Solve frac{sin210^{circ}*tan225^circ*cos315^circ}{sin(-45^circ)*cos(240^circ)}

$$\frac { \sin 210 ^ { \circ } \cdot \tan 225 ^ { \circ } \cdot \cos 315 ^ { \circ } } { \sin ( - 45 ^ { \circ } ) \cdot \cos ( 240 ^ { \circ } ) }$$

Evaluate

$$-\frac{4926969569187561\sqrt{2}}{2500000000000000}\approx -2.787114874$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{-0.9853939138375122}{\sin(-45) \cdot \cos(240)}$$

Use \sin(x)\cos(y)=\frac{1}{2}\left(\sin(x-y)+\sin(x+y)\right) to obtain the result.

$$\frac{-0.9853939138375122}{\frac{1}{2}\left(\sin(-45-240)+\sin(-45+240)\right)}$$

Subtract 240 from -45. Add 240 to -45.

$$\frac{-0.9853939138375122}{\frac{1}{2}\left(\sin(-285)+\sin(195)\right)}$$

Do the calculations.

$$\frac{-0.9853939138375122}{\frac{1}{4}\sqrt{2}}$$

Rationalize the denominator of \frac{-0.9853939138375122}{\frac{1}{4}\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{-0.9853939138375122\sqrt{2}}{\frac{1}{4}\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{-0.9853939138375122\sqrt{2}}{\frac{1}{4}\times 2}$$

Multiply \frac{1}{4} and 2 to get \frac{1}{2}.

$$\frac{-0.9853939138375122\sqrt{2}}{\frac{1}{2}}$$

Divide -0.9853939138375122\sqrt{2} by \frac{1}{2} by multiplying -0.9853939138375122\sqrt{2} by the reciprocal of \frac{1}{2}.

$$-0.9853939138375122\sqrt{2}\times 2$$

Multiply -0.9853939138375122 and 2 to get -1.9707878276750244.

$$-1.9707878276750244\sqrt{2}$$