Solve 4(sin150^{circ}cos135^circ+cos240^circsin135^circ)sin45^circ

$$4 ( \sin 150 ^ { \circ } \cos 135 ^ { \circ } + \cos 240 ^ { \circ } \sin 135 ^ { \circ } ) \sin 45 ^ { \circ }$$

Evaluate

$$-2$$

Solution Steps

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

$$4\left(\frac{1}{2}\left(\sin(150-135)+\sin(150+135)\right)+\cos(240)\sin(135)\right)\sin(45)$$

Subtract 135 from 150. Add 135 to 150.

$$4\left(\frac{1}{2}\left(\sin(15)+\sin(285)\right)+\cos(240)\sin(135)\right)\sin(45)$$

Do the calculations.

$$4\left(-\frac{1}{4}\sqrt{2}+\cos(240)\sin(135)\right)\sin(45)$$

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

$$4\left(-\frac{1}{4}\sqrt{2}+\frac{1}{2}\left(\sin(135-240)+\sin(135+240)\right)\right)\sin(45)$$

Subtract 240 from 135. Add 240 to 135.

$$4\left(-\frac{1}{4}\sqrt{2}+\frac{1}{2}\left(\sin(-105)+\sin(375)\right)\right)\sin(45)$$

Do the calculations.

$$4\left(-\frac{1}{4}\sqrt{2}-\frac{1}{4}\sqrt{2}\right)\sin(45)$$

Combine -\frac{1}{4}\sqrt{2} and -\frac{1}{4}\sqrt{2} to get -\frac{1}{2}\sqrt{2}.

$$4\left(-\frac{1}{2}\right)\sqrt{2}\sin(45)$$

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

$$-2\sqrt{2}\sin(45)$$

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

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

Cancel out 2 and 2.

$$-\sqrt{2}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$-2$$