###### Asked in Calculus

Calculus

# What is value of Cos pi?

## Related Questions

###### Asked in Math and Arithmetic, Calculus, Trigonometry

### What is the exact value of the expression cos 7pi over 12 cos pi over 6 -sin 7pi over 12 sin pi over 6?

cos(a)cos(b)-sin(a)sin(b)=cos(a+b)
a=7pi/12 and b=pi/6
a+b = 7pi/12 + pi/6 = 7pi/12 + 2pi/12 = 9pi/12
We want to find cos(9pi/12)
cos(9pi/12) = cos(3pi/4)
cos(3pi/4)= cos(pi-pi/4)
cos(pi)cos(pi/4)-sin(pi)sin(pi/4)
cos(pi)=-1
sin(pi)=0
cos(pi/4) = √2/2
sin(pi/4) =√2/2
cos(pi)cos(pi/4)-sin(pi)sin(pi/4) = - cos(pi/4) = -√2/2

###### Asked in Math and Arithmetic, Calculus, Trigonometry

### What is the exact value using a sum or difference formula of the expression cos 11pi over 12?

11pi/12 = pi - pi/12
cos(11pi/12) = cos(pi - pi/12)
cos(a-b) = cos(a)cos(b)+sin(a)sin(b)
cos(pi -pi/12) = cos(pi)cos(pi/12) + sin(pi)sin(pi/12)
sin(pi)=0
cos(pi)=-1
Therefore, cos(pi -pi/12) = -cos(pi/12)
pi/12=pi/3 -pi/4
cos(pi/12) = cos(pi/3 - pi/4)
= cos(pi/3)cos(pi/4)+sin(pi/3) sin(pi/4)
cos(pi/3)=1/2
sin(pi/3)=sqrt(3)/2
cos(pi/4)= sqrt(2)/2
sin(pi/4) = sqrt(2)/2
cos(pi/3)cos(pi/4)+sin(pi/3) sin(pi/4)
= (1/2)(sqrt(2)/2 ) + (sqrt(3)/2)( sqrt(2)/2)
= sqrt(2)/4 + sqrt(6) /4
= [sqrt(2)+sqrt(6)] /4
Therefore, cos(pi/12) = (sqrt(2)+sqrt(6))/4
-cos(pi/12) = -(sqrt(2)+sqrt(6))/4
cos(11pi/12) = -(sqrt(2)+sqrt(6))/4

###### Asked in Calculus, Trigonometry, Differential Equations

### What is the derivative of cos pi x plus sin pi y all to the 8th power equals 44?

(cos(pi x) + sin(pi y) )^8 = 44
differentiate both sides with respect to x
8 ( cos(pi x) + sin (pi y ) )^7 d/dx ( cos(pi x) + sin (pi y) =
0
8 ( cos(pi x) + sin (pi y ) )^7 (-sin (pi x) pi + cos (pi y) pi
dy/dx ) = 0
8 ( cos(pi x) + sin (pi y ) )^7 (pi cos(pi y) dy/dx - pi sin (pi
x) ) = 0
cos(pi y) dy/dx - pi sin(pi x) = 0
cos(pi y) dy/dx = sin(pi x)
dy/dx = sin (pi x) / cos(pi y)

###### Asked in Math and Arithmetic, Trigonometry

### What is sin23A minus sin7A upon sin2A plus sin14A if A equals pi upon 21?

Using the identity, sin(X)+sin(Y) =
2*sin[(x+y)/2]*cos[(x-y)/2]
the expression becomes
{2*sin[(23A-7A)/2]*cos[(23A+7A)/2]}/{2*sin[(2A+14A)/2]*cos[(2A-14A)/2]}
= {2*sin(8A)*cos(15A)}/{2*sin(8A)*cos(-6A)}
= cos(15A)/cos(-6A)}
= cos(15A)/cos(6A)} since cos(-x) = cos(x)
When A = pi/21,
15A = 15*pi/21
and 6A = 6*pi/21 = pi - 15pi/21
Therefore, cos(6A) = - cos(15A)
and hence the expression = -1.

###### Asked in Trigonometry

### What are the important points you are to use in graphing the cosine function?

The basic cosine function is bounded by -1 and 1. It is a
periodic function with a period of 2*pi radians (360 degrees).
cos(0) = 1, cos(pi/2) = 0, cos(pi) = -1, cos(3pi/2) = 0, cos(pi)
= 1. In between these values it forms a smooth curve. Also, it may
help to understand that when the curve crosses the x-axis, its
slope is 1 or -1.

###### Asked in Abstract Algebra

### Can anyone prove i to the power i where i is square root of -1 is rational?

Nobody can prove it because it is not true. It is a real number,
though.
eix = cos(x) + i*sin(x)
Therefore
ei*pi/2 = cos(pi/2) + i*sin(pi/2) = 0 + i*1 = i
Raising both sides to the ith power, (ei*pi/2)i = ii
So that ii = ei*i*pi/2 = e-pi/2 = 0.2079.
Gelfond proved that the above value is irrational
[transcendental, actually].