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Question: How do you find the value of \(\sin {20^ \circ }\) using the double angle identity?...

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

Explanation

Solution

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∘\sin {20^ \circ }then
Let – y=sin⁡20∘y = \sin {20^ \circ }
We know that in the trigonometry double angle identity formula are very few for sin⁡θ\sin \theta such as –
(1.)sin⁡2θ=2sin⁡θcos⁡θ (2.)sin⁡3θ=3sin⁡θ−4sin⁡3θ  (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∘\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⁡2θ=2sin⁡θcos⁡θ\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∘=0.6427\sin {40^ \circ } = 0.6427 and the value of cos⁡20∘=0.9396\cos {20^ \circ } = 0.9396
Now we will solve our given question we will use our first formula which is sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta
Now we have θ=20∘\theta = {20^ \circ }so we will put this value in our formula then we will get
⇒sin⁡2×20∘=2sin⁡20∘cos⁡20∘\Rightarrow \sin 2 \times {20^ \circ } = 2\sin {20^ \circ }\cos {20^ \circ }
Now we will calculate all value and we will get
⇒sin⁡40∘=2sin⁡20∘cos⁡20∘\Rightarrow \sin {40^ \circ } = 2\sin {20^ \circ }\cos {20^ \circ }
Now put the value of sin⁡40∘=0.6427\sin {40^ \circ } = 0.6427 and the value of cos⁡20∘=0.9396\cos {20^ \circ } = 0.9396 then we will get
⇒0.6427=2sin⁡20∘×0.9396\Rightarrow 0.6427 = 2\sin {20^ \circ } \times 0.9396
Now we will solve it completely then we will get
⇒0.6427=sin⁡20∘×1.8792\Rightarrow 0.6427 = \sin {20^ \circ } \times 1.8792
Now we will divide both sides by 1.87921.8792 so we will get the value of sin⁡20∘\sin {20^ \circ }
⇒sin⁡20∘=0.64271.8792\Rightarrow \sin {20^ \circ } = \dfrac{{0.6427}}{{1.8792}}
Now after completely solving this we will get our sin⁡20∘\sin {20^ \circ } value which is
⇒sin⁡20∘=0.34200\Rightarrow \sin {20^ \circ } = 0.34200

Therefore the value of the sin⁡20∘\sin {20^ \circ } is the 0.342000.34200 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.