Question

Evaluate 6 sin 20° - 8 sin³ 20°

Answer 1

Greetings! Thanks for the question :)

It's a question from trigonometry. We are going to evaluate the given expression by using identities.

We know that,
sin3x = 3sinx - 4sin³x.

Now,
6 sin 20° - 8 sin³ 20°

= 2 ( 3 sin 20° - 4 sin³ 20° )

= 2 ( sin (3 * 20)° )

= 2 ( sin60° )

= 2 ( √3 / 2 )

= √3 .

Therefore, 6 sin 20° - 8 sin³ 20° = √3 .

Answer 2

Heya,

The question would be easy if we will use the approach of multiple and submultiples.

we know,

$3 \sin( \alpha ) - 4 { \sin( \alpha ) }^{3} = \sin(3 \alpha )$ .........(i)

Now,

expression 6 sin20° - 8sin³20° can be written as 2(3sin20° - 4sin³20°).

=> 2(sin3 × 20°) ..........using (i)

=> 2 × sin60°

$2 \times \frac{ \sqrt{3} }{2} = \sqrt{3}$

Regards

KSHITIJ