Given the information:

• $\cos s = \frac{16}{17}$
• $\cos t = \frac{3}{5}$
• Both angles $s$ and $t$ are in the first quadrant, so $\sin s > 0$ and $\sin t > 0$.

Step 1: Find $\sin s$ and $\sin t$

We use the Pythagorean identity:
$\sin^2 \theta + \cos^2 \theta = 1$

For $s$:

$\sin^2 s + \left(\frac{16}{17}\right)^2 = 1$ $\sin^2 s + \frac{256}{289} = 1$ $\sin^2 s = 1 - \frac{256}{289} = \frac{289 - 256}{289} = \frac{33}{289}$ $\sin s = \sqrt{\frac{33}{289}} = \frac{\sqrt{33}}{17}$

For $t$:

$\sin^2 t + \left(\frac{3}{5}\right)^2 = 1$ $\sin^2 t + \frac{9}{25} = 1$ $\sin^2 t = 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25}$ $\sin t = \sqrt{\frac{16}{25}} = \frac{4}{5}$

Step 2: Find $\sin(s + t)$

We use the sum of angles formula for sine: $\sin(s + t) = \sin s \cos t + \cos s \sin t$

Substitute the known values: $\sin(s + t) = \frac{\sqrt{33}}{17} \cdot \frac{3}{5} + \frac{16}{17} \cdot \frac{4}{5}$ $\sin(s + t) = \frac{3\sqrt{33}}{85} + \frac{64}{85}$ $\sin(s + t) = \frac{3\sqrt{33} + 64}{85}$

Step 3: Find $\tan(s + t)$

We use the sum of angles formula for tangent: $\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t}$

First, find $\tan s$ and $\tan t$: $\tan s = \frac{\sin s}{\cos s} = \frac{\frac{\sqrt{33}}{17}}{\frac{16}{17}} = \frac{\sqrt{33}}{16}$ $\tan t = \frac{\sin t}{\cos t} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}$

Now substitute into the formula for $\tan(s + t)$: $\tan(s + t) = \frac{\frac{\sqrt{33}}{16} + \frac{4}{3}}{1 - \frac{\sqrt{33}}{16} \cdot \frac{4}{3}}$ Find a common denominator in the numerator and simplify: $\tan(s + t) = \frac{\frac{3\sqrt{33} + 64}{48}}{1 - \frac{4\sqrt{33}}{48}}$

Step 4: Determine the quadrant of $s + t$

Since $s$ and $t$ are in the first quadrant, and the sum $s + t$ is less than $180^\circ$ (as both are positive and less than $90^\circ$), the angle $s + t$ is in the first quadrant as well.

Final answers:

(a) $\sin(s + t) = \frac{3\sqrt{33} + 64}{85}$
(b) $\tan(s + t) = \frac{\frac{3\sqrt{33} + 64}{48}}{1 - \frac{4\sqrt{33}}{48}}$
(c) $s + t$ is in the first quadrant.

Would you like more details or explanations?

Here are 5 related questions to deepen your understanding:

1. What is the exact value of $\cos(s + t)$?
2. How does knowing the quadrant help in determining signs of trigonometric functions?
3. Can you find $\sin(2s)$ using the double-angle formula?
4. What would change if one of the angles were in a different quadrant?
5. How do trigonometric identities simplify calculations in physics problems?

Tip: Always use the Pythagorean identity when you know either $\sin$ or $\cos$ to quickly find the other.