Solve (sqrt{7})^2*sin45^circ/1+cos45^circ

$$( \sqrt { 7 } ) ^ { 2 } \times \frac { \sin 45 ^ { \circ } } { 1 + \cos 45 ^ { \circ } }$$

Evaluate

$$7\left(\sqrt{2}-1\right)\approx 2.899494937$$

Solution Steps

The square of \sqrt{7} is 7.

$$7\times \frac{\sin(45)}{1+\cos(45)}$$

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

$$7\times \frac{\frac{\sqrt{2}}{2}}{1+\cos(45)}$$

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

$$7\times \frac{\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$7\times \frac{\frac{\sqrt{2}}{2}}{\frac{2}{2}+\frac{\sqrt{2}}{2}}$$

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

$$7\times \frac{\frac{\sqrt{2}}{2}}{\frac{2+\sqrt{2}}{2}}$$

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

$$7\times \frac{\sqrt{2}\times 2}{2\left(2+\sqrt{2}\right)}$$

Cancel out 2 in both numerator and denominator.

$$7\times \frac{\sqrt{2}}{\sqrt{2}+2}$$

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

$$7\times \frac{\sqrt{2}\left(\sqrt{2}-2\right)}{\left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right)}$$

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

$$7\times \frac{\sqrt{2}\left(\sqrt{2}-2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}$$

Square \sqrt{2}. Square 2.

$$7\times \frac{\sqrt{2}\left(\sqrt{2}-2\right)}{2-4}$$

Subtract 4 from 2 to get -2.

$$7\times \frac{\sqrt{2}\left(\sqrt{2}-2\right)}{-2}$$

Express 7\times \frac{\sqrt{2}\left(\sqrt{2}-2\right)}{-2} as a single fraction.

$$\frac{7\sqrt{2}\left(\sqrt{2}-2\right)}{-2}$$

Use the distributive property to multiply 7\sqrt{2} by \sqrt{2}-2.

$$\frac{7\left(\sqrt{2}\right)^{2}-14\sqrt{2}}{-2}$$

The square of \sqrt{2} is 2.

$$\frac{7\times 2-14\sqrt{2}}{-2}$$

Multiply 7 and 2 to get 14.

$$\frac{14-14\sqrt{2}}{-2}$$