Find the value of 1/sin10° - 4 sin70°.

Question

Find the value of \(\frac{1}{sin10°} - 4\, sin70°.\)

Answer 1

sin(3 × 10°) = sin(30°) = \(\frac{1}{2}\)

⇒ 3 sin10° - 4 sin³10° = \(\frac{1}{2}\)

⇒ 8 sin³10° - 6 sin10° + 1 = 0

After solving this polynomial in sin10°, we get

⇒ sin10° = 0.17365

sin70° = sin(10° + 60°)

= sin10° cos60° + cos10° sin60°

= 0.17365 × \(\frac{1}{2}\) + 0.9848 × \(\frac{\sqrt3}{2}\)

= 0.086825 + 0.852868

= 0.93693

\(\frac{1}{sin 10°}\) - 4 sin70° = \(\frac{1}{0.17365} - 4(0.93693)\)

= 5.75871 - 3.74772

= 2.01099 (2.011 Approx)