We are given that:

$\tan \theta = \frac{4}{3}$

and we need to determine the exact values of $\sin \theta$ and $\cos \theta$.

Step 1: Use the Pythagorean Theorem

Since $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}$, we assign:

• Opposite side = 4
• Adjacent side = 3

Using the Pythagorean theorem to find the hypotenuse $r$:

$r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Step 2: Determine $\sin \theta$ and $\cos \theta$

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}$

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}$

Final Answer:

$\sin \theta = \frac{4}{5}, \quad \cos \theta = \frac{3}{5}$

Would you like a detailed explanation of any step?

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Related Questions:

1. If $\cos \theta = \frac{3}{5}$, what is $\sec \theta$?
2. How do we determine the quadrant when given a trigonometric ratio?
3. Can you find $\csc \theta$ using $\sin \theta$?
4. If $\sin \theta = \frac{4}{5}$, what is $\sin (90^\circ - \theta)$?
5. How does the Pythagorean Identity relate to this problem?

Tip: The Pythagorean Theorem is a powerful tool for solving right triangle problems—always use it to find missing sides before calculating trigonometric ratios.