# How to calculate the value of sin 50°????????????????????????????????????????????? ???????????????

Arun
25750 Points
6 years ago

Dear student

sin50=sin(90–40)

sin(90–40)=cos(40)….(we take the complementary trigonometric function if angle is in the form of n{π/2}±phi)

cos40=cosx

cos120=cos3x

cos3x=cos2xcosx-sin2xsinx

cos3x={cos²x-sin²x}(cosx)-2sinxcosx(sinx)

cos3x=cos³x-sin²xcosx-2sin²xcosx

cos3x=cos³x-3sin²xcosx

cos3x=cos³x-3cosx+3cos³x

cos3x=4cos³x-3cosx

cos 3x=cos 120=-cos 60=-1\2

8cos³x-6cosx+1=0

And by the courtesy of desmos graphing calculator, we have been able to obtain one negative and two positive solutions for cosx

There can't be a negative solution since all cosines are positive in quadrant 1,

sine function increases from 0 to 90, therefore sin50>sin45, and sin50>1/√2

So according to the next two conditions, 0.174 can't be a solution, and hence the only remaining solution is 0.766, which makes it our answer.

Sailik Banerjee
33 Points
6 years ago
Friends we can find out the value of sin50° through the following steps
Let;y=f(x)=sinx
and;x=45°;∆x=5°=(0.01745×5)=0.08725
Now;f(x)=sinx;f’(x)=cosx;
f(x+∆x)=f(x)+f’(x)×∆x
or;f(45°+5°)=f(45°)+f’(45°)×0.08725
or;f(50°)=sin45°+(cos45°×0.08725)
or;f(50°)=sin50°=0.7688