrant of the coordinate plane.
Using the Pythagorean theorem, we can find the radius (r) of the unit circle by using the coordinates of the point (-3, 0). Since the x-coordinate is -3, the y-coordinate is 0, and the distance from the origin to the point is the radius (r), we have:
r = sqrt((-3)^2 + 0^2)
r = sqrt(9 + 0)
r = sqrt(9)
r = 3
Therefore, the radius (r) of the unit circle is 3.
Now, let's find the trigonometric values.
a) csc(theta) stands for cosecant(theta), which is equal to 1/sin(theta).
In the third quadrant, the sine function is negative. Since the y-coordinate is 0, the value of sin(theta) is 0, which means that csc(theta) is undefined or infinite.
b) cot(theta) stands for cotangent(theta), which is equal to 1/tan(theta).
In the third quadrant, the tangent function is positive. Since the x-coordinate is -3 and the y-coordinate is 0, we have:
tan(theta) = y/x
tan(theta) = 0/-3
tan(theta) = 0
Therefore, cot(theta) is equal to 1/0, which is undefined or infinite.
So, in summary:
a) csc(theta) is undefined or infinite
b) cot(theta) is undefined or infinite