How do I evaluate this? cos 255 degress...?

Question

Cos 255degrees-cos195degrees

Answer 1

For this use this formula:

Cos (180 + a) = -Cos a
where a is any acute angle

255 = 180 + 75
Cos 225 = Cos(180 + 75)
= -Cos 75

195 = 180 + 15
Cos(195) = Cos (180 + 15)
= -Cos 15

Cos (255) - Cos (195)
= Cos 15 - Cos 75
= 0.70710678118654752440084436210485
= 0.71 (approximate)

Answer 2

In first place, see that 255 degrees is between 180 and
270 degrees. It means that the values of x-axis (the cosine projection) and y-axis (sine projection) are negatives.
So if both of them are negative, you do not have to worry about the sign anymore, but its numeric value. However,
if you substract 255 degrees less 180 degrees,
we realize that the equivalent in the Quadrant III is:

(255 - 180) degrees = 75 degrees

or in known angles can be also expressed as:

60 + (60 -45) degrees = 75 degrees

Note: I intended to combine 75 degrees or
decompose it into 0, 30, 45, 60 and 90 degrees
by means of adding them or subtracting them or
any.

Now, consider that AX=60 and BY=60 and CY=45
also that X = AX = 60 and Y = BY - CY = 15

So,
cos[X-Y] = cosX cosY - sinX sinY

where
cosX = cosAX
and
sinY = sin(BY - CY)
sin[BY - CY] = sinBY cosCY - cosBY sinCY
and
cos[BY - CY] = cosBY cosCY + sinBY sinCY

cos(60) = 1/2

sin(60 - 45) = sin(60) cos(45) - cos(60) sin(45)
= sqrt(2)*[sqrt(3) - 1]/4

sin(15) = sqrt(2) * [sqrt(3) - 1] / 4

cos(60 - 45) = cos(60) cos(45) + sin(60) sin(45)
= sqrt(2)*[sqrt(3) + 1]/4

cos(15) = sqrt(2) * [sqrt(3) + 1] / 4


So we have
cos[X+Y] = cosX cosY - sinX sinY
cos[60 + 15] = cos60 cos15 - sin60 sin15
= [1/2] * {sqrt(2) * [sqrt(3) + 1] / 4} - [sqrt(3) / 2] * {sqrt(2) * [sqrt(3) - 1] / 4}
= { sqrt(2) / 8 } * { sqrt(3) + 1 - [ sqrt(3) * ( sqrt(3) - 1 ) ] }
= { sqrt(2) / 8 } * { sqrt(3) + 1 - 3 + sqrt(3) }
= { sqrt(2) / 8 } * { 2 * sqrt(3) - 2 }
= { sqrt(2) / 4 } * [ sqrt(3) - 1]
= sin(15)

On other words, we verify other pretty important trigonometric
identity which is:
cos(90 - X) = sinX
where X is the angle.

Finally,
cos(255) = -cos(75) = -sin(15) = - { sqrt(2) / 4 } * [ sqrt(3) - 1] = -0.25881904510252...

Answer 3

Going by the zidan's answer, it suspiciously looks like it evaluates to 1/√2. Well done Sah!

Answer 4

cos255 = - cos 75= - sin15
cos 195 = - cos 15

sin30 = 1/2
cos30 = sqrt( 1 - 1/4) = sqrt(3)/2 = 2 cos^2 15 - 1
cos15 = sqrt( (1+sqrt(3)/2) / 2 )
sin15 = sqrt( 1 - (1+sqrt(3)/2)/2 ) = sqrt( (1-sqrt(3)/2)/2 )

cos255-cos195 =
-sqrt( (1-sqrt(3)/2)/2 ) -sqrt( (1+sqrt(3)/2)/2 )

Answer 5

cos(255)-cos(195)
= cos(225+30) - cos(225-30)
= -2sin(225)sin(30)
= -sin(225)
= sin(45)
= √2 / 2

Answer 6

You look in a table or use your calculator..