Questions: Find the exact value of the following trigonometric expression without the use of a calculator. sin((2π/3)-(π/4))

Solution Steps

To find the exact value of the trigonometric expression \(\sin \left(\frac{2 \pi}{3}-\frac{\pi}{4}\right)\), we can use the sine difference identity: \(\sin(a - b) = \sin a \cos b - \cos a \sin b\). First, identify \(a = \frac{2\pi}{3}\) and \(b = \frac{\pi}{4}\). Then, calculate \(\sin a\), \(\cos a\), \(\sin b\), and \(\cos b\) using known values from the unit circle. Finally, substitute these values into the identity to find the result.

We start with the expression \( \sin \left(\frac{2 \pi}{3} - \frac{\pi}{4}\right) \). Let \( a = \frac{2 \pi}{3} \) and \( b = \frac{\pi}{4} \).

Using known values from the unit circle:

• \( \sin a = \sin \left(\frac{2 \pi}{3}\right) = \frac{\sqrt{3}}{2} \approx 0.8660 \)
• \( \cos a = \cos \left(\frac{2 \pi}{3}\right) = -\frac{1}{2} \approx -0.5000 \)
• \( \sin b = \sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \)
• \( \cos b = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \)

Using the sine difference identity: \[ \sin(a - b) = \sin a \cos b - \cos a \sin b \] Substituting the values: \[ \sin \left(\frac{2 \pi}{3} - \frac{\pi}{4}\right) = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(-\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) \]

Calculating the terms: \[ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Final Answer