Question

How do you find the value of $\sin {20^ \circ }$ using the double angle identity?

Answer 1

Hint: If in the question, it is not mentioned which formula we need to use we can use any formula but here it is mentioned that we have to use the double angle identity formula, so we have to use it and by using those formulas we will solve our question.

Complete step by step Solution:
Given that –
We have to find the value of $\sin {20^ \circ }$then
Let – $y = \sin {20^ \circ }$
We know that in the trigonometry double angle identity formula are very few for $\sin \theta $ such as –
$
  (1.)\sin 2\theta = 2\sin \theta \cos \theta \\
  (2.)\sin 3\theta = 3\sin \theta - 4{\sin ^3}\theta \\
 $
We can use any one formula for finding the value of $\sin {20^ \circ }$ because both formulae are easy to use and we use them for finding the quick solution so we will use first formula which is the $\sin 2\theta = 2\sin \theta \cos \theta $ but we have to remember some angle values so some value which we need to remember are $\sin {40^ \circ } = 0.6427$ and the value of $\cos {20^ \circ } = 0.9396$
Now we will solve our given question we will use our first formula which is $\sin 2\theta = 2\sin \theta \cos \theta $
Now we have $\theta = {20^ \circ }$so we will put this value in our formula then we will get
$ \Rightarrow \sin 2 \times {20^ \circ } = 2\sin {20^ \circ }\cos {20^ \circ }$
Now we will calculate all value and we will get
$ \Rightarrow \sin {40^ \circ } = 2\sin {20^ \circ }\cos {20^ \circ }$
Now put the value of $\sin {40^ \circ } = 0.6427$ and the value of $\cos {20^ \circ } = 0.9396$ then we will get
$ \Rightarrow 0.6427 = 2\sin {20^ \circ } \times 0.9396$
Now we will solve it completely then we will get
$ \Rightarrow 0.6427 = \sin {20^ \circ } \times 1.8792$
Now we will divide both sides by $1.8792$ so we will get the value of $\sin {20^ \circ }$
$ \Rightarrow \sin {20^ \circ } = \dfrac{{0.6427}}{{1.8792}}$
Now after completely solving this we will get our $\sin {20^ \circ }$ value which is
$ \Rightarrow \sin {20^ \circ } = 0.34200$

Therefore the value of the $\sin {20^ \circ }$ is the $0.34200$ which is the required answer to our question.

Note: Always remember that in the trigonometry chapter in all classes we have to remember some points and special values of angles so we can solve any question very easily and quickly compared to others.