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

3 Answers
Oct 25, 2015

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

Feb 10, 2016

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.

#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#