Solve sin75^circ/sqrt{2+sqrt{6}}

$$\frac { \sin 75 ^ { \circ } } { \sqrt { 2 } + \sqrt { 6 } }$$

Evaluate

$$\frac{9659258262890683\left(\sqrt{6}-\sqrt{2}\right)}{40000000000000000}\approx 0.25$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{0.9659258262890683}{\sqrt{2} + \sqrt{6}}$$

Rationalize the denominator of \frac{0.9659258262890683}{\sqrt{2}+\sqrt{6}} by multiplying numerator and denominator by \sqrt{2}-\sqrt{6}.

$$\frac{0.9659258262890683\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right)}$$

Consider \left(\sqrt{2}+\sqrt{6}\right)\left(\sqrt{2}-\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{0.9659258262890683\left(\sqrt{2}-\sqrt{6}\right)}{\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Square \sqrt{2}. Square \sqrt{6}.

$$\frac{0.9659258262890683\left(\sqrt{2}-\sqrt{6}\right)}{2-6}$$

Subtract 6 from 2 to get -4.

$$\frac{0.9659258262890683\left(\sqrt{2}-\sqrt{6}\right)}{-4}$$

Divide 0.9659258262890683\left(\sqrt{2}-\sqrt{6}\right) by -4 to get -0.241481456572267075\left(\sqrt{2}-\sqrt{6}\right).

$$-0.241481456572267075\left(\sqrt{2}-\sqrt{6}\right)$$

Use the distributive property to multiply -0.241481456572267075 by \sqrt{2}-\sqrt{6}.

$$-0.241481456572267075\sqrt{2}+0.241481456572267075\sqrt{6}$$