Question

If $\sin x=\dfrac{4}{5}$, how do you find $\sin 2x$ ?

Answer 1

We have been given an equation of $\sin x$. We use the multiple angle formula of $\sin 2x=2\sin x\cos x$. It’s given that $\sin x=\dfrac{4}{5}$. From that we find the other ratio of cos at $x$ of that angle to find the solution. We also use the representation of a right-angle triangle with height and hypotenuse ratio being $\dfrac{4}{5}$ and the angle being $x$.

Complete step by step answer:
The given equation of $\sin x$ is $\sin x=\dfrac{4}{5}$. We try to convert $\sin 2x$ using the multiple angle formula of $\sin 2x=2\sin x\cos x$. Therefore, to find the answer of $\sin 2x$, we need to find the answer of $\cos x$.

We use the identity formula of ${{\left( \sin x \right)}^{2}}+{{\left( \cos x \right)}^{2}}=1$.
We can put the value of $\sin x=\dfrac{4}{5}$ in the equation of ${{\left( \sin x \right)}^{2}}+{{\left( \cos x \right)}^{2}}=1$.
Putting the value of $\sin x$, we get ${{\left( \dfrac{4}{5} \right)}^{2}}+{{\left( \cos x \right)}^{2}}=1$.
Now we perform the binary operations.
${{\left( \dfrac{4}{5} \right)}^{2}}+{{\left( \cos x \right)}^{2}}=1 \\\ \Rightarrow {{\left( \cos x \right)}^{2}}=1-\dfrac{16}{25}=\dfrac{9}{25} \\\ \Rightarrow \left( \cos x \right)=\pm \dfrac{3}{5} \\\$

The two different values of $\cos x$ is dependent on the value of the angle $x$.Now we put the values of $\sin x=\dfrac{4}{5}$ and $\left( \cos x \right)=\pm \dfrac{3}{5}$ in the equation of $\sin 2x=2\sin x\cos x$.So,
$\sin 2x=2\left( \dfrac{4}{5} \right)\left( \pm \dfrac{3}{5} \right)\\\ \therefore\sin 2x=\pm \dfrac{24}{25}$

Therefore, the value of $\sin 2x$ is $\pm \dfrac{24}{25}$.

Note: The trigonometric functions of multiple angles are the multiple angle formula. Double and triple angles formulas are there under the multiple angle formulas. Sine, tangent and cosine are the general functions for the multiple angle formula.