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`
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`
This material is intended as a summary. Use your textbook for detail explanation.
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