Question

find the value of cosπ/32​

Answer 1

Answer:

Go on applying the half angles formula, cos(x/2)=sqrt((1+cos(x))/2) as follows.

• Step 1: Cos(pi/4)=1/sqrt(2) that’s easy and all know that.
• Step 2 : Find cos(pi/8) using value of cos(pi/4) using above formula with x=pi/4
• Step 3: Find cos(pi/16) using value of cos(pi/8) using above formula with x=pi/4
• Step 4 : Find cos(pi/32) using value of cos(pi/16) using above formula with x=pi/16

If you want directly without going through this mess, then use the infinite series expansion of cos(x) in terms of ‘x’ and put x=pi/32 in it and take as much terms of this as possible according to your desired accuracy. For example. If you want 5 digits of accuracy, then find the minimum value of ‘t’ through trial and error such that, (pi/32)^(2(t+1))/(2(t+1))! < 0.000001 . The value of ‘t’ obtained in such a manner will give you the minimum number of terms of infinite series you should take into consideration for calculations for the desired accuracy up-to ‘5’ digits. Another point to note here is that, every term in this calculation of series must be of at-least of 6 digit accuracy without rounding off.

There are many other numerical methods also to calculate such value as it requires solving the equation cos(pi/32)=x or arc-cos(x)=pi/2. e.g. You can use Newton-Raphson method taking x=1 as initial guess., etc.

Here, arc-cos(x) is inverse cosine function.

I hope that this answer helps u :-)

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