if (0,-7) lies on the terminal side of angle theta in standard position, find each value sec(theta) , cot(theta) and sin(theta)
To find the values of sec(theta), cot(theta), and sin(theta) when a point lies on the terminal side of angle theta in standard position, we need to determine the coordinates of the point (0,-7) and then apply the trigonometric identities.
The point (0, -7) lies on the y-axis at a distance of 7 units below the origin. We can use this information to determine the values of the trigonometric functions.
To find sec(theta), we need to find the reciprocal of the cosine function.
First, we need to calculate the length of the hypotenuse using the Pythagorean theorem:
hypotenuse = sqrt((0^2) + (-7)^2) = sqrt(49) = 7
Now, we can calculate sec(theta) by taking the reciprocal of the cosine function:
sec(theta) = 1/cos(theta) = 1/hypotenuse = 1/7
To find cot(theta), we need to find the reciprocal of the tangent function.
Since the x-coordinate is 0, the adjacent side is also 0. Therefore, cot(theta) is undefined.
Finally, to find sin(theta), we need to calculate the ratio of the opposite side to the hypotenuse:
sin(theta) = opposite/hypotenuse = -7/7 = -1
Therefore, the values of sec(theta), cot(theta), and sin(theta) when (0, -7) lies on the terminal side of angle theta in standard position are:
sec(theta) = 1/7
cot(theta) = undefined
sin(theta) = -1