Questions: Use a sum or difference formula to find the exact value of the trigonometric function. sin(11π/12) sin(11π/12)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the exact value of \(\sin \frac{11\pi}{12}\), we can use the sum or difference formula for sine. We can express \(\frac{11\pi}{12}\) as a sum of angles whose sine values are known, such as \(\frac{11\pi}{12} = \frac{6\pi}{12} + \frac{5\pi}{12} = \frac{\pi}{2} + \frac{\pi}{3}\). Then, apply the sine sum formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\).

To find \(\sin \frac{11\pi}{12}\), we can express the angle as a sum of two known angles: \[ \frac{11\pi}{12} = \frac{\pi}{2} + \frac{\pi}{3} \]

Using the sine sum formula, we have: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] where \(a = \frac{\pi}{2}\) and \(b = \frac{\pi}{3}\).

We calculate the sine and cosine values:

• \(\sin \frac{\pi}{2} = 1\)
• \(\cos \frac{\pi}{2} \approx 6.1232 \times 10^{-17}\) (very close to 0)
• \(\sin \frac{\pi}{3} \approx 0.8660\)
• \(\cos \frac{\pi}{3} = \frac{1}{2}\)

Substituting these values into the sine sum formula gives: \[ \sin \frac{11\pi}{12} = 1 \cdot \cos \frac{\pi}{3} + \cos \frac{\pi}{2} \cdot \sin \frac{\pi}{3} \] This simplifies to: \[ \sin \frac{11\pi}{12} = 1 \cdot \frac{1}{2} + 0 \cdot 0.8660 = \frac{1}{2} \]

Final Answer