1. For `P(3,4)`, find value of all six trigonometric functions
Solution:
`P(3,4)`
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=3` and `y=4`
In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`
`=3^2 + 4^2`
`=9 + 16`
`=25`
`:.r=sqrt(25)=5`
So, `x=3,y=4 and r=5`
`(1)` `sin(theta)=y/r=(4)/(5)=4/5`
`(2)` `cos(theta)=x/r=(3)/(5)=3/5`
`(3)` `tan(theta)=y/x=(4)/(3)=4/3`
`(4)` `csc(theta)=r/y=(5)/(4)=5/4`
`(5)` `sec(theta)=r/x=(5)/(3)=5/3`
`(6)` `cot(theta)=x/y=(3)/(4)=3/4`
This material is intended as a summary. Use your textbook for detail explanation.
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