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The problem asks for the exact values of trigonometric functions for specific angles. Let's solve each part:

### a) \(\sin\left(\frac{7\pi}{6}\right)\)
The angle \(\frac{7\pi}{6}\) is in the third quadrant, where sine is negative. The reference angle is \(\pi/6\).

\[
\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right)
\]

Since \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\), we get:

\[
\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}
\]

### b) \(\cos\left(\frac{7\pi}{4}\right)\)
The angle \(\frac{7\pi}{4}\) is in the fourth quadrant, where cosine is positive. The reference angle is \(\pi/4\).

\[
\cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)
\]

Since \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), we get:

\[
\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]

### c) \(\tan\left(\frac{\pi}{3}\right)\)
The angle \(\frac{\pi}{3}\) is in the first quadrant, where tangent is positive.

\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]

### Final Answers:
- a) \(\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}\)
- b) \(\cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
- c) \(\tan\left(\frac{\pi}{3}\right) = \sqrt{3}\)

Do you need further explanation or any clarifications?

Here are 5 related questions:
1. What is the value of \(\cos\left(\frac{7\pi}{6}\right)\)?
2. How do you find the reference angle for any trigonometric function?
3. What is the exact value of \(\tan\left(\frac{7\pi}{6}\right)\)?
4. Why is \(\sin\) negative in the third quadrant?
5. How do the signs of sine and cosine change across the four quadrants?

**Tip:** Always remember the ASTC rule (All Students Take Calculus) to determine the signs of trigonometric functions in each quadrant.

Quelles sont les valeurs exactes ? a) sin(7π/6) b) cos(7π/4) c) tan(π/3)

Learn how to find the exact trigonometric values of sin(7π/6), cos(7π/4), and tan(π/3). This problem covers core trigonometric concepts like sine, cosine, and tangent in different quadrants. Suitable for high school students studying advanced trigonometry.

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