Find the minimum value of $$\cos \theta + \cos 2\theta $$for all real value of $$\theta $$

Question

Find the minimum value of $$\cos \theta  + \cos 2\theta $$for all real value of $$\theta $$

Vedantu Content Team · Accepted Answer

Hint:  The cosine function is a periodic function which is very important in trigonometry. The simplest way to understand the cosine function is to use the unit circle. For a given angle measure (-) draw a unit circle on the coordinate plane and draw the angle centered at the origin with one side as the positive $$x - axiz$$ the $$x - $$ coordinate of the point where the other side of the angle intersects the circle is $$\cos (\theta )$$, and the y coordinate is $$\sin (\theta )$$The amplitude of a wave is basically a measure of its height. Because that height is constantly changing amplitude can be different form moment to moment. If the wave has a regular up and down shape, like a cosine, sine wave. The amplitude is defined as the fact act distance the wave gets from its center. Which shows thatMax value of $$\cos \,\,\,\,\,\, + 1$$Min value of $$\cos \,\,\,\,\,\, - 1$$Complete  step by step answer: Given function is$$ = \cos 2\theta  + \cos \theta $$$$ = 2{\cos ^2}\theta  - 1 + \cos \theta $$	$$\left[ {\cos 2\theta  - 2{{\cos }^2}\theta  - 1} \right]$$$$ = {\cos ^2}\theta  - \dfrac{1}{2}\cos \theta  - \dfrac{1}{2}$$$$\left[ {dividing\;by\,2} \right]$$$$ = ({\cos ^2}\theta  + \dfrac{1}{2}\cos \theta  + \dfrac{1}{{16}}) - \dfrac{1}{{16}} - \dfrac{1}{2}$$		$$\left[ {For\,factorize\,add\& substract\,\dfrac{1}{{16}}} \right]$$$$ = {(\cos \theta  + \dfrac{1}{2})^2} - (\dfrac{{1 + 8}}{{16}})$$			$$\left[ {Making\,the\,whole\,square} \right]$$$$ = {(\cos \theta  + \dfrac{1}{4})^2} - (\dfrac{9}{{16}})$$Now for minimum value if $$\cos \theta  = 0$$i.e.$$ = \dfrac{{ - 9}}{{16}}$$Required minimum value of  $$\cos \theta  + \cos 2\theta $$ for all real values of $$\theta $$ is $$ = \dfrac{{ - 9}}{{16}}$$$$(\cos  + \dfrac{1}{2})\dfrac{1}{2}O$$$$\cos \theta  = \dfrac{1}{2}$$Note: If we take negative value of $$\cos \theta $$ because it’s a square so at last it become positive.Minimum value of $$\cos \theta $$ is -1 when $$\theta  = {180^0}$$ Maximum value $$\cos \theta $$ 1 when$$\theta $$ is $$O,360$$ Min-Max table.

Find the minimum value of \[\cos \theta + \cos 2\theta \]for all real value of \[\theta \]

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Find the minimum value of \[\cos \theta + \cos 2\theta \]for all real value of \[\theta \]

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