Given that $\sin t = \frac{7}{9}$ and $t$ is in the first quadrant, we can find the exact values of $\sin(2t)$, $\cos(2t)$, and $\tan(2t)$ using trigonometric identities.

Step 1: Determine $\cos t$

Since $t$ is in the first quadrant, $\cos t$ is positive. Using the Pythagorean identity: $\sin^2 t + \cos^2 t = 1$ Substitute $\sin t = \frac{7}{9}$: $\left(\frac{7}{9}\right)^2 + \cos^2 t = 1$ $\frac{49}{81} + \cos^2 t = 1$ $\cos^2 t = 1 - \frac{49}{81} = \frac{81}{81} - \frac{49}{81} = \frac{32}{81}$ $\cos t = \sqrt{\frac{32}{81}} = \frac{\sqrt{32}}{9} = \frac{4\sqrt{2}}{9}$

Step 2: Calculate $\sin(2t)$

Use the double-angle identity for sine: $\sin(2t) = 2 \sin t \cos t$ Substitute $\sin t = \frac{7}{9}$ and $\cos t = \frac{4\sqrt{2}}{9}$: $\sin(2t) = 2 \times \frac{7}{9} \times \frac{4\sqrt{2}}{9} = \frac{56\sqrt{2}}{81}$

Step 3: Calculate $\cos(2t)$

Use the double-angle identity for cosine: $\cos(2t) = \cos^2 t - \sin^2 t$ Substitute $\cos^2 t = \frac{32}{81}$ and $\sin^2 t = \frac{49}{81}$: $\cos(2t) = \frac{32}{81} - \frac{49}{81} = \frac{-17}{81}$

Step 4: Calculate $\tan(2t)$

Use the identity for tangent: $\tan(2t) = \frac{\sin(2t)}{\cos(2t)}$ Substitute $\sin(2t) = \frac{56\sqrt{2}}{81}$ and $\cos(2t) = \frac{-17}{81}$: $\tan(2t) = \frac{\frac{56\sqrt{2}}{81}}{\frac{-17}{81}} = \frac{56\sqrt{2}}{-17} = -\frac{56\sqrt{2}}{17}$

Final Results:

$\sin(2t) = \frac{56\sqrt{2}}{81}, \quad \cos(2t) = \frac{-17}{81}, \quad \tan(2t) = -\frac{56\sqrt{2}}{17}$

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the double-angle identities for sine and cosine?
2. What are the conditions for a function to have a double angle identity?
3. Can you find $\sin(2t)$, $\cos(2t)$, and $\tan(2t)$ if $t$ is in a different quadrant?
4. How do you solve for $t$ directly given $\sin t = \frac{7}{9}$?
5. What is the geometric interpretation of these trigonometric values in the unit circle?
6. How would the results differ if $\sin t$ was negative?
7. How can the Pythagorean identity be derived from the unit circle?
8. What is the significance of the exact values of trigonometric functions?

Tip: Remember that the signs of trigonometric functions depend on the quadrant in which the angle lies.