Question

Find all solutions in the interval [0, 2π).
7 tan3x - 21 tan x = 0

Answer 1

is this the answer?

Answer 2

http://openstudy.com/study#/updates/50f1ca0ce4b0694eaccf36a0 there ya go

Answer 3

so you have \[7 \cdot \tan(3x)-21 \cdot \tan(x)=0\]
We will probably have to attempt to write tan(3x) in terms of tan(x) somehow using identities

Answer 4

like if possible that is

Answer 5

we can start out first with dividing both sides by 7
\[\tan(3x)-3 \tan(x)=0\]

Answer 6

My first thought it to rewrite tan(3x)=tan(2x+x)
using the sum identity for tan

Answer 7

how did you get to tan(3x)=tan(2x+x)

Answer 8

2x+x is 3x

Answer 9

oh ok

Answer 10

use the sum formula for tan(2x+x)

Answer 11

then use the double angle formula for tan(2x)

Answer 12

before we continue
are you sure you didn't mean to write
7*tan^3(x)-21tan(x)=0?

Answer 13

yeah that's the right equation sorry

Answer 14

oh that makes thing a lot more easier

Answer 15

\[7 \tan^3(x)-21\tan(x)=0\\ \text{ divide both sides by } 7 \\ \tan ^3(x)-3 \tan(x)=0\]
now just factor the left hand side

Answer 16

notice that both terms have a tan(x) in common

Answer 17

have you done that yet ? if so can you type here please ?

Answer 18

tanx = 0 or tan^2 x = 3 ?

Answer 19

let's see you have
\[\tan(x)(\tan^2(x)-3)=0 \\ \tan(x)=0 \text{ or } \tan^2(x)=3 \\ \tan(x)=0 \text{ or } \tan(x)=\sqrt{3} \text{ or } \tan(x)=-\sqrt{3}\]
you have 3 equations to solve in the interval [0,2pi]

Answer 20

tan(x)=sqrt(3) happens when sin(x)=sqrt(3)/2 and cos(x)=1/2
or when sin(x)=-sqrt(3)/2 and cos(x)=-1/2
(you can find when tan(x)=-sqrt(3) in a similar way; just make sin and cos differ in sign)


tan(x)=0 happens when sin(x)=0

Answer 21

can you solve sin(x)=0?

Answer 22

how do I do that

Answer 23

use the unit circle

Answer 24

https://www.mathsisfun.com/geometry/images/circle-unit-radians.gif
look for which angles the y-coordinate is 0

Answer 25

for example you see the angle 0 has y-coordinate 0
there is another angle so that the y-coordinate is 0, what is it?

Answer 26

pi/2?

Answer 27

at pi/2 the y-coordinate is 1

Answer 28

0?

Answer 29

the y-coordinate is the second number in the ordered pairs

Answer 30

I already said at the angle 0 the y-coordinate is 0
I'm asking you to give me another answer looking at the unit circle

Answer 31

another angle that the y-coordinate is 0? I see two points (1,0) and (-1,0) anf pi and 0

Answer 32

yes! bingo!

Answer 33

tan(x)=0 has solutions x=0 or x=pi

Answer 34

now you have to solve tan(x)=sqrt(3)

then solve solve tan(x)=-sqrt(3)

Answer 35

tan(x)=sqrt(3) only happens when tan is positive
which only happens when sin and cos have the same signs
so look in the first and third quadrant
look for when the sin value there is either sqrt(3)/2 or -sqrt(3)/2

Answer 36

for tan(x)=-sqrt(3)
this happens only in the 2nd and fourth quadrants (you know where sin and cos have opposite signs)
so here you would look for where again sin value is either sqrt(3)/2 or -sqrt(3)/2

Answer 37

the sin value being the y-coordinates

Answer 38

ok so are all those the solutions

Answer 39

well you only found two solutions
there are 4 more

Answer 40

none of those are choices though

Answer 41

I see 0,pi
you haven't even attempted to find the other four solutions

Answer 42

I told you how above but I'm not sure why you haven't tried to

Answer 43

for example tan(x)=sqrt(3)
the hint I gave to find your solutions to this was to look in the first and third quadrant and find when the sin value was either sqrt(3)/2 or -sqrt(3)/2

that happens at the angles pi/3 and 4pi/3

Answer 44

you try to solve tan(x)=-sqrt(3)

Answer 45

tanx = tan (π/6) or tan (π + π/6)

Answer 46

Is this the next question?

Answer 47

Or are you trying to solve tan(x)=-sqrt(3)?

Answer 48

Ok the sin value is the y-coordinate
do you see anywhere in the 2nd or 4th quadrant that the y-coordinate is sqrt(3)/2 or -sqrt(3)/2

Answer 49

yes

Answer 50

at what angles?

Answer 51

Do you see the angle for which the y-coordinate is sqrt(3)/2 in the 2nd quadrant?
Do you see the angle for which the y-coordinate is -sqrt(3)/2 in the 4th quadrant?

Answer 52

\[\text{ on } [0,2 \pi] \\ \tan(x)=0 \text{ when } x=0,\pi \\ \tan(x)=\sqrt{3} \text{ when } x=\frac{\pi}{3}, \frac{4 \pi}{3} \\ \tan(x)=-\sqrt{3} \text{ when } x=\frac{2 \pi}{3}, \frac{5\pi}{3}\]

Answer 53

ok then what

Answer 54

I don't think we can do anything further :p

Answer 55

What do you want to do further? Besides solve the equation?

Answer 56

oh ok thank you so much!

Answer 57

Like all I seen where choices to the equation above which I already found for you

Answer 58

I think you really need to work on learning how to read the unit circle.

Answer 59

like for example sin(x)=1/2
means look for the angles where you have the y-coordinate is 1/2

Answer 60

so you go to your unit circle

Answer 61

yeah I think I have somewhat of an understanding now

Answer 62

and you see it happens at pi/6 or also 5pi/6

Answer 63

yeah

Answer 64

so the solutions to sin(x)=1/2 is therefore x=pi/6 or 5pi/6

Answer 65

that had nothing to do with your question
it was just another example

Answer 66

thanks that was really helpful

Answer 67

np have a lovely day