Evaluate all six trigonometric functions at t, where the given point lies on the terminal side of an angle of t radians in standard position. (40/41, -9/41)

draw the triangle, but first, scale it (that wont change the angle). (40,-9) works. That makes the hypotenuse sqrt(40^2+9^2)-=41

so you have a triangle, in the fourth quadrant sides 40, -9, with a hypotenuse of 41

sin= -9/41
cos= 40/41

You take it from here.

You know you are in quadrant IV

by CAST, the cosine, (and the secant), are positive in that quadrant. All other trig ratios are negative.
You also know you are dealing with a 9-40-41 right-angled triangle, since 9^2 + 40^2 = 41^2.
so x=40, y = -9 , and r = 41

Now you can state the other 5 trig ratios.

e.g. tan t = -9/41 (because tanØ = y/x, etc)

e.g. tan t = -9/40

To evaluate the six trigonometric functions at the given point (40/41, -9/41), we can use the Pythagorean theorem and the definitions of the trigonometric functions:

First, let's find the value of the hypotenuse of the right-angled triangle formed. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

The given point (40/41, -9/41) represents the coordinates of a point on the terminal side of an angle. We can use these coordinates as the lengths of the sides.

Let's define the opposite side as -9/41 and the adjacent side as 40/41. To find the length of the hypotenuse, we calculate:

hypotenuse = √((opposite)^2 + (adjacent)^2)
= √((-9/41)^2 + (40/41)^2)
= √((81/1681) + (1600/1681))
= √(1681/1681)
= 1

Now we have the length of the hypotenuse, which is 1. Next, we can evaluate the six trigonometric functions using the given point and the lengths of the sides:

1. Sine (sin(t)) = opposite / hypotenuse = (-9/41) / 1 = -9/41
2. Cosine (cos(t)) = adjacent / hypotenuse = (40/41) / 1 = 40/41
3. Tangent (tan(t)) = opposite / adjacent = (-9/41) / (40/41) = -9/40
4. Cosecant (csc(t)) = 1 / sine(t) = 1 / (-9/41) = -41/9
5. Secant (sec(t)) = 1 / cosine(t) = 1 / (40/41) = 41/40
6. Cotangent (cot(t)) = 1 / tangent(t) = 1 / (-9/40) = -40/9

Therefore, the values of the trigonometric functions at the given point (40/41, -9/41) are:
sin(t) = -9/41
cos(t) = 40/41
tan(t) = -9/40
csc(t) = -41/9
sec(t) = 41/40
cot(t) = -40/9