Find the exact values of the six trigonometric functions of theta if the terminal side of theta in standard position contains the point (-5/-4)

well, the points give you the opposite side, and the adjacent side. The hypotenuse must then be sqrt (25+16). So now one knows all three sides.

Use the formulas for sine, cosine, and tangent, then invert them for the other three.

so R would be sqrt41 and then sin=-4/sqrt41 cos=-5/sqrt41 and so on and so forth right?

The values of the six trig. functions should be:

sin(theta) = -4
-------
sqrt(41)

cos(theta) = -5
-------
sqrt(41)
tang(theta) = 4/5
csc(theta = sqrt(41)
--------
-4
sec(theta) = sqrt(41)
--------
-5
cot(theta)= 5/4

To find the values of the six trigonometric functions (sine, cosine, tangent, secant, cosecant, and cotangent) of an angle theta, we need to know the positions of the terminal side of theta relative to the unit circle.

Given that the point (-5, -4) lies on the terminal side of theta, we can determine the position of this point on the unit circle by finding the corresponding angle.

Step 1: Determine the radius of the unit circle
The radius of the unit circle is always 1. So, the radius for this case is 1.

Step 2: Determine the angle
Since the given point (-5, -4) lies on the terminal side of theta, we can consider the right triangle formed by connecting the origin (0,0) to the given point and the x-axis. The length of the horizontal leg of the triangle is -5, and the length of the vertical leg is -4.

Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
hypotenuse = sqrt((-5)^2 + (-4)^2)
= sqrt(25 + 16)
= sqrt(41)

Now, we can determine the angle using the inverse trigonometric functions:
cos(theta) = adjacent/hypotenuse
= -5/sqrt(41)

The terminal side of theta intersects the fourth quadrant, where cosine is negative. Thus, we have:
theta = arccos(-5/sqrt(41))

Finally, to find the values of the trigonometric functions of theta, we can substitute the angle theta into the formulas:

sin(theta) = y-coordinate/hypotenuse = -4/sqrt(41)
tan(theta) = sin(theta)/cos(theta)
= -4/sqrt(41) / (-5/sqrt(41))
= 4/5

cosec(theta) = 1/sin(theta) = sqrt(41)/(-4)
sec(theta) = 1/cos(theta) = sqrt(41)/(-5)
cot(theta) = 1/tan(theta) = 5/4

Therefore, the exact values of the six trigonometric functions of theta are:
sin(theta) = -4/sqrt(41)
cos(theta) = -5/sqrt(41)
tan(theta) = 4/5
cosec(theta) = sqrt(41)/(-4)
sec(theta) = sqrt(41)/(-5)
cot(theta) = 5/4