3. For `P(-8,6)`, find value of all six trigonometric functions
Solution:
`P(-8,6)`
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=-8` and `y=6`
In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`
`=(-8)^2 + 6^2`
`=64 + 36`
`=100`
`:.r=sqrt(100)=10`
So, `x=-8,y=6 and r=10`
`(1)` `sin(theta)=y/r=(6)/(10)=3/5=3/5`
`(2)` `cos(theta)=x/r=(-8)/(10)=(-4)/5=-4/5`
`(3)` `tan(theta)=y/x=(6)/(-8)=(-3)/4=-3/4`
`(4)` `csc(theta)=r/y=(10)/(6)=5/3=5/3`
`(5)` `sec(theta)=r/x=(10)/(-8)=(-5)/4=-5/4`
`(6)` `cot(theta)=x/y=(-8)/(6)=(-4)/3=-4/3`
This material is intended as a summary. Use your textbook for detail explanation.
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