Find two solutions to solve sin15 + sin75 .
3 Answers
hala718 | High School Teacher | (Level 1) Educator Emeritus
sin15+sin75
Method (1)
We know that sinx +siny = 2sin (x+y)/2 * cos(x-y)/2
sin15+sin75 = 2sin(45) * cos (-30)
= 2(sqrt(2)/2) (sqrt(3)/2)
= sqrt(6)/2
Method (2)
sin15+sin75
we will rewrite sin75 as sin(90-15)= cos15
sin15+cos15= = sin(45-30) + cos(45-30)
= sin45cos30-sin30cos45+ sin45sin30+cos45cos30
= sqrt2/2 * sqrt(3)/2 - 1/2 *sqrt(2)/2 + sqrt(2)/2 *1/2 +sqrt(2)/2 *sqrt(3)/2
= sqrt(6)/4 -sqrt(2)/4 + sqrt(2)/4 + sqrt(6)/4
= 2sqrt(6)/4= sqrt(6)/2
neela | High School Teacher | (Level 3) Valedictorian
To find 2 solutions to sin15+sin75
Solution:
sin15+sin75 = sin75+sin 15 = 2sin (75+15)/2 * cos (75-15)/2 = 2sin 45*cos30
=2 (1/sqrt2)*(sqrt3/2) = 2(sqrt2*sqrt3)/4 = (sqrt6)/2.
2nd metod to solve:
sin15 + sin75 = sin15+ sin (90 -15) = sin15+cos15 = {(1/sqrt2)sin15 + (1/sqrt2) cos 15}sqrt2
(sin45*sin15 + cos45cos15)sqrt2
= cos (45-15) sqrt2
= (cos 30 )sqrt2
= {(sqrt3)/2} sqrt2
= (sqrt6)/2
One method would be to consider the fact that being an addition of two alike functions, we'll transform the addition into a product, in this way:
sin a + sin b = 2sin [(a+b)/2]cos [(a-b)/2]
sin 15 + sin 75 = 2sin[(15+75)/2] cos [(15-75)/2]
sin 15 + sin 75 = 2sin45cos30
sin 15 + sin 75 = 2*[(sqrt2)/2]*[(sqrt3)/2]
sin 15 + sin 75 = sqrt(2*3)/2=sqrt(6)/2
Another manner of solving would be to write the angles:
15 = 45 - 30
75 = 45 + 30
sin (45 - 30) = sin45*cos30 - sin30*cos45
=(sqrt2/2)(sqrt3/2) - sqrt2/4
= (sqrt6 - sqrt2)/4
sin (45 + 30) = sin45*cos30 + sin30*cos45
= (sqrt6 + sqrt2)/4
So,
sin 15 + sin 75 = (sqrt6 - sqrt2+sqrt6 + sqrt2)/4
sin 15 + sin 75 = 2sqrt6/4
sin 15 + sin 75 = sqrt6/2