Question

Solve the equation
cos(θ + 60◦) = 2 sin θ
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦.

Answer 1

use addition angle formula for first one is probably a good start

Answer 2

Not sure what that is... can someone show it step by step? I'm not familiar with these equations?

Answer 3

\[\cos(x+60)=\cos(x)\cos(60)-\sin(x)\sin(60)\]
\[=\frac{1}{2}\cos(x)-\frac{\sqrt{3}}{2}\sin(x)\] hmmm maybe this was not such a good start

Answer 4

wow, ok i am stuck. this is how wolfram does it, and i'll be darned if i see an easy method
http://www.wolframalpha.com/input/?i=cos%28x%2Bpi%2F3%29%3D2sin%28x%29
what class is this for?

Answer 5

It's for A-levels, pure mathematics 3.
I'll check it out and tell you if I need help.

Answer 6

I have a stepped solution, but am not sure what it all means.. If I give you that, maybe you can help me explain each step?

Answer 7

ok i have an idea, but the solutution would require a calculator

Answer 8

first we can write
\[=\frac{1}{2}\cos(x)-\frac{\sqrt{3}}{2}\sin(x)=2\sin(x)\] then
\[\cos(x)-\sqrt{3}\sin(x)=4\sin(x)\]
\[\cos(x)=(4-\sqrt{3})\sin(x)\]
\[\frac{\cos(x)}{\sin(x)}=4-\sqrt{3}\] or
\[\frac{\sin(x)}{\cos(x)}=\frac{1}{4-\sqrt{3}}\]

Answer 9

then we can take the inverse tangent to get a numerical answer

Answer 10

if you have a better way please let me know, i am curious

Answer 11

I'm lost.. Sorry! I'm not at all good with these things... if you can explain from the easiest possible step, I'd be much obliged! :)

Answer 12

well i don't really know, that is the problem

Answer 13

3 Attempt use of cos(A + B) formula to obtain an equation in cos θ and sin θ M1 Use trig formula to obtain an equation in tan θ (or cos θ, sin θ or cot θ) M1 Obtain tan θ = 1/(4 + 3 ) or equivalent (or find cos θ, sin θ or cot θ) A1 Obtain answer θ = 9.9° A1
Obtain θ = 189.9°, and no others in the given interval [Ignore answers outside the given interval. Treat answers in radians as a misread (0.173, 3.31).]

This is what it says... not sure what it all means?

Answer 14

first i used the addtion angle formula

Answer 15

ok just what it says

Answer 16

Can you explain this to me?

Answer 17

yes.
first we use this
\[\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)\] on the left hand side
we have
\[\cos(x+60)=\cos(x)\cos(60)-\sin(x)\sin(60)\] now x is a variable, but 60 is a number, so evaluating we get
\[\frac{1}{2}\cos(x)-\frac{\sqrt{3}}{2}\sin(x)=2\sin(x)\]

Answer 18

so far so good?

Answer 19

let me see.

Answer 20

Ok, I'm fine up until the very last step. How did you get that?

Answer 21

are you talking about how did he know
\[\cos(60)=\frac{1}{2} ; \sin(60)=\frac{\sqrt{3}}{2}?\]

Answer 22

\[\cos(60)=\frac{1}{2}\] and
\[\sin(60)=\frac{\sqrt{3}}{2}\]

Answer 23

i evaluated the funtions at 60 degrees

Answer 24

Oh, Ok.

Answer 25

We can actually proved this by drawing an equilateral triagle :)

Answer 26

Ok, so next step?

Answer 27

then i multiplied everything by 2 (both sides) to get
\[\cos(x)-\sqrt{3}\sin(x)=4\sin(x)\]

Answer 28

myininiaya is a triangle junkie. likes them almost as much as kit kats

Answer 29

Triangle jukie? is this a new kind of addition?

Answer 30

then put cosine by itself on the left and got
\[\cos(x)=4\sin(x)+\sqrt{3}\sin(x)\]
factored and got
\[\cos(x)=(4+\sqrt{3})\sin(x)\]

Answer 31

it is an addiction addition

Answer 32

then divided by sine to get
\[\frac{\cos(x)}{\sin(x)}=4+\sqrt{3}\]

Answer 33

Ah Ok. Triangles... Wish I was a junkie in them! Ok, next! :)

Answer 34

\[\cot(x)=4+\sqrt{3} \]

Answer 35

then took the reciprocal and got
\[\tan(x)=\frac{1}{4+\sqrt{3}}\]

Answer 36

because my calculator does not have a cotangent button on it

Answer 37

lol okay satellite

Answer 38

so we are looking for approximations not exact solutions?

Answer 39

and finally
\[x=\tan^{-1}\left( \frac{1}{4+\sqrt{3}}\right)\]

Answer 40

@myininaya
yes i guess so, since i can't think of another way

Answer 41

i thought maybe there was some trick to get an exact answer, but i can't see it

Answer 42

I think that's the only way to get it. Thanks for the help! I understand it now! :)

Answer 43

for more solutions:
\[\tan(x+180 n)=4+\sqrt{3}\]

Answer 44

I don't know if I could get the answer w/o the steps first being placed.... Must get the hang of it!

Answer 45

where n is an integer

Answer 46

\[x=\tan^{-1}(4+\sqrt{3})-180 n , n \in \mathbb{Z} \]

Answer 47

what myininaya said. you are asked for all solutions in [0, 360)

Answer 48

oh didn't see the interval

Answer 49

but still you can use the formula i have to find all the solutions in that interval

Answer 50

actually it is
\[x=\tan^{-1}(\frac{1}{4+\sqrt{3}})-180 n , n \in \mathbb{Z}\]

Answer 51

or something like that

Answer 52

yes satellite is right

Answer 53

Hmm. is that part of the answer?

Answer 54

2 to 2200

Answer 55

you get 9.896° so that is one answer

Answer 56

then add 180 degrees for the other answer

Answer 57

Ok. Thanks!