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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