4. If `sin(x)=7/25`, find other trigonometry functions `sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)`

Solution:
`sin(x)=7/25`, in Quadrant-1


Opposite side `(y)`, adjacent side `(x)` and hypotenuse `(r)`

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

`sin(theta) = "opposite"/"hypotenuse" = y/r`

`cos(theta) = "adjacent"/"hypotenuse" = x/r`

`tan(theta) = "opposite"/"adjacent" = y/x`

`csc(theta) = "hypotenuse"/"opposite" = r/y`

`sec(theta) = "hypotenuse"/"adjacent" = r/x`

`cot(theta) = "adjacent"/"opposite" = x/y`

`sin(x) = "opposite"/"hypotenuse" = y/r = 7/25`

Here `y=7` and `r=25`

In triangle ABC, by Pythagoras' theorem
`r^2=x^2+y^2`

`:.x^2=r^2-y^2`

`=25^2-7^2`

`=625-49`

`=576`

`:.x=sqrt(576)=24` (`:'` x is +ve in Quadrant-1)

So, `x=24,y=7 and r=25`

`(1)` `sin(x)=y/r=(7)/(25)=7/25`

`(2)` `cos(x)=x/r=(24)/(25)=24/25`

`(3)` `tan(x)=y/x=(7)/(24)=7/24`

`(4)` `csc(x)=r/y=(25)/(7)=25/7`

`(5)` `sec(x)=r/x=(25)/(24)=25/24`

`(6)` `cot(x)=x/y=(24)/(7)=24/7`

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Second Method
`sin(x)=7/25`, in Quadrant-1

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

`=1-(7/25)^2`

`=1-49/625`

`=(625-49)/625`

`=576/625`

`:.cos(x)=sqrt(576/625)=24/25=24/25`


`(2)` `tan(x)=sin(x)/cos(x)=(7/25)/(24/25)=7/25 xx 25/24=7/24=7/24`


`(3)` `csc(x)=1/sin(x)=1/(7/25)=25/7=25/7`


`(4)` `sec(x)=1/cos(x)=1/(24/25)=25/24=25/24`


`(5)` `cot(x)=1/tan(x)=1/(7/24)=24/7=24/7`

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