Question

-3sin(0.5piN)+4cos(0.5piN)= 5cos(0.5piN+0.6435) ..how i can convert from left to right side?

Answer 1

So we want
\[-\sin(C)\sin(0.5\pi N)+\cos(C)\cos(0.5 \pi N)\]
So we can write this as
\[\cos(0.5 \pi N+C)\]

\[\frac{\sqrt{(3)^2+(4)^2}}{\sqrt{(3)^2+(4)^2}}(-3 \sin(0.5 \pi N)+4 \cos(0.5 \pi N))\]
\[5(\frac{-3}{5}\sin(0.5 \pi N)+\frac{4}{5} \cos(0.5 \pi N))\]

Answer 2

But we want sin(C)=3/5 and cos(C)=4/5

Answer 3

which can happen in a (3,4,5) triangle :)

Answer 4

\[5\cos(0.5 \pi N+C)=-3 \sin(0.5 \pi N)+4 \cos(0.5 \pi N)\]

Answer 5

Divide by 5 on the LHS first

Answer 6

You need to find angle C between 0 and pi/2

Answer 7

you can use either of those equations I mentioned

Answer 8

Then use @ZombiePig 's diagram express 3/5 and 4.5 in terms of cos C and sin C

Answer 9

*4/5

Answer 10

Then apply
cos a . cos b - sin a . sin b = cos (a+b)

Answer 11

Can this be solved without electronic aid? (Don't say any chart or table)

Answer 12

yep unless you want to approximate C

Answer 13

But still we can guesstimate a value between 0 and pi/2 without a calculator

Answer 14

I got it ! thanks all
:)

Answer 15

\[C=\arccos(\frac{4}{5}) \text{ exact value}\]

Answer 16

\[{-3 \over 5}\sin(0.5piN)+{4 \over 5}\cos(0.5piN)\]
\[{-\sin C}\sin(0.5piN)+{\cos C}\cos(0.5piN)\]
\[\cos ( 0.5piN + C)\]
\[\cos ( 0.5piN + \tan^{-1} { 3 \over 4})\]

Answer 17

The question ask for proof.

Answer 18

@FoolForMath , didn't get what you are trying to convey ?

Answer 19

0.6435 that's C isn't? can we compute this value without electronic aid?

Answer 20

I don't think so. @FoolForMath

Answer 21

the question is not asking for the proof , it's just simplifying the answer.