Solve 4(sin(150)cos(135)+cos(240)sin(135))

$$4( \sin ( 150 ) \cos ( 135 ) + \cos ( 240 ) \sin ( 135 ) )$$

Evaluate

$$-2\sqrt{2}\approx -2.828427125$$

Solution Steps

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

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

Subtract 135 from 150. Add 135 to 150.

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

Do the calculations.

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

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

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

Subtract 240 from 135. Add 240 to 135.

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

Do the calculations.

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

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

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

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

$$-2\sqrt{2}$$