Solve 6(cos75^circ+1sin75^circ)/2(cos30^circ+1sin30^circ)

$$\frac { 6 ( \cos 75 ^ { \circ } + 1 \sin 75 ^ { \circ } ) } { 2 ( \cos 30 ^ { \circ } + 1 \sin 30 ^ { \circ } ) }$$

Evaluate

$$\frac{45927932677184589\left(\sqrt{3}-1\right)}{12500000000000000}\approx 2.689726417$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{6 {(0.25881904510252074 + 1 \cdot 0.9659258262890683)}}{2 {(\cos(30) + 1 \sin(30))}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{3\left(0.25881904510252074+0.9659258262890683\right)}{\sin(30)+\cos(30)}$$

Add 0.25881904510252074 and 0.9659258262890683 to get 1.22474487139158904.

$$\frac{3\times 1.22474487139158904}{\sin(30)+\cos(30)}$$

Multiply 3 and 1.22474487139158904 to get 3.67423461417476712.

$$\frac{3.67423461417476712}{\sin(30)+\cos(30)}$$

Get the value of \sin(30) from trigonometric values table.

$$\frac{3.67423461417476712}{\frac{1}{2}+\cos(30)}$$

Get the value of \cos(30) from trigonometric values table.

$$\frac{3.67423461417476712}{\frac{1}{2}+\frac{\sqrt{3}}{2}}$$

Since \frac{1}{2} and \frac{\sqrt{3}}{2} have the same denominator, add them by adding their numerators.

$$\frac{3.67423461417476712}{\frac{1+\sqrt{3}}{2}}$$

Divide 3.67423461417476712 by \frac{1+\sqrt{3}}{2} by multiplying 3.67423461417476712 by the reciprocal of \frac{1+\sqrt{3}}{2}.

$$\frac{3.67423461417476712\times 2}{1+\sqrt{3}}$$

Rationalize the denominator of \frac{3.67423461417476712\times 2}{1+\sqrt{3}} by multiplying numerator and denominator by 1-\sqrt{3}.

$$\frac{3.67423461417476712\times 2\left(1-\sqrt{3}\right)}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}$$

Consider \left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3.67423461417476712\times 2\left(1-\sqrt{3}\right)}{1^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 1. Square \sqrt{3}.

$$\frac{3.67423461417476712\times 2\left(1-\sqrt{3}\right)}{1-3}$$

Subtract 3 from 1 to get -2.

$$\frac{3.67423461417476712\times 2\left(1-\sqrt{3}\right)}{-2}$$

Multiply 3.67423461417476712 and 2 to get 7.34846922834953424.

$$\frac{7.34846922834953424\left(1-\sqrt{3}\right)}{-2}$$

Divide 7.34846922834953424\left(1-\sqrt{3}\right) by -2 to get -3.67423461417476712\left(1-\sqrt{3}\right).

$$-3.67423461417476712\left(1-\sqrt{3}\right)$$

Use the distributive property to multiply -3.67423461417476712 by 1-\sqrt{3}.

$$-3.67423461417476712+3.67423461417476712\sqrt{3}$$