Trigonometric Equations

Section 4.6



Question 50

tan x = sqrt{3}

arctan(tan x) = arctan(sqrt{3}

x = pi/3

To find all solutions in the interval [-2pi, 2pi], I can use the unit circle, right?

Question 52

cot x = 1

arccot(cot x) = arccot(1)

x = pi/4

I can find all solutions by using the unit circle, right?

 

use the unit circle and find angles

 

Ok. Thanks.

 

I tried using the unit circle but can't seem to find all the solutions between -2pi and 2pi. Please, do 49 as a guide for me to do the rest.

 

given: tan (x) = sqrt(3), -2pi<=x<= 2pi

using unit circle you need to find where sin(x)/cos(x)=sqrt(3)

you will see that 60° angle ends at point (1/2, sqrt(3)/2)

so, sin(x)/cos(x)=(sqrt(3)/2)/(1/2) =sqrt(3)

same is with angle 240°: (-1/2, -sqrt(3)/2)

sin(x)/cos(x)=(-sqrt(3)/2)/(-1/2) =sqrt(3)

120° degrees:(-1/2, sqrt(3)/2)
(sqrt(3)/2)/(-1/2) = -sqrt(3) -> angle we need is -120°

300° degrees:(1/2, -sqrt(3)/2)

(-sqrt(3)/2)/(1/2) = -sqrt(3)->angle we need is -300°


or do it this way:

tan (x) = sqrt(3), -2pi<=x<= 2pi

x = tan^-1 (sqrt(3))
x=pi/3

in given interval, -2pi<=x<= 2pi , solutions are

x=pi/3 +pi*n
and, your solutions are:
x= pi/3
x= 4pi/3
x=- 2pi/3
x=-5pi/3

Degrees:
x=60°
x=240°
x=-120°
x=-300°

 

52.
cot(x)=1
Solutions for the range :-2pi<=x<=2pi

x=pi/4, x= 5pi/4,x= -3pi/4, x= - 7pi/4 not x= 3pi/4, x= 7pi/4
be careful with a signs
cot(3pi/4) = 1 is false
cot(7pi/4) = 1 is false

 

At least I try, right?