Question

What is the value of $\cos \left( 2\pi +x \right)$ if $\sin x=0.3$?

Answer 1

We first use the formula of associative angle formula to find the simplified form of $\cos \left( 2\pi +x \right)$ and then use the identity relation where ${{\sin }^{2}}x+{{\cos }^{2}}x=1$. As the value of $\left| x \right|\le \dfrac{\pi }{2}$, we take the positive value of the solution only.

Complete step by step answer:
For $\cos \theta$ we assume $\theta =k\times \dfrac{\pi }{2}+\alpha$, $k\in \mathbb{Z}$. Here we took the addition of $\alpha$. We also need to remember that $\left| \alpha \right|\le \dfrac{\pi }{2}$.
Now we take the value of k. If it’s even then keep the ratio as cos and if it’s odd then the ratio changes to sin ratio from cos.
Then we find the position of the given angle as a quadrant value measured in counter clockwise movement from the origin and the positive side of the X-axis.
If the angel falls in the first or fourth quadrant then the sign remains positive but if it falls in the second or third quadrant then the sign becomes negative.
The final form becomes $\cos \left( 2\pi +x \right)=\cos \left( 4\times \dfrac{\pi }{2}+x \right)=\cos x$.
We have the identity relation where ${{\sin }^{2}}x+{{\cos }^{2}}x=1$. We get $\cos x=\pm \sqrt{1-{{\sin }^{2}}x}$.
Placing the value, we get $\cos x=\pm \sqrt{1-{{\left( 0.3 \right)}^{2}}}=\pm \sqrt{0.91}=\pm 0.954$.
As the value of $\left| x \right|\le \dfrac{\pi }{2}$, we only can only take the value $\cos x=0.954$.

Note: We need to remember that the easiest way to avoid the change of ratio thing is to form the multiple of $\pi$ instead of $\dfrac{\pi }{2}$. It makes the multiplied number always even. In that case we don’t have to change the ratio. If $x=k\times \pi +\alpha =2k\times \dfrac{\pi }{2}+\alpha$. Value of $2k$ is always even.