2. For `P(15,8)`, find value of all six trigonometric functions
Solution:
`P(15,8)`
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=15` and `y=8`
In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`
`=15^2 + 8^2`
`=225 + 64`
`=289`
`:.r=sqrt(289)=17`
So, `x=15,y=8 and r=17`
`(1)` `sin(theta)=y/r=(8)/(17)=8/17`
`(2)` `cos(theta)=x/r=(15)/(17)=15/17`
`(3)` `tan(theta)=y/x=(8)/(15)=8/15`
`(4)` `csc(theta)=r/y=(17)/(8)=17/8`
`(5)` `sec(theta)=r/x=(17)/(15)=17/15`
`(6)` `cot(theta)=x/y=(15)/(8)=15/8`
This material is intended as a summary. Use your textbook for detail explanation.
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