prove that cos 52 + cos 68 + cos172 = 0
first of all cos 172 = -cos 8
So now we have
LS
= cos 52 + cos 68 - cos 8
one of the lesser known identities is
cos A − cos B = −2 sin ½ (A + B) sin ½ (A − B)
(here is good list of them)
http://www.themathpage.com/atrig/trigonometric-identities.htm
let's look at our cos 52 - cos8, here let's A = 52, B=8
cos 52 - cos 8 = -2sin[60/2]sin[44/2]
cos52 -cos8 = - sin 22
so
cos 52 + cos 68 + cos172
= cos52 + cos68 - cos8
= cos52 - cos8 + cos68
= - sin22 + cos68
but by the complementary relationship
sin 22 = cos68 , (since 22+68 = 90)
so finally
= -cos 68 + cos 68
= 0
= RS
Q.E.D.
To prove that cos 52 + cos 68 + cos 172 = 0, we'll use the trigonometric identity:
cos(A) + cos(B) = 2 * cos((A + B) / 2) * cos((A - B) / 2)
First, let's simplify cos 52 + cos 68:
cos 52 + cos 68 = 2 * cos((52 + 68) / 2) * cos((68 - 52) / 2)
= 2 * cos(120) * cos(8)
Now let's simplify cos 172:
cos 172 = cos(180 - 8) [using the property cos(A) = cos(-A)]
= -cos(8) [using the property cos(A) = -cos(-A)]
Now we can substitute these values back into the original expression:
cos 52 + cos 68 + cos 172 = 2 * cos(120) * cos(8) + (-cos(8))
= 2 * (-1/2) * cos(8) - cos(8)
= -cos(8) - cos(8)
= -2 * cos(8)
Since -2 * cos(8) = 0, we have proven that cos 52 + cos 68 + cos 172 = 0.
To prove that cos 52 + cos 68 + cos 172 = 0, let's use the property of the cosine function called cosine rule. The cosine rule states that:
cos (A + B) = cos A * cos B - sin A * sin B
Using this rule, we can rewrite the equation as follows:
cos 52 + cos 68 + cos 172 = cos (52 + 68) + cos 172
= (cos 52 * cos 68 - sin 52 * sin 68) + cos 172
Now, we need to simplify this expression further. Here's how to evaluate each term individually:
1. cos 52 * cos 68:
Using a calculator or trigonometric tables, we can find the values for cos 52 and cos 68. Multiply these two values together.
2. sin 52 * sin 68:
Since cos A = sin (90 - A), sin 52 * sin 68 is equivalent to cos (90 - 52) * cos (90 - 68). Calculate these two values individually and multiply them together.
3. cos 172:
Using a calculator or trigonometric tables, find the value for cos 172.
Once you have evaluated these terms, substitute the values back into the equation and simplify. If the final result is 0, then the proof will be complete.