4. The Equation for the terminal Side `theta` is `sqrt(3)x+y=0` and `x<=0`. Find value of all six trigonometric functions
Solution:
`sqrt(3)x+y=0`
`y=-sqrt(3)x`
Here, `x<=0`
Let `x=-1`
So `y=-sqrt(3)xx-1=sqrt(3)`
`P(-1,sqrt(3))`
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=-1` and `y=sqrt(3)`
In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`
`=(-1)^2 + sqrt(3)^2`
`=1 + 3`
`=4`
`:.r=sqrt(4)=2`
So, `x=-1,y=sqrt(3) and r=2`
`(1)` `sin(theta)=y/r=(sqrt(3))/(2)=0.86603`
`(2)` `cos(theta)=x/r=(-1)/(2)=-1/2`
`(3)` `tan(theta)=y/x=(sqrt(3))/(-1)=-sqrt(3)=-1.73205`
`(4)` `csc(theta)=r/y=(2)/(sqrt(3))=(2sqrt(3))/3=1.1547`
`(5)` `sec(theta)=r/x=(2)/(-1)=-2`
`(6)` `cot(theta)=x/y=(-1)/(sqrt(3))=(-sqrt(3))/3=-0.57735`
This material is intended as a summary. Use your textbook for detail explanation.
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