Question

27. If sin (A+B) = sin Acos B+cos A sin B
and cos (A - B) = cos Acos B+sin Asin B,
find the values of (i) sin 75° and (ii) cos 15º.​

Answer 1

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Answer 2

Given :-

$Sin 75^{\circ}$

$Cos15^{\circ}$

To find :-

Their values using suitable formula.

Solution:-

• It can be written as.

→$\mathsf{Sin 75^{\circ}}$

→$\mathsf{ Sin ( 30 + 45)}$

→$\mathsf{Sin30 .Cos45 + Cos 30 . Sin 45}$

→$\mathsf{\left(\dfrac{1}{2} \times \dfrac{1}{\sqrt{2}} \right)+ \left(\dfrac{\sqrt{3}}{2}\times \dfrac{1}{\sqrt{2}}\right)}$

→$\mathsf{ \left(\dfrac{1}{2\sqrt{2}}\right)+ \left(\dfrac{\sqrt{3}}{2\sqrt{2}}\right)}$

→$\mathsf{\dfrac{1 + \sqrt{3}}{2\sqrt{2}}}$

hence,

The value is $\mathsf{\dfrac{1 + \sqrt{3}}{2\sqrt{2}}}$

→$\mathsf{Cos 15^{\circ}}$

→$\mathsf{Cos (45 -30) }$

→$\mathsf{Cos 45 .Cos 30 + Sin45 .Sin 30}$

→$\mathsf{\left(\dfrac{1}{2\sqrt{2}}\times \dfrac{\sqrt{3}}{2}\right)+ \left(\dfrac{1}{2\sqrt{2}}\times{\dfrac{1}{2}}\right)}$

→$\mathsf{\left(\dfrac{\sqrt{3}}{2\sqrt{2}}\right)+\left(\dfrac{1}{2\sqrt{2}}\right)}$

→$\mathsf{ \dfrac{\sqrt{3}+1}{2\sqrt2}}$

hence,

The value is $\mathsf{ \dfrac{\sqrt{3}+1}{2\sqrt2}}$

• Formula used :-

→$\mathsf{sin (A+B) = sin A.cos B+cos A. sin B }$

→$\mathsf{cos (A - B) = cos A.cos B+sin A.sin B}$