Question

$\mathrm{\sin \left(75^{\circ \:}\right)\cos \left(15^{\circ \:}\right)}$

solve the question

Answer 1

Answer:

Sin75°=sin(45°+30°)

=Sin45°cos30°+cos45°sin30°

= >[Sin(A+B)=sinA cosB+cosA sinB]

=1/√2.√3/2+1/√2.1/2 [sin45°=cos45°=1/√2]

=√3+1/2√2 Ans.

Cos15°=cos(45°–30°)

=> [Cos(A-B)=cosAcosB+sinAsinB]

Cos(45°-30°)=cos45°cos30°+sin45°sin30°

=1/√2. √3/2+1/√2. 1/2

[sin30°=1/2 & cos30°=√3/2]

=√3/2√2+1/2√2

=√3+1/2√2 Ans.

Answer 2

$\sin \left(75^{\circ \:}\right)\cos \left(15^{\circ \:}\right)=\frac{2+\sqrt{3}}{4}\quad \begin{pmatrix}\mathrm{Decimal:}&0.93301\dots \end{pmatrix}$

$\mathrm{Use\:the\:following\:identity}:\quad \cos \left(t\right)\sin \left(s\right)=\frac{\sin \left(s+t\right)+\sin \left(s-t\right)}{2}$

$\sin \left(75^{\circ \:}\right)\cos \left(15^{\circ \:}\right)=\frac{\sin \left(75^{\circ \:}+15^{\circ \:}\right)+\sin \left(75^{\circ \:}-15^{\circ \:}\right)}{2}$

$=\frac{\sin \left(75^{\circ \:}+15^{\circ \:}\right)+\sin \left(75^{\circ \:}-15^{\circ \:}\right)}{2}$

$\mathrm{Simplify}$

$=\frac{\sin \left(90^{\circ \:}\right)+\sin \left(60^{\circ \:}\right)}{2}$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(90^{\circ \:}\right)=1$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(60^{\circ \:}\right)=\frac{\sqrt{3}}{2}$

$=\frac{1+\frac{\sqrt{3}}{2}}{2}$

$\frac{1+\frac{\sqrt{3}}{2}}{2}=\frac{2+\sqrt{3}}{4}$