Prove sin(x+60degrees)=sin(120degrees-x)?

Question

any suggestions appreciated :)

question is
Prove sin(x+60degrees)=sin(120degrees-x)

question is
Prove sin(x+60degrees)=
sin(120degrees-x)

Answer 1

There's a pretty basic identity, sin (180 - A) = sin A

then sin (x+ 60) = sin [180 - (x+60)] = sin (120 - x) , as required.

Note that angles are in degrees.
The initial identity is evident from the projection definition of the sine function.

Answer 2

Imagine a line OP and this is the radius of a circle and we call it r. O is the centre of the circle and OP rotates about O. Also, let O be the origin of x and y coordinates. You can see that as OP rotates about O we can descibe the position of P at any point on the circumference of the circle by the relevant values of x and y.

Now, when OP makes an angle of 60 degrees above the positive x axis, sin 60 degrees = y/r

Now rotate OP through the 90 degree position to the point where OP makes an angle of 120 degrees to the x axis. The position of P now is also decribed by y/r.

So, sin 60 = y/r and sin 120 = y/r

Hence sin 60 = sin 120

and sin (180 - 60) must also = sin 120

In general sin(180 - theta) = sin theta

Answer 3

This is an example of a math problem that you will find easier to prove graphically than algebraically. Remember what ratio sin actually measures.....y/r....that means that x has nothing to do with this ration, so it does not matter where the angle is related to x.

Therefore....because 180-60 = 120 you can see that you have the same angle from both right and left.

Answer 4

Sin 60 and Sin 120 are the same

Answer 5

required to prove: sin(x+60) =sin(120-x)

here we are using degrees

now, sin (A+B) = sinA.cosB + cosA.sinB (identity)

let A= x and B =60

sin(x+60) = sinx.cos60+cosx.sin60

= (1/2)sinx+((root3)/2) cosx.....................(1)

now, sin (A-B) = sinA.cosB - cosA.sinB (identity)

let A= 120 and B = x

sin(120-x)= sin120.cosx-cos120.sinx

= ((root3)/2)cosx-(-1/2sinx)

= 1/2sinx+((root3)/2)cosx...(2)

equations (1) and (2) are equivalent,

therefore, sin(x+60) = sin(120-x) as required

i only wish that i could draw a diagram to explain
this better

note: sin(x+60) in the first quadrant is equal to
sin(180-(x+60)) =sin(120-x) in the second quadrant
-another simpler way of proving this identity

Answer 6

How 'bout x = 60 degrees since
sin(60) = sin(120) = (√3)/2


Doug

Answer 7

I hope I have the formula right because I'm going on memory, but I think there's a formula that says
sin (x+y) = sinxcosy +sinycosx
sin (x-y) = sinxcosy - siny cosx

sin(x+60)= sinxcos60 + sin60cosx = ..5sinx+.866cosx

sin(120-x) = sin120cosx-sinxcos120 = .866cosx+.5sinx

Since cos120 is negative, it will make the expression positive.

Answer 8

It has taken you three attempts to even write the question correctly..Do you know of any property of sine function that me be of use here.?Does the presence of 120 and 60 give you any clue?Pl reply

Answer 9

L.H.S = a million+sinx-cosx/a million+sinx+cosx = a million+sinx-cosx(a million+sinx+cosx)/a million+sinx+cosx(a million+s... =(a million+sinx)^2-(cosx)^2/(a million+sinx+cosx)^2 =2sinx+2sin^2x/2+2sinx+cosx+2sinxcosx =2sinx(a million+sinx)/2(a million+sinx)+2cosx(a million+sinx) =sinx/2cosx =tanx/2