Make use of a trig identitiy to find the exact value of cos 75 degrees.?

Answer 1

The radians are more familiar regarding the unit circle, so let's convert 75 degrees to radians. To convert to radians, you just multiply pi/180 by the degrees.

75 * pi/180 = 75pi/180, which reduces to 5pi/12

The trig identity that we will be taking advantage of is going to be one of the following:

cos (a + b) = cos(a)cos(b) - sin(a)sin(b)
-or-
cos (a - b) = cos(a)cos(b) + sin(a)sin(b)

Our goal here is to split up 5pi/12 into two parts that appear on our known values of the unit circle (everything in the form pi/2, pi/3, pi/4, and so forth).

To split up 5pi/12, we just have to think of splitting the 5.

What two numbers does 5 split up into? The answer to that is:
4+1
3+2

Try 4+1: 5pi/12 = 4pi/12 + pi/12. 4pi/12 = pi/3, is a known value on your unit circle. pi/12, however, is not.

Try 3+2: 5pi/12 = 3pi/12 + 2pi/12 = pi/4 + pi/6. Both pi/4 and pi/6 are known values on our unit circle, so that will be our decomposition.

cos(5pi/12) = cos (pi/4 + pi/6)

And now we use that formula

cos(pi/4 + pi/6) = cos(pi/4)cos(pi/6) - sin(pi/4)sin(pi/6)

The coordinates of the point pi/4 on the unit circle are:
(sqrt(2)/2, sqrt(2),2), so the cosine is the first value and the sine is the second value.

The coordinates of the point pi/6 on the unit circle are:
(sqrt(3)/2, 1/2).

So it's now pretty much a plugfest.

cos (pi/4 + pi/6) = [sqrt(2)/2] [sqrt(3)/2] - [sqrt(2)/2] [1/2]
= sqrt(6)/4 - sqrt(2)/4
= [sqrt(6) - sqrt(2)]/4

Answer 2

The cosine is a good function so cos(-x) = cos(x): cos(-seventy 5) = cos(seventy 5) cos(seventy 5) = cos(40 5 + 30) cos(40 5 + 30) = cos(40 5)cos(30) - sin(40 5)sin(30) cos(40 5)cos(30) - sin(40 5)sin(30) = (?2/2)cos(30) - (?2/2)sin(30) (?2/2)cos(30) - (?2/2)sin(30) = (?2/2){cos(30) - sin(30)} (?2/2){cos(30) - sin(30)} = (?2/2){?3/2 - a million/2} (?2/2){?3/2 - a million/2} = ?6/4 - ?2/4 ?6/4 - ?2/4 = (a million/4)(?6 - ?2)

Answer 3

Use the sum of angles formula

cos(a + b) = cos a cos b - sin a sin b

You find two angles you already know that add up to 75, then use this formula.

Answer 4

cos 75 degree=0.259

Answer 5

cos 75o = cos (30o + 45o) = cos 30o cos 45o - sin 30o sin 45o = (sqrt3/2) (sqrt2)/2 - (1/2) (sqrt 2)/2

So: cos 75o = (sqrt3-1)(sqrt2)/4

Anabel