find exact value of sin 165 degrees?

Answer 1

sin 165 = sin(180 - 15) = sin 15
sin 15 = sin(30/2) = Sqrt [ (1 - cos(30)) / 2 ]
= Sqrt [ (1 - Sqrt(3)/2 )/2 ]
= Sqrt [ (2 - Sqrt(3))/4 ]
= [Sqrt(2 - Sqrt(3))] / 2

Answer 2

Hi,

Using a calculator, sin(165 degrees) = .2588190451.

There are also half-angle formulas you could use to find this.
sin( 1/2A) = + or - sqrt[(1-cosA)/2] Since 165 is half of 330, this would be sin( 165) = + sqrt[(1-cos(330))/2]
The sign is a positive because in the second quadrant where 165 is located, sine is always positive. Cos(330) = sqrt(3)/2 = 1.732/2 = .866, so the formula becomes
sin( 165) = sqrt[(1-.866)/2]
sin( 165) = sqrt[(.134)/2]
sin( 165) = sqrt[.067]
sin(165) = .2588

A third way you could do this is by using the sum formula:
sin(A + B) = sinAcosB + sinBcosA

Since 165 = 45 + 120, this would be:
sin(165)=sin(45 + 120) = sin45cos120 + sin120cos45
sin(165)=sin(45 + 120) = 1/sqrt(2)*(-1/2) + sqrt(3)/2*1/sqrt(2)
sin(165) =1/1.414*(-1/2) + 1.732/2*1/1.414
sin(165)=.7072*-.5+.866*.7072
sin(165)=-.3536+.6124
sin(165)=.2588

That's 3 different ways to get there, depending on what section you might be working in.

I hope that helps!

Answer 3

165 degrees = 165pi/180 = 33pi/36 = 11pi/12

Note that 165 = 135 + 30, so

sin(165) = sin(135 + 30)

Use the sine addition identity
sin(a + b) = sin(a)cos(b) + sin(b)cos(a).

sin(135 + 30) = sin(135)cos(30) + sin(30)cos(135)

135 = 3pi/4 and 30 = pi/6, so

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

Answer 4

First of all sin165 = sin15 (not -sin15 as suggested).
And you find the exact value of sin15 by using the fact that sin30 = 1/2 and using the double angle formula for sine.

Answer 5

sin(165) = sin(135 + 30)

You can use sum and difference formulas to solve this.

sin(u + v) = sin(u)cos(v) + cos(u)sin(v)

sin(135 + 30) = sin(135)cos(30) + cos(135)sin(30)

sin(135 + 30) = [sqrt(2)/2][sqrt(3)/2] + [-sqrt(2)/2][1/2]

sin(135 + 30) = [sqrt(2)/4] * [sqrt(3) - 1]

Answer 6

Same as sin of 15 deg exept it's negative because it's in the 4th quadrant

Answer 7

0.9977972794

Answer 8

0.25881904510252076234889883762405