compute..
Cos(1degree)+cos(3degree)+cos(5degree)+...+Cos(179degree)
plz show working even an hint can,t help me.Have been do maths alday my brain is fried..Ah thanks
cute problem
recall that cos x = - cos(180 - x)
that is
cos(1) = - cos(179)
cos(3) = - cos(177)
..
cos(89) = - cos(91)
cos(91) = - cos(89)
...
cos(177) = - cos(3)
cos(179) = - cos(1)
so :
cos1 + cos3 + ... + cos 177 + cos 179
= (cos1+cos179) + (cos3+cos177) + ...+ (cos89+cos91)
= 0+0+...+0+0
= 0
To compute the given expression, we can use the formula for the sum of a finite arithmetic series. The formula is:
Sn = (n/2)(a + l)
Where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, the first term (a) is cos(1 degree), and the last term (l) is cos(179 degrees). Notice that the terms increase by 2 degrees each time, so the common difference (d) is 2 degrees.
To find the number of terms (n), we can use the formula for the nth term of an arithmetic series:
an = a + (n - 1)d
For this series, using the last term, we have:
cos(179 degrees) = cos(1 degree) + (n - 1)(2 degrees)
Simplifying the equation:
2n = 178
n = 89
Now that we know the number of terms, we can substitute the values into the sum formula:
Sn = (89/2)(cos(1 degree) + cos(179 degrees))
To compute cos(1 degree) and cos(179 degrees), we need to convert the angles to radians. The conversion factor is π/180.
cos(1 degree) = cos((1π)/180)
cos(179 degrees) = cos((179π)/180)
Then we can substitute these values into the sum formula and calculate the answer.
However, as you mentioned that even a hint won't help, the quickest way to compute this sum is by using a scientific calculator or a mathematical software tool, such as Python, MATLAB, or Wolfram Alpha.
Using Python, you can write a simple code to compute the sum:
```python
import math
sum = 0
for i in range(1, 180, 2):
angle = math.radians(i)
sum += math.cos(angle)
print(sum)
```
Running this code will give you the desired sum.