Question

Find the exact value of sin 105 degrees​

Answer 1

`(√(6)+√(2))/(4)`

Explanation:

I'm going to write 105 as a sum of numbers on the unit circle.

If I do that, I must use the sum identity for sine.

`\sin(105)=\sin(60+45)`

`\sin(60)\cos(45)+\sin(45)\cos(60)`

Plug in the values for sin(60),cos(45), sin(45),cos(60)

`(√(3))/(2)(√(2))/(2)+(√(2))/(2)(1)/(2)`

`(√(3)√(2)+√(2))/(4)`

`(√(6)+√(2))/(4)`

Answer 2

Sin 105 degrees is equivalent to (√6 - √2) / 4.

The exact value of sin 105 degrees can be determined using trigonometric identities. Knowing that sin (90 + θ) = cos θ, we can rewrite sin 105 degrees as sin (90 + 15) degrees.

Applying the identity, sin (90 + 15) degrees equals cos 15 degrees.

Utilizing the trigonometric values of common angles, cos 15 degrees can be expressed as (√6 - √2) / 4.

This value is derived from trigonometric relationships, providing an exact representation of sin 105 degrees without resorting to decimal approximations.