Question

14. Find the value of Cos30°. Cos 45º - Sin 30°. Sin45°​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\large{\boxed{\sf{\cos \left(30^{\circ \:}\right)\cos \left(45^{\circ \:}\right)-\sin \left(30^{\circ \:}\right)\sin \left(45^{\circ \:}\right)}}$

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\sf{\cos \left(30^{\circ \:}\right)\cos \left(45^{\circ \:}\right)-\sin \left(30^{\circ \:}\right)\sin \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{4}\quad \begin{pmatrix}\mathrm{Decimal:}&0.25881\dots \end{pmatrix}}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(30^{\circ \:}\right)=\dfrac{\sqrt{3}}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(30^{\circ \:}\right)=\dfrac{1}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(45^{\circ \:}\right)=\dfrac{\sqrt{2}}{2}$

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$\sf{=\dfrac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}-\dfrac{1}{2}\cdot \dfrac{\sqrt{2}}{2}}$

$\huge\boxed{\sf{=\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)}{4}}}$