Question

the value of cos10.cos50.sin160​

Answer 1

We have to find the value of cos10° cos50° sin160°

Solution : we know, cos(90° - x) = sinx

sin(180° - x) = sinx

so, cos10° = cos(90° - 80°) = sin80°

cos50° = cos(90° - 40°) = sin40°

sin160° = sin(180° - 20°) = sin20°

so, cos10° cos50° cos160° = sin80° sin40° sin20°

= 1/2sin80° (2sin40° sin20°)

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

= 1/2 sin80° {cos(40° - 20°) - cos(40° + 20°)}

= 1/2 sin80° {cos20° - cos60°}

= 1/2 sin80° cos20° - 1/2 sin80° cos60°

= 1/4 (2sin80° cos20°) - 1/2 × sin80° × 1/2

using formula, 2sinA cosB = sin(A + B) + sin(A - B)

= 1/4 {sin(80° + 20°) + sin(80° - 20°)} - 1/4 sin80°

= 1/4 (sin100° + sin60°) - 1/4 sin80°

= 1/4 {sin(180° - 80°) + √3/2} - 1/4 sin80°

= 1/4 sin80° + √3/8 - 1/4 sin80°

= √3/8

Therefore the value of cos10° cos50° sin160° is √3/8