Help solving Cos(115) degrees.?

Question

Using the half angle identity ( maybe some other ones idk) and without a calculator.

I could not get the exact number for cos(230) but i got.

sqrt{2+2cos(230)}/2 And I dont know if that is even right.
Is there a way to get a answer without any trig signs in it. (getting an exact number for cos(230)

Answer 1

The angle is 230. Try putting it into acute form.

230-180 = 50

cos 230 = - cos 50 (since it is in the third quadrant)

There no way to do it, unless you know what cos 5 and sin 5 are for example. cos 60, cos 45, cos 30 and cos 90 only get you so far.

Answer 2

I'd try to go cos(120 - 5) = cos120cos5 + sin120sin5.

5 degrees is the hard part.

First, get cos(15) using x=30 in the half-angle formula:
cos(x/2) = sqrt[(1+cosx)/2
→ cos15 = sqrt[1+sqrt3/2)/2]

Now use x=5 in the triple-angle formula here:
cos(3x) = cosx(1 - 4sin²x)
→ cos15 = cos5(1 - 4sin²15)
→ sqrt[1+sqrt3/2)/2] = cos5[4cos²5 - 3]

At this point you need to solve this natsy polynimial to find the value of cos5... It doesn't look easy.

Answer 3

You don't need a half angle identity.
Look: cos(115) = - cos(75)
(cos(180-x)= -cos x.)
cos(75) = cos(30+45) = cos 30 cos 45 - sin 30 sin 45.
Next, both cos 45 and sin 45 are equal to √2/2.
So we have √2/2(cos 30 - sin 30)=
√2/2(√3/2 - 1/2).
= 1/4(√6 -√2).
So cos(115) = -1/4(√6 - √2).

Answer 4

use a calculator

Answer 5

No
Please give me best answer thanks!