What-are-the-possible-value-of-cos-sin-if-sin-cos-1-2-

Question Number 135855 by liberty last updated on 16/Mar/21

W h a t a r e t h e p o s s i b l e v a l u e o f

\cos α \times \sin β i f \sin α \times \cos β = - \frac{1}{2}

Answered by EDWIN88 last updated on 16/Mar/21

We have - 1 ⩽ \sin (α + β) ⩽ 1 and - 1 ⩽ \sin (α - β) ⩽ 1

now from - 1 ⩽ \sin (α + β) ⩽ 1 we get

- 1 ⩽ \sin α \cos β + \cos α \sin β ⩽ 1

- 1 ⩽ - \frac{1}{2} + \cos α \sin β ⩽ 1 \Rightarrow - \frac{1}{2} ⩽ \cos α \sin β ⩽ \frac{3}{2} \dots (i)

from - 1 ⩽ \sin (α - β) ⩽ 1 we get

- 1 ⩽ \sin α \cos β - \cos α \sin β ⩽ 1

- 1 ⩽ - \frac{1}{2} - \cos α \sin β ⩽ 1

- \frac{3}{2} ⩽ \cos α \sin β ⩽ \frac{1}{2} \dots (ii)

combine equation (i) and (ii)

we get \Rightarrow - \frac{1}{2} ⩽ \cos α \sin β ⩽ \frac{1}{2}

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