5. The Equation for the terminal Side `theta` is `y=2x` and `sec(theta)>0`. Find value of all six trigonometric functions
Solution:
`y=2x`
Here, `sec(theta)>0`
`=>cos(theta)>0`
`=>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)=1.11803`
`(5)` `sec(theta)=r/x=(sqrt(5))/(1)=sqrt(5)=2.23607`
`(6)` `cot(theta)=x/y=(1)/(2)=1/2`
This material is intended as a summary. Use your textbook for detail explanation.
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