Find the exact value of the trigonometric expression without the use of a calculator. sin(pi/12) cos(7pi/12) + cos(pi/12) sin(7pi/12) =
The Correct Answer and Explanation is:
Correct Answer:
We are asked to find the exact value of:sin(π12)cos(7π12)+cos(π12)sin(7π12)\sin\left(\frac{\pi}{12}\right)\cos\left(\frac{7\pi}{12}\right) + \cos\left(\frac{\pi}{12}\right)\sin\left(\frac{7\pi}{12}\right)sin(12π)cos(127π)+cos(12π)sin(127π)
This expression matches the identity for the sine of a sum of angles:sin(A+B)=sin(A)cos(B)+cos(A)sin(B)\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)sin(A+B)=sin(A)cos(B)+cos(A)sin(B)
In this case:
- A=π12A = \frac{\pi}{12}A=12π
- B=7π12B = \frac{7\pi}{12}B=127π
Thus, applying the identity:sin(π12+7π12)=sin(8π12)=sin(2π3)\sin\left(\frac{\pi}{12} + \frac{7\pi}{12}\right) = \sin\left(\frac{8\pi}{12}\right) = \sin\left(\frac{2\pi}{3}\right)sin(12π+127π)=sin(128π)=sin(32π)
We know that:sin(2π3)=sin(π−π3)\sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi – \frac{\pi}{3}\right)sin(32π)=sin(π−3π)
Using the identity for sine of supplementary angles:sin(π−x)=sin(x)\sin(\pi – x) = \sin(x)sin(π−x)=sin(x)
Therefore:sin(2π3)=sin(π3)=32\sin\left(\frac{2\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}sin(32π)=sin(3π)=23
Final exact value:32\boxed{\frac{\sqrt{3}}{2}}23
Detailed Explanation:
This problem tests knowledge of trigonometric identities and simplifications involving radians. The given expression fits the standard identity for the sine of a sum of angles, which simplifies the calculation significantly.
First, identify the structure:
- The expression has sin\sinsin of one angle multiplied by cos\coscos of another and vice versa, which is the exact form of sin(A+B)\sin(A + B)sin(A+B).
- Plug in the angles: A=π12A = \frac{\pi}{12}A=12π and B=7π12B = \frac{7\pi}{12}B=127π.
- Their sum is 8π12=2π3\frac{8\pi}{12} = \frac{2\pi}{3}128π=32π.
To find sin(2π3)\sin\left(\frac{2\pi}{3}\right)sin(32π), note that 2π3\frac{2\pi}{3}32π lies in the second quadrant where sine is positive. Using reference angles, sin(2π3)\sin(\frac{2\pi}{3})sin(32π) simplifies to sin(π3)\sin(\frac{\pi}{3})sin(3π).
The known exact value for sin(π3)\sin(\frac{\pi}{3})sin(3π) is 32\frac{\sqrt{3}}{2}23.
Thus, without needing a calculator, the expression simplifies directly to 32\frac{\sqrt{3}}{2}23.
