Answer
424.8k+ views
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 that
Max 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.
![seo images](https://www.vedantu.com/question-sets/08cfd48c-6f28-46aa-bab4-e38328a132918908614336695186257.png)
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.
![seo images](https://www.vedantu.com/question-sets/27c30fb0-8a32-48c0-9eba-4b61b5c1469b6009083508438721933.png)
Which shows that
Max 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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)