Trigonometric Equations

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Section 4.6

20211116_032420.jpg


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?
 
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
 

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