Solve sin(pi+3pi/4)

$$\sin ( \pi +3 \frac{ \pi }{ 4 } )$$

Evaluate

$$-\frac{\sqrt{2}}{2}\approx -0.707106781$$

Short Solution Steps

Express 3\times \frac{\pi }{4} as a single fraction.

$$\sin(\pi +\frac{3\pi }{4})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \pi times \frac{4}{4}.

$$\sin(\frac{4\pi }{4}+\frac{3\pi }{4})$$

Since \frac{4\pi }{4} and \frac{3\pi }{4} have the same denominator, add them by adding their numerators.

$$\sin(\frac{4\pi +3\pi }{4})$$

Combine like terms in 4\pi +3\pi .

$$\sin(\frac{7\pi }{4})$$

Use \sin(x+y)=\sin(x)\cos(y)+\sin(y)\cos(x) where x=\frac{3\pi }{2} and y=\frac{\pi }{4} to obtain the result.

$$\sin(\frac{3\pi }{2}+\frac{\pi }{4})=\sin(\frac{3\pi }{2})\cos(\frac{\pi }{4})+\sin(\frac{\pi }{4})\cos(\frac{3\pi }{2})$$

Get the value of \sin(\frac{3\pi }{2}) from trigonometric values table.

$$-\cos(\frac{\pi }{4})+\sin(\frac{\pi }{4})\cos(\frac{3\pi }{2})$$

Get the value of \cos(\frac{\pi }{4}) from trigonometric values table.

$$-\frac{\sqrt{2}}{2}+\sin(\frac{\pi }{4})\cos(\frac{3\pi }{2})$$

Get the value of \sin(\frac{\pi }{4}) from trigonometric values table.

$$-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\cos(\frac{3\pi }{2})$$

Get the value of \cos(\frac{3\pi }{2}) from trigonometric values table.

$$-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\times 0$$

Do the calculations.

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