# I need a review or step by step explain how these two problems work. \cos

Suman Cole 2021-10-15 Answered
I need a review or step by step explain how these two problems work.
$$\displaystyle{\cos{{\left({14}\pi-{0}\right)}}}$$
Determine the limit as x approaches the given x-coordinate and the continuity of the function at the that x-coordinate.
$$\displaystyle\lim_{{{x}\to{15}\frac{\pi}{{8}}}}{\cos{{\left({4}{x}-{\cos{{\left({4}{x}\right)}}}\right)}}}$$

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wheezym
To find: The value of $$\displaystyle{\cos{{\left({14}\pi-{0}\right)}}}$$.
Concept used: The value of $$\displaystyle{\cos{{2}}}\pi{n}$$ is 1 where the value of n is 0,1,2,3...
Calculation: Find the value of $$\displaystyle{\cos{{\left({14}\pi-{0}\right)}}}$$ as,
$$\displaystyle{\cos{{\left({14}\pi-{0}\right)}}}={\cos{{\left({14}\pi\right)}}}$$
$$\displaystyle={\cos{{\left({7}{\left({2}\pi\right)}\right)}}}$$
$$\displaystyle={1}$$
Answer: The value of $$\displaystyle{\cos{{\left({14}\pi-{0}\right)}}}$$ is 1.
To find: $$\displaystyle\lim_{{{x}\to{\frac{{{15}\pi}}{{{8}}}}}}{\cos{{\left({4}{x}-{\cos{{\left({4}{x}\right)}}}\right)}}}$$
The trigonometric identity is $$\displaystyle{\cos{{\left({x}\right)}}}={\sin{{\left({\frac{{\pi}}{{{2}}}}-{x}\right)}}}$$ and $$\displaystyle{\sin{{\left(-{x}\right)}}}=-{\sin{{\left({x}\right)}}}$$
Calculation:
Solve by plugging $$\displaystyle{x}={\frac{{{15}\pi}}{{{8}}}}$$ as,
$$\displaystyle\lim_{{{x}\to{\frac{{{15}\pi}}{{{8}}}}}}{\cos{{\left({4}{x}-{\cos{{\left({4}{x}\right)}}}\right)}}}={\cos{{\left({4}\cdot{\frac{{{15}\pi}}{{{8}}}}-{\cos{{\left({4}\cdot{\frac{{{15}\pi}}{{{8}}}}\right)}}}\right)}}}$$
$$\displaystyle={\cos{{\left({\frac{{{15}\pi}}{{{2}}}}-{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}\right)}}}\right)}}}$$
Find the value of $$\displaystyle{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}\right)}}}$$ as,
$$\displaystyle{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}\right)}}}={\sin{{\left({\frac{{\pi}}{{{2}}}}-{\frac{{{15}\pi}}{{{2}}}}\right)}}}$$
$$\displaystyle={\sin{{\left(-{7}\pi\right)}}}$$
$$\displaystyle=-{\sin{{\left(-{7}\pi\right)}}}$$
$$\displaystyle={0}$$
Now apply the value $$\displaystyle{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}\right)}}}={0}$$ in the expression $$\displaystyle{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}-{\cos{{\left({\frac{{{15}}}{{\pi}}}\right)}}}\right)}}}$$
$$\displaystyle{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}-{\cos{{\left({\frac{{{15}\pi}}{{{2}}}}\right)}}}\right)}}}={\cos{{\left({\frac{{{15}\pi}}{{{2}}}}-{0}\right)}}}$$
Answer: The value of $$\displaystyle\lim_{{{x}\to{\frac{{{15}\pi}}{{{8}}}}}}{\cos{{\left({4}{x}-{\cos{{\left({4}{x}\right)}}}\right)}}}$$ is 0.