SOLUTION: Find the values of the six trigonometric functions for the angle in standard position determined by the point ​(​1,-5)

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Question 1175465: Find the values of the six trigonometric functions for the angle in standard position determined by the point ​(​1,-5)
Found 2 solutions by Edwin McCravy, MathLover1:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
We draw a right triangle with an angle in Q4, since (1,-5) is in Q4, we put x=1
on the adjacent side, and y=-5 on the opposite side. 



Then we find the hypotenuse r by the Pythagorean theorem:



Since the hypotenuse r is always taken positive so we put √26 on the hypotenuse (r):



So now we know that





tan%28X%29=opposite%2Fadjacent+=+y%2Fx=%28-5%29%2F1=-5

sec%28X%29=hypotenuse%2Fopposite+=+r%2Fx=sqrt%2826%29%2F1=sqrt%2826%29

csc%28X%29=hypotenuse%2Fopposite+=+r%2Fy=sqrt%2826%29%2F%28-5%29=-sqrt%2826%29%2F5

cot%28X%29=adjacent%2Fopposite+=+x%2Fy=1%2F%28-5%29=-1%2F5

Edwin

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
the point ​(​1,-5)= ​(​x,y)
x=1
y=-5

r=sqrt%281%5E2%2B%28-5%29%5E2%29
r=sqrt%281%2B25%29
r=sqrt%2826%29


sin%28x%29=y%2Fr
sin%28x%29=-5%2Fsqrt%2826%29
sin%28x%29=-5sqrt%2826%29%2F26


cos%28x%29=x%2Fr
cos%28x%29=1%2Fsqrt%2826%29
cos%28x%29=sqrt%2826%29%2F26


tan%28x%29=sin%28x%29%2Fcos%28x%29
tan%28x%29=%28-5sqrt%2826%29%2F26%29%2F%28sqrt%2826%29%2F26%29
tan%28x%29=-5

cot%28x%29=cos%28x%29%2Fsin%28x%29
cot%28x%29=%28sqrt%2826%29%2F26%29%2F%28-5sqrt%2826%29%2F26%29
cot%28x%29=-1%2F5


sec%28x%29=1%2Fcos%28x%29
sec%28x%29=1%2F%28sqrt%2826%29%2F26%29
sec%28x%29=26%2Fsqrt%2826%29
sec%28x%29=26sqrt%2826%29%2F26
sec%28x%29=sqrt%2826%29


csc%28x%29=1%2Fsin%28x%29
csc%28x%29=1%2F%28-5sqrt%2826%29%2F26%29
csc%28x%29=26%2F%28-5sqrt%2826%29%29
csc%28x%29=26sqrt%2826%29%2F%28-5%2A26%29
csc%28x%29=-sqrt%2826%29%2F5