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Question: How do you find the exact value of \[\sin 105{}^\circ \]?...

How do you find the exact value of sin⁡105∘\sin 105{}^\circ ?

Explanation

Solution

In this question as it is given that the angle is 105∘105{}^\circ and we do not know the value of sin⁡105∘\sin 105{}^\circ then we can split the angle in two parts whose value are known to us and that are 45∘45{}^\circ and 60∘60{}^\circ . After this we can use the trigonometric formula forsin⁡\sin function.

Formula used:
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A+B)=\sin A\cos B+\cos A\sin B

Complete step-by-step answer:
Here, comparing the above formula with the given question
⇒A+B=105∘,A=60∘,B=45∘\Rightarrow A+B=105{}^\circ { , A=60}{}^\circ { , B=45}{}^\circ
Now substituting these values in the formula
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A+B)=\sin A\cos B+\cos A\sin B
⇒sin⁡(60∘+45∘)=sin⁡60∘cos⁡45∘+cos⁡60∘sin⁡45∘\Rightarrow \sin (60{}^\circ +45{}^\circ )=\sin 60{}^\circ \cos 45{}^\circ +\cos 60{}^\circ \sin 45{}^\circ
And we know that
⇒sin⁡60∘=32,cos60∘=12,sin45∘=cos45∘=12\Rightarrow \sin 60{}^\circ =\dfrac{\sqrt{3}}{2}{ , cos60}{}^\circ {=}\dfrac{1}{2}{ , sin45}{}^\circ {=cos45}{}^\circ {=}\dfrac{1}{\sqrt{2}}
Putting these values
⇒32.12+12.12\Rightarrow \dfrac{\sqrt{3}}{2}.\dfrac{1}{\sqrt{2}}+\dfrac{1}{2}.\dfrac{1}{\sqrt{2}}
⇒3+122\Rightarrow \dfrac{\sqrt{3}+1}{2\sqrt{2}}
Now rationalising the denominator
⇒2+64\Rightarrow \dfrac{\sqrt{2}+\sqrt{6}}{4}
Hence, sin⁡105∘=2+64\sin 105{}^\circ =\dfrac{\sqrt{2}+\sqrt{6}}{4}

Note: When we have to find the value of the trigonometric functions and the values of that angles are not known to us then we split the angles in such a way that the separate value of sin angles are known to us for example 30∘,45∘,60∘,90∘30{}^\circ ,45{}^\circ ,60{}^\circ ,90{}^\circ .