Natalie N.

asked • 11/19/23

Solve the equation. (Find all solutions of the equation in the interval [0, 2𝜋). Enter your answers as a comma-separated list.)

4 tan(2x) − 4 cot(x) = 0

2 Answers By Expert Tutors

By:

Patrick F. answered • 11/19/23

Tutor
5 (14)

Patient and Inspiring Math Tutor
About this tutor ›
About this tutor ›


Upvote • 0 Downvote
More
Report

Doug C.

I think pi/2 and 3pi/2 are also solutions. Solving using the double angle identity for tan(2x) does not lead to those solutions. However converting to sin(2x)/cos(2x) then using double angle identity for sin and cos does lead to those solutions. Wondering if there is some indicator that that is the case.
Report

11/19/23

Patrick F.

tutor
You're right. Missing are the cases where tan(2x) = cot(x) =0. (pi/2, 3pi/2). I'll have to think more about why that is. Thanks!
Report

11/19/23

Andrew S.

tutor
How did you animate this explanation? Pretty cool.
Report

12/07/23

Patrick F.

tutor
Thanks. I use Manim which is a library in Python.
Report

12/07/23

Raymond B. answered • 11/19/23

Tutor
5 (2)

Math, microeconomics or criminal justice

See tutors like this
See tutors like this

one method is graph tan2x-cotx

such as using online Desmos graphing or a handheld graphing calculator

it shows 6 intersections with the x axis between 0 and 2pi

pi/6, pi/2, 5p/6, 7pi/6, 3pi/2, and 11pi/6

those are all solutions to the equation


algebraically use the double angle formula:


4tan2x -4cotx=0


divide by 4

tan2x-cotx=0

tan2x= cotx= 1/tanx


2tanx/(1-tan^2(x))=1/tanx

cross multiply

2tan^2(x)=1-tan^2(x)

3tan^2(x)=1

tan^2(x)=1/3

tanx =+/-sqr(1/3)=+/-1/sqr3

x= 30, 150, 210, 330 degrees

or

x=pi/6, 5pi/6, 7pi/6, 11pi/6



try pi/2

tan(2pi/2) - cot(pi/2)

=tanpi - cot(pi/2)

= 0 -0

= 0


try 3pi/2

tan (3pi) -cot(3pi/2)

= 0-0

= 0 - 0


pi/2 and 3pi/2 are also solutions


pi/6, pi/2, 5pi/6, 7pi/6, 3pi/2, 11pi/6


use the half angle formula

solve for 2x= pi and 3pi

x = pi/2 and 3pi/2

Upvote • 0 Downvote
More
Report

Doug C.

I think pi/2 and 3pi/2 are also solutions. Solving using the double angle identity for tan(2x) does not lead to those solutions. However converting to sin(2x)/cos(2x) then using double angle identity for sin and cos does lead to those solutions. Wondering if there is some indicator that that is the case.
Report

11/19/23

Still looking for help? Get the right answer, fast.

Ask a question for free

Get a free answer to a quick problem.
Most questions answered within 4 hours.

OR

Find an Online Tutor Now

Choose an expert and meet online. No packages or subscriptions, pay only for the time you need.