If sint=7/9, and t is in quadrant I, find the exact value of sin(2t), cos(2t), and tan(2t) algebraically without solving for t.

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.

Find the exact values of sin(2t), cos(2t), and tan(2t) algebraically for a given sine value and quadrant I angle. Learn the step-by-step process using trigonometric identities and the Pythagorean identity. Suitable for high school students studying trigonometry.

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