Solve sin30^circ+cos45^circ/tan60^circ

$$\frac { \sin 30 ^ { \circ } + \cos 45 ^ { \circ } } { \tan 60 ^ { \circ } }$$

Evaluate

$$\frac{\sqrt{3}\left(\sqrt{2}+1\right)}{6}\approx 0.696923425$$

Solution Steps

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

$$\frac{\frac{1}{2}+\cos(45)}{\tan(60)}$$

Get the value of \cos(45) from trigonometric values table.

$$\frac{\frac{1}{2}+\frac{\sqrt{2}}{2}}{\tan(60)}$$

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

$$\frac{\frac{1+\sqrt{2}}{2}}{\tan(60)}$$

Get the value of \tan(60) from trigonometric values table.

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

Express \frac{\frac{1+\sqrt{2}}{2}}{\sqrt{3}} as a single fraction.

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

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

$$\frac{\left(1+\sqrt{2}\right)\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

$$\frac{\left(1+\sqrt{2}\right)\sqrt{3}}{6}$$

Use the distributive property to multiply 1+\sqrt{2} by \sqrt{3}.

$$\frac{\sqrt{3}+\sqrt{2}\sqrt{3}}{6}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{3}+\sqrt{6}}{6}$$