How do you evaluate #sin (pi / 12) * cos (3 pi / 4) - cos (pi / 12) * sin (3 pi / 4)#?

Question

Emery Adkins · Accepted Answer

#-sqrt3/2#

Recall

#sin(A-B)=sinAcosB-cosAsinB#

#sin(pi/12)cos((3pi)/4)-cos(pi/12)sin((3pi)/4)=sin(pi/12-(3pi)/4)#

#sin(pi/12-(9pi)/12)#

Recall

#sin(-x)=-sin(x)#

#sin(-(2pi)/3)=-sin((2pi)/3)#

#-sqrt3/2#

Caroline Adams · Answer

undefined To evaluate $ \sin\left(\frac{\pi}{12}\right) \cdot \cos\left(\frac{3\pi}{4}\right) - \cos\left(\frac{\pi}{12}\right) \cdot \sin\left(\frac{3\pi}{4}\right) $, follow these steps:

1. Calculate the values of $ \sin\left(\frac{\pi}{12}\right) $, $ \cos\left(\frac{3\pi}{4}\right) $, $ \cos\left(\frac{\pi}{12}\right) $, and $ \sin\left(\frac{3\pi}{4}\right) $.
  
2. Substitute the calculated values into the expression:

$$ \sin\left(\frac{\pi}{12}\right) \cdot \cos\left(\frac{3\pi}{4}\right) - \cos\left(\frac{\pi}{12}\right) \cdot \sin\left(\frac{3\pi}{4}\right) $$

3. Perform the multiplication and subtraction operations to find the result.

Let's calculate the values:

$$ \sin\left(\frac{\pi}{12}\right) \approx 0.2588 $$
$$ \cos\left(\frac{3\pi}{4}\right) = \cos\left(135^\circ\right) = -\frac{\sqrt{2}}{2} $$
$$ \cos\left(\frac{\pi}{12}\right) \approx 0.9659 $$
$$ \sin\left(\frac{3\pi}{4}\right) = \sin\left(135^\circ\right) = \frac{\sqrt{2}}{2} $$

Substitute these values:

$$ 0.2588 \cdot \left(-\frac{\sqrt{2}}{2}\right) - 0.9659 \cdot \frac{\sqrt{2}}{2} $$

Now, perform the multiplication and subtraction:

$$ \approx -0.3656 - 0.6831 $$

$$ \approx -1.0487 $$

Therefore, $ \sin\left(\frac{\pi}{12}\right) \cdot \cos\left(\frac{3\pi}{4}\right) - \cos\left(\frac{\pi}{12}\right) \cdot \sin\left(\frac{3\pi}{4}\right) $ is approximately $ -1.0487 $.