Solve cos(pi/3)*(cos(pi/4)+sin(pi/4))-4cos(pi/6)sin(frac{pi}{3})

$$\cos ( \frac{ \pi }{ 3 } ) \times ( \cos ( \frac{ \pi }{ 4 } ) + \sin ( \frac{ \pi }{ 4 } ) ) -4 \cos ( \frac{ \pi }{ 6 } ) \sin ( \frac{ \pi }{ 3 } )$$

Evaluate

$$\frac{\sqrt{2}}{2}-3\approx -2.292893219$$

Solution Steps

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

$$\frac{1}{2}\left(\cos(\frac{\pi }{4})+\sin(\frac{\pi }{4})\right)-4\cos(\frac{\pi }{6})\sin(\frac{\pi }{3})$$

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

$$\frac{1}{2}\left(\frac{\sqrt{2}}{2}+\sin(\frac{\pi }{4})\right)-4\cos(\frac{\pi }{6})\sin(\frac{\pi }{3})$$

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

$$\frac{1}{2}\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\right)-4\cos(\frac{\pi }{6})\sin(\frac{\pi }{3})$$

Combine \frac{\sqrt{2}}{2} and \frac{\sqrt{2}}{2} to get \sqrt{2}.

$$\frac{1}{2}\sqrt{2}-4\cos(\frac{\pi }{6})\sin(\frac{\pi }{3})$$

Get the value of \cos(\frac{\pi }{6}) from trigonometric values table.

$$\frac{1}{2}\sqrt{2}-4\times \frac{\sqrt{3}}{2}\sin(\frac{\pi }{3})$$

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

$$\frac{1}{2}\sqrt{2}-4\times \frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{2}$$

Multiply \frac{\sqrt{3}}{2} and \frac{\sqrt{3}}{2} to get \left(\frac{\sqrt{3}}{2}\right)^{2}.

$$\frac{1}{2}\sqrt{2}-4\times \left(\frac{\sqrt{3}}{2}\right)^{2}$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1}{2}\sqrt{2}-4\times \frac{\left(\sqrt{3}\right)^{2}}{2^{2}}$$

Express 4\times \frac{\left(\sqrt{3}\right)^{2}}{2^{2}} as a single fraction.

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

The square of \sqrt{3} is 3.

$$\frac{1}{2}\sqrt{2}-\frac{4\times 3}{2^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{1}{2}\sqrt{2}-\frac{12}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{1}{2}\sqrt{2}-\frac{12}{4}$$

Divide 12 by 4 to get 3.

$$\frac{1}{2}\sqrt{2}-3$$