Use the fact that 36 = 120 -84 to find the exact value of x

[hndalama]

Use the fact that 36 = 120 -84 to find the exact value of x given that sinx = sin84 - sin36

i solved this by rewriting 36 = 120 - 84 as 84 = 120 - 36 and then solving as follows

sinx = sin(120 - 36) - sin36
sinx = sin120cos36 - cos120sin36 - sin36
sinx = sin120cos36 + 0.5sin36 - sin36
sinx = sin120cos36 - 0.5sin36
sinx= sin120cos36 + cos120sin36
sinx =sin(120+36) = sin(156)

the answer given is 24 which I get by 180 - 156 =24

I want to know if the question can be solved by substituting sin(120 - 84) for sin36, when i use a similar method as above I get stuck. My work is as follows:

sinx = sin84 - sin(120 - 84)
sinx = sin84 - (sin120cos84 - cos120sin84)
sinx = sin84 - (sin120cos84 + 0.5sin84)
sinx = 0.5sin84 - sin120cos84
???

 

[Ishuda]

Use the fact that 36 = 120 -84 to find the exact value of x given that sinx = sin84 - sin36

i solved this by rewriting 36 = 120 - 84 as 84 = 120 - 36 and then solving as follows

sinx = sin(120 - 36) - sin36
sinx = sin120cos36 - cos120sin36 - sin36
sinx = sin120cos36 + 0.5sin36 - sin36
sinx = sin120cos36 - 0.5sin36
sinx= sin120cos36 + cos120sin36
sinx =sin(120+36) = sin(156)

the answer given is 24 which I get by 180 - 156 =24

I want to know if the question can be solved by substituting sin(120 - 84) for sin36, when i use a similar method as above I get stuck. My work is as follows:

sinx = sin84 - sin(120 - 84)
sinx = sin84 - (sin120cos84 - cos120sin84)
sinx = sin84 - (sin120cos84 + 0.5sin84)
sinx = 0.5sin84 - sin120cos84
???

Unless one restricts x in some way, this question does not have an answer. As a simple example suppose we want the exact value of x when sin(x) = 0. Well
sin(n\(\displaystyle \pi\)) = 0
so choose an n and get an x.

Thus in your first working you would get
x = 156 + 360 n
or, since sin(\(\displaystyle \pi\)-x)=sin(x).
x = 24 + 360 n

Continuing with what you have, your second working gives
sinx = 0.5sin84 - sin120cos84
sinx = -cos120sin84 - sin120cos84
sinx = -[cos120sin84 + sin120cos84] = - sin(204)
= -sin(204-360)=-sin(-156) = sin(156)
which is what you have in the first working.

 

[hndalama]

Unless one restricts x in some way, this question does not have an answer. As a simple example suppose we want the exact value of x when sin(x) = 0. Well
sin(n\(\displaystyle \pi\)) = 0
so choose an n and get an x.

Thus in your first working you would get
x = 156 + 360 n
or, since sin(\(\displaystyle \pi\)-x)=sin(x).
x = 24 + 360 n

Continuing with what you have, your second working gives
sinx = 0.5sin84 - sin120cos84
sinx = -cos120sin84 - sin120cos84
sinx = -[cos120sin84 + sin120cos84] = - sin(204)
= -sin(204-360)=-sin(-156) = sin(156)
which is what you have in the first working.

Thank you

 

[Deleted member 4993]

Use the fact that 36 = 120 -84 to find the exact value of x given that sinx = sin84 - sin36???

You did a problem like this:

http://www.freemathhelp.com/forum/threads/97368-prove-this-identity-sin(3x)-sin(x)-2sin(2x)cos(x)

Use that "principle".

(84+36)/2 = 60

sin(84) - sin(36)

= sin (60 + 24) - sin(60 - 24)

= 2*sin(24) * cos(60) =sin(24) → x = 24, 156, (2π ± 24) , (2π ± 156)

 

[hoosie]

Using Trig angle addition formula

Example:

Find the exact value of x given that √3cosx = cos48° + cos12°

Use addition angle formulas:
cos(A + B) = cosAcosB - sinAsinB
cos(A - B) = cosAcosB + sinAsinB

Let A = 30°, B = 18°

cos48°
= cos(30°+ 18°)
= cos30°cos18°- sin30°sin18°

cos12°
= cos(30°- 18°)
= cos30°cos18° + sin30°sin18°




√3cosx
= cos48°+ cos12°
= cos30°cos18°- sin30°sin18°+ cos30°cos18°+ sin30°sin18°
= 2cos30°cos18°
= √3cos18° since cos30° = √3/2

Therefore x = 18