Question

How do you find the exact value of $\sin (2x)$ using the double angle formula?

Answer 1

In order to find the solution to this particular numerical the student should know what double angle formula is. How to simplify the double angle formula to its normal form. Basically when the angle of the trigonometric function is doubled, it is called double angle. But this doesn’t mean that the value would also double for example $\sin (60) \ne 2 \times \sin (30)$. In order to evaluate the double angle , there are separate formulae for each trigonometric function. In this particular case we will have to use the double angle formula for $\sin (\theta )$.

Complete step-by-step answer:
In order to find the exact value of $\sin (2x)$, we will first have to apply the double angle formula to this $\sin ()$ function. The double angle formula is as follows:
$\sin (2x) = 2\sin (x)\cos (x)$
Now in order to find the value of $\sin (x)$ or $\cos (x)$ we need to know only $1$ value out of $\sin (x)$ or $\cos (x)$. In order to know the other value we can use the formula
${\sin ^2}x + {\cos ^2}x = 1$.
Consider an example where we are only given the value of $\sin (x)$ and we have to find the value of $\sin (2x)$. We can follow the below given method
$\sin (2x) = 2\sin (x)\cos (x).........(1)$
Since we are given the value of $\sin (x)$, we can bring $\cos (x)$ in terms of $\sin (x)$
$\cos (x) = \pm \sqrt {1 - {{\sin }^2}x} ...........(2)$
Thus the simplification would become
$\sin (2x) = \pm 2\sin (x)\sqrt {1 - {{\sin }^2}x} .........(3)$
Thus with the above equation we can find the exact value of $\sin (2x)$.

Note: The students should remain cautious while calculating the value using the double angle formula. It is advisable that the students should learn the double angle formula or all trigonometric functions. The questions which could be asked would be of the following type: Find the value of $\sin (2x)$ ,given $\cos (x) = \dfrac{{ - 25}}{7}$.The students should also learn the basic triplets which would be used for calculating the values of the hypotenuse and then use them to calculate $\sin (x)\& \cos (x)$.