Solve cos(pi/4+3pi/2)

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{3\pi }{2} times \frac{2}{2}.

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

Since \frac{\pi }{4} and \frac{2\times 3\pi }{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \pi +2\times 3\pi .

$$\cos(\frac{\pi +6\pi }{4})$$

Combine like terms in \pi +6\pi .

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

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

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

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

$$0\cos(\frac{\pi }{4})-\sin(\frac{\pi }{4})\sin(\frac{3\pi }{2})$$

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

$$0\times \frac{\sqrt{2}}{2}-\sin(\frac{\pi }{4})\sin(\frac{3\pi }{2})$$

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

$$0\times \frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\sin(\frac{3\pi }{2})$$

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

$$0\times \frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\left(-1\right)$$

Do the calculations.

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