Let (-7, -4) be a point on the terminal side of (theta). Find the exact values of sin(theta), csc(theta), and cot(theta).

Answer 1

To find the values of the trigonometric functions, we need to find the values of the opposite side, adjacent side, and hypotenuse in a right triangle with the point (-7, -4) on the terminal side of theta.

Since (-7, -4) is in the third quadrant, the x-coordinate (-7) will be negative, and the y-coordinate (-4) will also be negative.

We can find the hypotenuse by using the Pythagorean theorem:
hypotenuse = sqrt((-7)^2 + (-4)^2) = sqrt(49 + 16) = sqrt(65)

The opposite side is the y-coordinate (-4), and the adjacent side is the x-coordinate (-7).

Now we can find the values of the trigonometric functions:

sin(theta) = opposite/hypotenuse = -4/sqrt(65)

csc(theta) = 1/sin(theta) = sqrt(65)/-4 = -sqrt(65)/4

cot(theta) = adjacent/opposite = -7/-4 = 7/4

Therefore, the exact values of sin(theta), csc(theta), and cot(theta) are:

sin(theta) = -4/sqrt(65)
csc(theta) = -sqrt(65)/4
cot(theta) = 7/4