# JEE Main & Advanced Mathematics Trigonometrical Ratios and Identities Maximum and Minimum Value of a $\mathbf{cos}\,\,\mathbf{\theta }\,\,\mathbf{+}\,\mathbf{b}\,\,\mathbf{sin}\,\,\mathbf{\theta }$

## Maximum and Minimum Value of a $\mathbf{cos}\,\,\mathbf{\theta }\,\,\mathbf{+}\,\mathbf{b}\,\,\mathbf{sin}\,\,\mathbf{\theta }$

Category : JEE Main & Advanced

Let  $a=r\cos \alpha$    .....(i)   and   $b=r\sin \alpha$         .....(ii)

Squaring and adding (i) and (ii), then ${{a}^{2}}+{{b}^{2}}={{r}^{2}}$ or, $r=\sqrt{{{a}^{2}}+{{b}^{2}}}$

$\therefore$          $a\sin \theta +b\cos \theta =r(\sin \theta \cos \alpha +\cos \theta \sin \alpha )=r\sin (\theta +\alpha )$

But $-1\le \sin \theta <1$ So, $-1\le \sin (\theta +\alpha )\le 1$;

Then $-r\le r\sin (\theta +\alpha )\le r$

Hence, $\sqrt{2}-1$

Then the greatest and least values of $a\sin \theta +b\cos \theta$ are respectively $\sqrt{{{a}^{2}}+{{b}^{2}}}$ and $-\sqrt{{{a}^{2}}+{{b}^{2}}}$.

Therefore, ${{\sin }^{2}}x+c\text{ose}{{\text{c}}^{\text{2}}}x\ge 2,$ for every real $x$.

${{\cos }^{2}}x+{{\sec }^{2}}x\ge 2,$ for every real $x$.

${{\tan }^{2}}x+{{\cot }^{2}}x\ge 2$, for every real $x$.

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