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How do you evaluate sin(pi/5)?

source : socratic.org

How do you evaluate sin(pi/5)?

Let #theta=pi/5#, then #5theta=pi#

and #3theta=pi-2theta#. Note #theta) is an acute angle.

Hence #sin3theta=sin(pi-2theta)# but as #sin(pi-A)=sinA#

This can be written as

#sin3theta=sin2theta# expanding them

or #3sintheta-4sin^3theta=2sinthetacostheta#

as #theta=pi/5# we have #sintheta!=0# and dividing by it we get

#3-4sin^2theta=2costheta# or

#3-4(1-cos^2theta)=2costheta#

or #4cos^2theta-2costheta-1=0#

and using quadratic formula #costheta=(2+-sqrt(2^2-4*4*(-1)))/(2*4)#

= #(2+-sqrt(20))/8=(1+-sqrt5)/4#.

But as #(1-sqrt5)/4# is negative and #costheta# cannot take this value, hence

#costheta=(1+sqrt5)/4# and

#sintheta=sqrt(1-((1+sqrt5)/4)^2)=sqrt(1-((1+5+2sqrt5)/16))#

= #sqrt((16-6-2sqrt5)/16)=sqrt(10-2sqrt5)/4#

As #theta=pi/5#

#sin(pi/5)=sqrt(10-2sqrt5)/4#

Online calculation with the function sin according to the sin(pi)

Online calculation with the function sin according to the sin(pi) – To calculate sine online of `pi/6`, enter sin(`pi/6`), after calculation, the result `1/2` is returned. Note that the sine function is able to recognize some special angles and make the calculations with special associated values in exact form.sin(pi/5). Extended Keyboard. Upload.How To Find cos(pi/5), sin(pi/5) | MSM mathIn this video I showed how to solve exact value of cos(pi/5) and sin(pi/5) by used relation angles (pi-(2pi/5)…

sin(pi/5) – Wolfram|Alpha – In this case, $x=\pi/5$. Setting the above two equations equal to each other results in the quadratic equation in questionHence #sin3theta=sin(pi-2theta)# but as #sin(pi-A)=sinA#. This can be written as. #sin3theta=sin2theta# expanding them.Because there is sin18 but there will be cos 18. =4 sin to the power of 4 pi/5.cos square pi/5.

sin(pi/5) - Wolfram|Alpha

How To Find cos(pi/5), sin(pi/5) | MSM math – YouTube – sin( °) = 0. Таблица синусов в радианах.tan pi/2. undefined. sin 2pi.For all values of angle A and B `(i)sin(A-B)=sinAcosB-cosAsinB (ii)sin(A+B)=sinAcosB+cosAsinB`. Geometric representation of sine and cosine of angles.

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