SOLUTION: P(6,-4) is a point on the terminal side of theta in standard form. Find the exact values of the trigonometric functions of theta

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Question 1202445: P(6,-4) is a point on the terminal side of theta in standard form. Find the exact values of the trigonometric functions of theta
Answer by math_tutor2020(3817) About Me  (Show Source):
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Answers:
sin%28theta%29+=+%28-2%2Asqrt%2813%29%29%2F13
cos%28theta%29+=+%283%2Asqrt%2813%29%29%2F13
tan%28theta%29+=+-2%2F3
csc%28theta%29+=+%28-sqrt%2813%29%29%2F2
sec%28theta%29+=+%28sqrt%2813%29%29%2F3
cot%28theta%29+=+-3%2F2


Explanation:

The soh cah toa formulas are
sin%28theta%29+=+opposite%2Fhypotenuse
cos%28theta%29+=+adjacent%2Fhypotenuse
tan%28theta%29+=+opposite%2Fadjacent
A shorter format would be
sin%28theta%29+=+y%2Fr
cos%28theta%29+=+x%2Fr
tan%28theta%29+=+y%2Fx
This shorter format is very useful when we're given coordinates of the terminal point.

In this case we have (x,y) = (6,-4) that is the terminal point.
Use the pythagorean theorem to find the radius of the circle, which will give the hypotenuse of the right triangle.
a%5E2%2Bb%5E2+=+c%5E2
x%5E2%2By%5E2+=+r%5E2
r+=+sqrt%28x%5E2%2By%5E2%29
r+=+sqrt%286%5E2%2B%28-4%29%5E2%29
r+=+sqrt%2852%29
r+=+sqrt%284%2A13%29
r+=+sqrt%284%29%2Asqrt%2813%29
r+=+2%2Asqrt%2813%29

Then,


tan%28theta%29+=+y%2Fx+=+-4%2F6+=+-2%2F3

The other three trig ratios are the reciprocals of those previously mentioned three ratios.
csc = 1/sin
sec = 1/cos
cot = 1/tan
meaning that
csc = r/y
sec = r/x
cot = x/y
csc%28theta%29+=+r%2Fy+=+%282%2Asqrt%2813%29%29%2F%28-4%29+=+%28-sqrt%2813%29%29%2F2
sec%28theta%29+=+r%2Fx+=+%282%2Asqrt%2813%29%29%2F6+=+%28sqrt%2813%29%29%2F3
cot%28theta%29+=+x%2Fy+=+6%2F%28-4%29+=+-3%2F2