4. For `P(7,-24)`, find value of all six trigonometric functions
Solution:
`P(7,-24)`
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`
Here `x=7` and `y=-24`
In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`
`=7^2 + (-24)^2`
`=49 + 576`
`=625`
`:.r=sqrt(625)=25`
So, `x=7,y=-24 and r=25`
`(1)` `sin(theta)=y/r=(-24)/(25)=-24/25`
`(2)` `cos(theta)=x/r=(7)/(25)=7/25`
`(3)` `tan(theta)=y/x=(-24)/(7)=-24/7`
`(4)` `csc(theta)=r/y=(25)/(-24)=(-25)/24=-25/24`
`(5)` `sec(theta)=r/x=(25)/(7)=25/7`
`(6)` `cot(theta)=x/y=(7)/(-24)=(-7)/24=-7/24`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
