Question

what is general solution for tan(θ/4)=1?

Answer 1

do you agree that

tan(x) = sin(x)/cos(x)

??

Answer 2

yes

Answer 3

so that means if tan(theta) = 1, then

sin(theta) = cos(theta)

Answer 4

look on the unit circle for points that have the same x and y coordinates

Answer 5

do you mean like (√2/2, √2,2)

Answer 6

yes, what is the angle that corresponds to that point?

Answer 7

45

Answer 8

good, or pi/4 radians

Answer 9

where else?

Answer 10

hint: look in Q3

Answer 11

5pi/4

Answer 12

so would those be for the two equations

Answer 13

so if theta = pi/4 or theta = 5pi/4, then tan(theta) = 1

Answer 14

now these aren't the only 2 solutions. There are infinitely more

Answer 15

we have to account for all of the coterminal angles

Answer 16

eg:

theta = pi/4
add on 2pi
(pi/4) + 2pi = pi/4 + 8pi/4 = 9pi/4

in general, we just write

\[\Large \frac{\pi}{4} + 2\pi*k\]

where k is any integer (whole number)

Answer 17

we do the same for 5pi/4 to get
\[\Large \frac{5\pi}{4} + 2\pi*k\]

Answer 18

with me so far?

Answer 19

yes this is making much more sense

Answer 20

ok so the theta inside the tangent is actually theta/4

which means

-------------------

if

\[\Large \tan\left(\frac{\theta}{4}\right) = 1\]

then

\[\Large \frac{\theta}{4} = \frac{\pi}{4}+2\pi*k\]
or
\[\Large \frac{\theta}{4} = \frac{5\pi}{4}+2\pi*k\]


agreed?

Answer 21

yes

Answer 22

the next step is to multiply both sides by 4 to completely isolate theta
So let's do that to the first equation
\[\Large \frac{\theta}{4} = \frac{\pi}{4}+2\pi*k\]
\[\Large 4*\frac{\theta}{4} = 4*\left(\frac{\pi}{4}+2\pi*k\right)\]
\[\Large \theta = \pi+8\pi*k\]
and then to the other equation as well
\[\Large \frac{\theta}{4} = \frac{5\pi}{4}+2\pi*k\]
\[\Large 4*\frac{\theta}{4} = 4*\left(\frac{5\pi}{4}+2\pi*k\right)\]
\[\Large \theta = 5\pi+8\pi*k\]
----------------------
So, in the end, we get
\[\Large \theta = \pi+8\pi*k\]
or
\[\Large \theta = 5\pi+8\pi*k\]

Answer 23

wow thanks so much!!

Answer 24

you're welcome