Solve sin(150-120)=sin(150)*cos(120)-sin(120)*cos(150)

$$\sin ( 150-120 ) = \sin ( 150 ) \times \cos ( 120 ) - \sin ( 120 ) \times \cos ( 150 )$$

Verify

Solution Steps

Subtract 120 from 150 to get 30.

$$\sin(30)=\sin(150)\cos(120)-\sin(120)\cos(150)$$

Get the value of \sin(30) from trigonometric values table.

$$\frac{1}{2}=\sin(150)\cos(120)-\sin(120)\cos(150)$$

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

$$\frac{1}{2}=\frac{1}{2}\left(\sin(150-120)+\sin(150+120)\right)-\sin(120)\cos(150)$$

Subtract 120 from 150. Add 120 to 150.

$$\frac{1}{2}=\frac{1}{2}\left(\sin(30)+\sin(270)\right)-\sin(120)\cos(150)$$

Get the value of \sin(30) from trigonometric values table.

$$\frac{1}{2}=\frac{1}{2}\left(\frac{1}{2}+\sin(270)\right)-\sin(120)\cos(150)$$

Get the value of \sin(270) from trigonometric values table.

$$\frac{1}{2}=\frac{1}{2}\left(\frac{1}{2}-1\right)-\sin(120)\cos(150)$$

Do the calculations.

$$\frac{1}{2}=-\frac{1}{4}-\sin(120)\cos(150)$$

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

$$\frac{1}{2}=-\frac{1}{4}-\frac{1}{2}\left(\sin(120-150)+\sin(120+150)\right)$$

Subtract 150 from 120. Add 150 to 120.

$$\frac{1}{2}=-\frac{1}{4}-\frac{1}{2}\left(\sin(-30)+\sin(270)\right)$$

Use the property \sin(-x)=-\sin(x).

$$\frac{1}{2}=-\frac{1}{4}-\frac{1}{2}\left(-\sin(30)+\sin(270)\right)$$

Get the value of \sin(30) from trigonometric values table.

$$\frac{1}{2}=-\frac{1}{4}-\frac{1}{2}\left(-\frac{1}{2}+\sin(270)\right)$$

Get the value of \sin(270) from trigonometric values table.

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

Do the calculations.

$$\frac{1}{2}=-\frac{1}{4}-\left(-\frac{3}{4}\right)$$

The opposite of -\frac{3}{4} is \frac{3}{4}.

$$\frac{1}{2}=-\frac{1}{4}+\frac{3}{4}$$

Add -\frac{1}{4} and \frac{3}{4} to get \frac{1}{2}.

$$\frac{1}{2}=\frac{1}{2}$$

Compare \frac{1}{2} and \frac{1}{2}.

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