Respuesta :

sin 165 = sin(120 + 45) =sin 120 cos 45 + cos 120 sin 45 =
[sqrt(3)/2 * sqrt(2)/2] + (-1/2)(sqrt(2)/2) =
[sqrt(6) - sqrt(2)] / 4



By the help of trignometry properties the exact value of  [tex]\rm sin(165^\circ)=0.2588[/tex].

Solution :

  • In right triangle, [tex]\rm sin\theta[/tex] is equal to the ratio of the length of the opposite side to the length of the hypotenuse.
  • The most common trigonometric function is sine function.
  • It is written as 'sin' without the 'e' in the formula.

Now from trignometry properties the formula of sin(A + B) is given by:

[tex]\rm sin(A+B)=sinAcosB+cosAsinB[/tex] ---- (1)

Now using equation (1) we can find the value of [tex]\rm sin(165^\circ)[/tex] .

[tex]\rm sin(120^\circ+45^\circ)=sin(120^\circ)cos(45^\circ)+cos(120^\circ)sin(45^\circ)[/tex]

[tex]\rm sin(165^\circ) = sin(90^\circ+30^\circ)cos(45^\circ)+cos(90^\circ+30^\circ)sin(45^\circ)[/tex]

[tex]\rm sin(165^\circ) = cos(30^\circ)cos(45^\circ)-sin(30^\circ)sin(45^\circ)[/tex]

[tex]\rm sin(165^\circ) = \dfrac{\sqrt{3} }{2}\times \dfrac{1}{\sqrt{2} }-\dfrac{1}{2}\times\dfrac{1}{\sqrt{2} }[/tex]

[tex]\rm sin(165^\circ) = \dfrac{\sqrt{3}-1 }{2\sqrt{2} }=0.2588[/tex]

For more information, refer the link given below

https://brainly.com/question/19731462