Find the exact value of cos 1 degree + cos 2 degrees + cos 3 degrees + ... + cos 357 + cos 358 degrees + cos 359 degrees.
recall that cos(180-x) = -cos(x)
and just pair up the angles. The result will be quite simple.
To find the exact value of cos 1 degree + cos 2 degrees + cos 3 degrees + ... + cos 357 + cos 358 degrees + cos 359 degrees, we can use the formula for the sum of cosines.
The formula for the sum of cosines is:
cos(a) + cos(a + d) + cos(a + 2d) + ... + cos(a + (n-1)d) = [cos((n-1)d/2) * cos(a + (n-1)d/2)] / sin(d/2),
where a is the first angle, d is the common difference between angles, and n is the number of terms.
In this case, a = 1 degree, d = 1 degree (since each term increases by 1 degree), and n = 359.
Plugging these values into the formula, we have:
cos(1) + cos(2) + cos(3) + ... + cos(358) + cos(359) = [cos(359/2) * cos(1 + 359/2)] / sin(1/2).
Now let's calculate the value using a calculator.
To find the exact value of cos 1 degree + cos 2 degrees + cos 3 degrees + ... + cos 357 + cos 358 degrees + cos 359 degrees, we can use the trigonometric identity:
cos(A + B) = cos A * cos B - sin A * sin B
Let's write the sum of the angles as follows:
cos 1 degree + cos 2 degrees + cos 3 degrees + ... + cos 357 + cos 358 degrees + cos 359 degrees
= cos 1 degree + cos 359 degrees + cos 2 degrees + cos 358 degrees + cos 3 degrees + cos 357 degrees + ...
Notice that the angles are in pairs that sum up to 360 degrees. Therefore, we can pair each cosine term with another cosine term to simplify the expression:
(cos 1 degree + cos 359 degrees) + (cos 2 degrees + cos 358 degrees) + (cos 3 degrees + cos 357 degrees) + ...
Now, using the cos(A + B) identity we mentioned earlier, we can simplify each pair:
cos 360 degrees = 1
So, cos 1 degree + cos 359 degrees = 2 * cos 180 degrees = 2 * (-1) = -2
cos 2 degrees + cos 358 degrees = 2 * cos 180 degrees = 2 * (-1) = -2
cos 3 degrees + cos 357 degrees = 2 * cos 180 degrees = 2 * (-1) = -2
We can see that each pair always adds up to -2. Since there are 179 pairs, the total sum is:
-2 * 179 = -358
Therefore, the exact value of cos 1 degree + cos 2 degrees + cos 3 degrees + ... + cos 357 + cos 358 degrees + cos 359 degrees is -358.