Let (-7, -4) be a point on the terminal side of (theta). Find the exact values of sin(theta), csc(theta), and cot(theta).
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
To find the exact values of sin(theta), csc(theta), and cot(theta), we first need to find the values of the trigonometric functions in the given quadrant.
We are given that the point (-7, -4) is on the terminal side of theta. To find the values of the trigonometric functions, we can use the coordinates of the point to determine the lengths of the sides of the right triangle formed by the point and the x and y axes.
Let's denote the length of the horizontal side of the triangle (adjacent side) as x and the length of the vertical side (opposite side) as y.
From the given information, we have:
x = -7
y = -4
Using the Pythagorean theorem, we can find the length of the hypotenuse (r) of the triangle.
r² = x² + y²
r² = (-7)² + (-4)²
r² = 49 + 16
r² = 65
Taking the square root of both sides, we find:
r = √65
Now, we can find the trigonometric functions:
1. sine (sin(theta)):
sin(theta) = y / r
sin(theta) = -4 / √65
2. cosecant (csc(theta)):
csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (-4 / √65)
csc(theta) = -√65 / 4
3. cotangent (cot(theta)):
cot(theta) = x / y
cot(theta) = -7 / -4
cot(theta) = 7/4
Therefore, the exact values of sin(theta), csc(theta), and cot(theta) are:
sin(theta) = -4 / √65
csc(theta) = -√65 / 4
cot(theta) = 7/4