Question

find value of sin pi/15 * sin 4pi/15 * sin 3pi/10

Answer 1

we have to find the value of sin(π/15) . sin(4π/15) . sin(3π/10)

let's put π = 180°

so, we have to find the value of sin(12°) sin(48°) sin(54°)

= 1/2[2sin(12°) sin(48°)] sin(54°)

using formula, 2sinA sinB = cos(A - B) - cos(A + B)

= 1/2[ cos36° - cos60°]sin(54°)

= 1/2[cos36° sin54° - cos60°. sin54°]

= 1/2[cos36°. sin(90° - 36°) - 1/2 sin(90° - 36°)]

= 1/2 [cos²36° - 1/2cos36°]

= 1/4 [2cos²36° - cos36°]

we know, cos36° = (√5 + 1)/4

= 1/4[2{(√5 + 1)/4}² - (√5 + 1)/4]

= 1/4[(6 + 2√5)/8 - (√5 + 1)/4]

=1/4[(6 + 2√5 - 2√5 - 2)/8]

= 1/4[ 1/2]

= 1/8

therefore value of sin(π/15). sin(4π/15) . sin(3π/10) is 1/8