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Problem: If sin(x)=5/13 then find cos(x),tan(x),csc(x),sec(x),cot(x) [ Calculator, Method and examples ]

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Your problem `->` If sin(x)=5/13 then find cos(x),tan(x),csc(x),sec(x),cot(x)


`sin(x)=5/13`

Opposite side `(a)`, adjacent side `(b)` and hypotenuse `(c)`

`sin(x), cos(x), tan(x)` fromula


`sin(x) = "opposite"/"hypotenuse" = a/c`

`cos(x) = "adjacent"/"hypotenuse" = b/c`

`tan(x) = "opposite"/"adjacent" = a/b`

`csc(x) = "hypotenuse"/"opposite" = c/a`

`sec(x) = "hypotenuse"/"adjacent" = c/b`

`cot(x) = "adjacent"/"opposite" = b/a`


`sin(x) = "opposite"/"hypotenuse" = a/c = 5/13`

Here `a=5` and `c=13`

In triangle ABC, by Pythagoras' theorem
`c^2 = a^2 + b^2`

`:. b^2 = c^2 - a^2`

`= 13^2 - 5^2`

`= 169 - 25`

`= 144`

`:. b = sqrt(144) = 12`

So, `a=5,b=12 and c=13`

`(1)` `cos(x)=b/c=12/13`

`(2)` `tan(x)=a/b=5/12`

`(3)` `csc(x)=c/a=13/5`

`(4)` `sec(x)=c/b=13/12`

`(5)` `cot(x)=b/a=12/5`


Second Method
`sin(x)=5/13`

`(1)` `cos^2(x)=1-sin^2(x)`

`=1-(5/13)^2`

`=1-25/169`

`=(169-25)/169`

`=144/169`

`:.cos(x)=sqrt(144/169)=12/13`


`(2)` `tan(x)=sin(x)/cos(x)=(5/13)/(12/13)=5/12`


`(3)` `csc(x)=1/sin(x)=1/(5/13)=13/5`


`(4)` `sec(x)=1/cos(x)=1/(12/13)=13/12`


`(5)` `cot(x)=1/tan(x)=1/(5/12)=12/5`









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