Question

What is the value of $\sin \infty$ and $\cos \infty$ ?

Answer 1

We have to find the value of $\sin \infty$ and $\cos \infty$. We will solve this question stating the intervals in which the value of sine and cosine function lies . We also give the explanation for the value of the infinity function of cosine and sine function . We can also state the periodicity of the cosine and sine function . We should also have the knowledge of ranges of trigonometric functions .

Complete step-by-step answer :
We know that , the maximum value of a $\sin$ function is always $1$ for every $\left( {{\text{ }}4n{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }} \times {\text{ }}\dfrac{\pi }{2}$ , where $n$ is a whole number
whereas the minimum value of the $\sin$ function is always $- 1$ for every $\left( {{\text{ }}4n{\text{ }} - 1{\text{ }}} \right){\text{ }} \times {\text{ }}\dfrac{\pi }{2}$ , where $n$ is a whole number .
So ,
$- 1{\text{ }} \leqslant {\text{ }} \sin{\text{ }}x{\text{ }} \leqslant {\text{ }}1$
Hence , the value of $\sin{\text{ }}x$ lies in between the interval $\left[ {{\text{ }} - 1{\text{ }},{\text{ }}1{\text{ }}} \right]$ where $x$ is between the interval $\left[ {{\text{ }}\dfrac{{ - \pi }}{2}{\text{ }},{\text{ }}\dfrac{\pi }{2}} \right]$
For any of the value of $x$ it will lie in between the interval $\left[ {{\text{ }} - 1{\text{ }},{\text{ }}1{\text{ }}} \right]$. The exact value of $\sin \infty$ cannot be calculated but it will only lie in between the interval $\left[ {{\text{ }} - {\text{ }}1{\text{ }},{\text{ }}1{\text{ }}} \right]{\text{ }}.$
Similarly for $\cos \infty$
We know that , the maximum value of a $\cos$ function is always $1$ for every $\left( {{\text{ }}2n{\text{ }}} \right){\text{ }} \times {\text{ }}\pi$ , where $n$ is a whole number
whereas the minimum value of the $sin$function is always $- 1$ for every $\left( {{\text{ }}2n{\text{ }} + {\text{ }}1{\text{ }}} \right){\text{ }} \times {\text{ }}\pi$ , where $n$ is a whole number .
So ,
$- 1{\text{ }} \leqslant {\text{ }}cos{\text{ }}x{\text{ }} \leqslant {\text{ }}1$
Hence , the value of $cos{\text{ }}x$ lies in between the interval $\left[ {{\text{ }} - 1{\text{ }},{\text{ }}1{\text{ }}} \right]$ where $x$ is between the interval $\left[ {{\text{ }}0{\text{ }},{\text{ }}\pi {\text{ }}} \right]$ .
For any of the value of $x$ it will lie in between the interval $\left[ {{\text{ }} - 1{\text{ }},{\text{ }}1{\text{ }}} \right]$. The exact value of $\cos \infty$ cannot be calculated but it will only lie in between the interval $\left[ {{\text{ }} - {\text{ }}1{\text{ }},{\text{ }}1{\text{ }}} \right]$ .
Hence , the value of $\sin \infty$ and $\cos \infty$ lies in the interval $\left[ {{\text{ }} - 1{\text{ }},{\text{ }}1{\text{ }}} \right]$.

Note : The periodic value of $\cos$ and $\sin$ function is $2\pi$ I.e. the value of $\cos$ and $\sin$ function repeats after an interval of $2\pi$ .
The expansions of the trigonometric terms :
$\cos x = 1 - (\dfrac{{{x^2}}}{{2!}}) + (\dfrac{{{x^4}}}{{4!}}) - (\dfrac{{{x^6}}}{{6!}}) + .......................$
$\sin x = x - (\dfrac{{{x^3}}}{{3!}}) + (\dfrac{{{x^5}}}{{5!}}) - (\dfrac{{{x^7}}}{{7!}}) + .................$