Question

How do you find the exact value of sin 105 degrees?

Answer 1

Find exact value of sin (105)

Ans: `(sqrt(2 + sqrt3)/2)`

Explanation:

sin (105) = sin (15 + 90) = cos 15.
First find (cos 15). Call cos 15 = cos x
Apply the trig identity:`cos 2x = 2cos^2 x - 1.`
cos 2x = cos (30) `= sqrt3/2 = 2cos^2 x - 1`
2cos^2 x = 1 + sqrt3/2 = (2 + sqrt3)/2
cos^2 x = (2 + sqrt3)/4
cos x = cos 15 = (sqrt(2 + sqrt3)/2. (since cos 15 is positive)

`sin (105) = cos (15) = sqrt(2 + sqrt3)/2.`
Check by calculator.
sin (105) = cos 15 = 0.97
`sqrt(2 + sqrt3)/2 = 1.93/2 = 0.97.` OK

Answer 2

Use `sin 105 = sin (60 + 45) = sin 60 cos 45 + cos 60 sin 45`
`=(sqrt3/2)(1/sqrt2)+(1/2)(1/sqrt2)=sqrt2/4((sqrt3+1)=0.9656` nearly.

Explanation:

sin 45, cos 45 and sin 60 are irrational. So, the answer is a surd.

Answer 3

`sin105^@=(sqrt6+sqrt2)/4`

Explanation:

We know `sin(A+B)=sinAcosB+cosAsinB`

Hence `sin105^@`

= `sin(60^@+45^@)`

= `sin60^@cos45^@+cos60^@sin45^@`

= `sqrt3/2xx1/sqrt2+1/2xx1/sqrt2`

= `(sqrt3+1)/(2sqrt2)xxsqrt2/sqrt2`

= `(sqrt6+sqrt2)/4`