Question

Solve: cos2Ɵ = sin Ɵ

Answer 1

\[\cos(2 \theta)-\sin(\theta)=0\]
Hint put in terms of sine
-------------------------

\[\cos(2 \theta)=\cos^2(\theta)- \sin^2(\theta)=(1-\sin^2(\theta))-\sin^2(\theta)=1-2\sin^2(\theta)\]

Answer 2

is 1−2sin2(θ) the answer ?

Answer 3

No

Answer 4

how do i go from there?

Answer 5

\[\cos(2 \theta)=1 - 2 \sin^2(\theta) \text{ is an identity}\]
we need to use it in our problem

Answer 6

\[1-2\sin^2(\theta)-\sin(\theta)=0\]
you need to solve this for theta

Answer 7

\[-2 \sin^2(\theta)-\sin(\theta)+1=0 \]


Can you solve:
\[-2u^2-u+1=0\]
for u?

Answer 8

if so the relationship btw the equations I put is
\[u=\sin(\theta)\]
so find what u is then replace u with sin(theta)

Answer 9

the problem above −2u2−u+1=0 you just add u to the other side right?

Answer 10

No solve that for u

Answer 11

\[-2u^2-u+1=0\]
Solve this for u

Answer 12

u = √1/2

Answer 13

You should had got two solutions.

Answer 14

Square roots can be both negative and positive. Those would be the two solutions.

Answer 15

oh i see

Answer 16

so it's u= +-√1/2 ?

Answer 17

Yes

Answer 18

\[(-2u+1)(u+1)=0\]

Answer 19

is it 2u + 1 = 0?

Answer 20

You have a*b=0 => either a=0 , b=0 or both=0
so we have
-2u+1=0 or u+1=0
Solve both of these linear equations.

Answer 21

-2u+1 i knew it :|

Answer 22

u = 1/2 and u = -1

Answer 23

Write now remember our relationship
\[u=\sin(\theta)\]

Answer 24

So we have
\[\sin(\theta)=\frac{1}{2} \text{ or } \sin(\theta)=-1\]

Answer 25

Oops right* not write*
sorry wrong right/write lol

Answer 26

it's ok

Answer 27

so I can plug in to the calculator : inverse sin (1/2) then

Answer 28

after that multiply by pi/180 to get pi/6 right ?

Answer 29

30 or Pi/6

Answer 30

Well I would take off points if you used a calculator

Answer 31

oh ok

Answer 32

It doesn't give you all the solutions and plus sometimes it gives you an approximation

Answer 33

depending on what calculator you use

Answer 34

There is the unit circle though

Answer 35

it's really confusing :|

Answer 36

but if i use calculator then I'm fine with that :)

Answer 37

when is sine, 1/2?

Answer 38

brainshot gave you one value that satisfies the equation

Answer 39

@myininaya Are you a teacher?

Answer 40

Myininaya is the awsomest teacher of all time :D

Answer 41

Ha yeah, but I meant in reality.

Answer 42

Oh yes in reality.

Answer 43

What do you teach?

Answer 44

Calculus to undegrad students in a university.

Answer 45

Lol, I am talking for her :P :D

Answer 46

@brainshot3 I am.
\[\sin(\theta)=\frac{1}{2}\]
\[\theta=\frac{\pi}{6}+2 n \pi \text{ this } n 2 \pi \text{ part is the n times we go around the circle}\]
But we also have another value in the interval [0,2pi] such that \[\sin(\theta)=\frac{1}{2}\]
\[\theta=\frac{5 \pi}{6} +2 n \pi\]
And by the way n is an integer

Answer 47

http://www.google.com/imgres?imgurl=http://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/300px-Unit_circle_angles_color.svg.png&imgrefurl=http://en.wikipedia.org/wiki/Unit_circle&h=300&w=300&sz=38&tbnid=zCoPII8QibzEpM:&tbnh=86&tbnw=86&prev=/search%3Fq%3Dunit%2Bcircle%26tbm%3Disch%26tbo%3Du&zoom=1&q=unit+circle&docid=D09vYD7H2hKF-M&sa=X&ei=ja17T7qbDuOh2QW1r7WrAw&ved=0CC8Q9QEwAA&dur=438

Answer 48

and a phd student in mathematics and yet find time to help people here lol

Answer 49

Wow! What do you plan to get a job in?

Answer 50

You really need to familiarize yourself with the unit circle. It is your best friend in trig.

Answer 51

thanks for the link!

Answer 52

You still need to solve the other equation.

Answer 53

\[\sin(\theta)=-1 \]

Answer 54

http://www.khanacademy.org/math/trigonometry/v/the-unit-circle-definition-of-trigonometric-function

Answer 55

i got cos^2 θ = 0 ?

Answer 56

thanks for the link @FoolForMath

Answer 57

\[ \sin(\theta)=-1=\sin (-\frac \pi 2) \implies \theta = n\pi +(-1)^n (-\frac \pi 2) \]

Answer 58

Btw, Myin, I thought that definition of \(2n\pi \) is only for \( \cos \), am I missing something?

Answer 59

This question is really hard for me, I will look over it again tonight thanks guys!

Answer 60

Btw can you guys help me with another one ?

Answer 61

I kinda in a rush :|

Answer 62

\[ \sin(\theta)=-1=\sin (-\frac \pi 2) \implies \theta = n\pi +(-1)^n (-\frac \pi 2) \]\[= n\pi +(-1)^{n+1} \left( \frac \pi 2\right)\]