Solve 50*cos45/70-50sin45

$$\frac { 50 \times \cos 45 } { 70 - 50 \sin 45 }$$

Evaluate

$$\frac{35\sqrt{2}+25}{73}\approx 1.020513352$$

Solution Steps

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

$$\frac{50\times \frac{\sqrt{2}}{2}}{70-50\sin(45)}$$

Cancel out 2, the greatest common factor in 50 and 2.

$$\frac{25\sqrt{2}}{70-50\sin(45)}$$

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

$$\frac{25\sqrt{2}}{70-50\times \frac{\sqrt{2}}{2}}$$

Cancel out 2, the greatest common factor in 50 and 2.

$$\frac{25\sqrt{2}}{70-25\sqrt{2}}$$

Rationalize the denominator of \frac{25\sqrt{2}}{70-25\sqrt{2}} by multiplying numerator and denominator by 70+25\sqrt{2}.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{\left(70-25\sqrt{2}\right)\left(70+25\sqrt{2}\right)}$$

Consider \left(70-25\sqrt{2}\right)\left(70+25\sqrt{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}.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{70^{2}-\left(-25\sqrt{2}\right)^{2}}$$

Calculate 70 to the power of 2 and get 4900.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{4900-\left(-25\sqrt{2}\right)^{2}}$$

Expand \left(-25\sqrt{2}\right)^{2}.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{4900-\left(-25\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -25 to the power of 2 and get 625.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{4900-625\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{4900-625\times 2}$$

Multiply 625 and 2 to get 1250.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{4900-1250}$$

Subtract 1250 from 4900 to get 3650.

$$\frac{25\sqrt{2}\left(70+25\sqrt{2}\right)}{3650}$$

Divide 25\sqrt{2}\left(70+25\sqrt{2}\right) by 3650 to get \frac{1}{146}\sqrt{2}\left(70+25\sqrt{2}\right).

$$\frac{1}{146}\sqrt{2}\left(70+25\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{146}\sqrt{2} by 70+25\sqrt{2}.

$$\frac{35}{73}\sqrt{2}+\frac{25}{146}\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{35}{73}\sqrt{2}+\frac{25}{146}\times 2$$

Multiply \frac{25}{146} and 2 to get \frac{25}{73}.

$$\frac{35}{73}\sqrt{2}+\frac{25}{73}$$