Solve 2sin(pi/2)+3sin(pi)+4sin(3pi/2)/4cos(frac{pi{2})+3cos(pi)+5cos(3pi/2)}

$$\frac{ 2 \sin ( \frac{ \pi }{ 2 } ) +3 \sin ( \pi ) +4 \sin ( \frac{ 3 \pi }{ 2 } ) }{ 4 \cos ( \frac{ \pi }{ 2 } ) +3 \cos ( \pi ) +5 \cos ( \frac{ 3 \pi }{ 2 } ) }$$

Evaluate

$$\frac{2}{3}\approx 0.666666667$$

Solution Steps

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

$$\frac{2\times 1+3\sin(\pi )+4\sin(\frac{3\pi }{2})}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Multiply 2 and 1 to get 2.

$$\frac{2+3\sin(\pi )+4\sin(\frac{3\pi }{2})}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

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

$$\frac{2+3\times 0+4\sin(\frac{3\pi }{2})}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Multiply 3 and 0 to get 0.

$$\frac{2+0+4\sin(\frac{3\pi }{2})}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Add 2 and 0 to get 2.

$$\frac{2+4\sin(\frac{3\pi }{2})}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

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

$$\frac{2+4\left(-1\right)}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Multiply 4 and -1 to get -4.

$$\frac{2-4}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Subtract 4 from 2 to get -2.

$$\frac{-2}{4\cos(\frac{\pi }{2})+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

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

$$\frac{-2}{4\times 0+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Multiply 4 and 0 to get 0.

$$\frac{-2}{0+3\cos(\pi )+5\cos(\frac{3\pi }{2})}$$

Get the value of \cos(\pi ) from trigonometric values table.

$$\frac{-2}{0+3\left(-1\right)+5\cos(\frac{3\pi }{2})}$$

Multiply 3 and -1 to get -3.

$$\frac{-2}{0-3+5\cos(\frac{3\pi }{2})}$$

Subtract 3 from 0 to get -3.

$$\frac{-2}{-3+5\cos(\frac{3\pi }{2})}$$

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

$$\frac{-2}{-3+5\times 0}$$

Multiply 5 and 0 to get 0.

$$\frac{-2}{-3+0}$$

Add -3 and 0 to get -3.

$$\frac{-2}{-3}$$

Fraction \frac{-2}{-3} can be simplified to \frac{2}{3} by removing the negative sign from both the numerator and the denominator.

$$\frac{2}{3}$$

Factor

$$\frac{2}{3} = 0.6666666666666666$$