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a<=3cosx+5sin(x-30)<=b

find a and b

aayush agarwal , 12 Years ago

Grade

2 Answers

jagdish singh singh

$\hspace{-16}$ Let $\bf{ y = 3 \cos x +5\sin (x-30^{0})}$\\\\\\ $\bf{y = 3\cos x +5 \left\{\sin x\cdot \cos (30^0)-\cos x \cdot \sin (30^0)\right\}}$\\\\\\ $\bf{y = 3\cos x+5\left\{\sin x.\frac{\sqrt{3}}{2}-\cos x .\frac{1}{2}\right\}}$\\\\\\ $\bf{y = \frac{1}{2}.\cos x+\frac{5.\sqrt{3}}{2}.\sin x}$\\\\\\ Now Using the formula::\\\\\\ $\bf{-\sqrt{a^2+b^2}\leq a\sin x+b \cos x \leq \sqrt{a^2+b^2}}$\\\\\\ $\bf{-\frac{1}{2}.\sqrt{1+75}\leq \frac{1}{2}.\cos x+\frac{5.\sqrt{3}}{2}.\sin x\leq \frac{1}{2}\cdot \sqrt{1+75}}$\\\\\\ So $\bf{-\frac{\sqrt{76}}{2}\leq y \leq \frac{\sqrt{76}}{2}}$\\\\\\ So $\bf{a = -\frac{\sqrt{76}}{2}}$ and $\bf{b = +\frac{\sqrt{76}}{2}}$$

Last Activity: 12 Years ago

India VK

Given expression = 3 cos x + 5 (sin x . (root3)/2 - cos x. 1/2) = -1/2 cos x + 5 . root(3)/2 sin x

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