Question

Prove
sin 10 + sin 20 + sin 30 + sin 40 + sin 50 = sin 70 sin 80​

Answer 1

Step-by-step explanation:

L.H.S. = 2sin15cos5+2sin45cos5 [ using sin C+sin D= 2sin C+D/2 cos C-D/2 for sin10+sin20 & sin40+sin50]

= 2cos5 (sin15+sin45)

= 2cos5 (2sin30cos15) [ using sin C+sin D= 2sin C+D/2 cos C-D/2 ]

= 2cos5 (2 x 1/2 x cos15)

= 2cos5 cos15

R.H.S. = sin70+sin80

= 2sin75cos5 [ using sin C+sin D= 2sin C+D/2 cos C-D/2 ]

sin75 = sin(90-15) = cos 15

L.H.S = 2cos5 cos15

R.H.S. = 2cos15 cos5 [ since, sin75 = cos15 ]

Answer 2

Answer:

$\sin( 10) + \sin(20) + \sin(30) + \sin(40) + \sin(50) \\ = 0.17365 + 0.34202 + 0.5 + 0.64279 + 0.76604 \\ = 2.4245$

$\sin(70) \sin(80) \\ = 0.93969 \times 0.98481 \\ = 0.92542$