1. The Equation for the terminal Side `theta` is `2x+y=0` and `x>=0`. Find value of all six trigonometric functions

Solution:
`2x+y=0`

`y=-2x`

Here, `x>=0`

Let `x=1`

So `y=-2`

`P(1,-2)`


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=-2`

In triangle ABC, by Pythagoras' theorem
`r^2 = x^2 + y^2`

`=1^2 + (-2)^2`

`=1 + 4`

`=5`

`:.r=sqrt(5)`

So, `x=1,y=-2 and r=sqrt(5)`

`(1)` `sin(theta)=y/r=(-2)/(sqrt(5))=(-2sqrt(5))/5=-0.89443`

`(2)` `cos(theta)=x/r=(1)/(sqrt(5))=(sqrt(5))/5=0.44721`

`(3)` `tan(theta)=y/x=(-2)/(1)=-2`

`(4)` `csc(theta)=r/y=(sqrt(5))/(-2)=(-sqrt(5))/2=-1.11803`

`(5)` `sec(theta)=r/x=(sqrt(5))/(1)=sqrt(5)=2.23607`

`(6)` `cot(theta)=x/y=(1)/(-2)=(-1)/2=-1/2`

---

---