Solve 1/2+sqrt{3}+|sin30^circ-1|

$$\frac { 1 } { 2 + \sqrt { 3 } } + | \sin 30 ^ { \circ } - 1 |$$

Evaluate

$$\frac{5}{2}-\sqrt{3}\approx 0.767949192$$

Solution Steps

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

$$\frac{2-\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}+|\sin(30)-1|$$

Consider \left(2+\sqrt{3}\right)\left(2-\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}.

$$\frac{2-\sqrt{3}}{2^{2}-\left(\sqrt{3}\right)^{2}}+|\sin(30)-1|$$

Square 2. Square \sqrt{3}.

$$\frac{2-\sqrt{3}}{4-3}+|\sin(30)-1|$$

Subtract 3 from 4 to get 1.

$$\frac{2-\sqrt{3}}{1}+|\sin(30)-1|$$

Anything divided by one gives itself.

$$2-\sqrt{3}+|\sin(30)-1|$$

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

$$2-\sqrt{3}+|\frac{1}{2}-1|$$

Subtract 1 from \frac{1}{2} to get -\frac{1}{2}.

$$2-\sqrt{3}+|-\frac{1}{2}|$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -\frac{1}{2} is \frac{1}{2}.

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

Add 2 and \frac{1}{2} to get \frac{5}{2}.

$$\frac{5}{2}-\sqrt{3}$$