### (a)

**Step 1 — Interpret the given ratio.**
$$\tan A = \frac{\text{perpendicular}}{\text{base}} = \frac{4}{3}$$
So perpendicular $= 4k$, base $= 3k$.

**Step 2 — Find the hypotenuse using Pythagoras theorem.**
$$\text{hyp} = \sqrt{(4k)^2 + (3k)^2} = \sqrt{16k^2 + 9k^2} = \sqrt{25k^2} = 5k$$

**Step 3 — Write the required ratios.**
$$\sin A = \frac{\text{perp}}{\text{hyp}} = \frac{4k}{5k} = \frac{4}{5}$$
$$\cos A = \frac{\text{base}}{\text{hyp}} = \frac{3k}{5k} = \frac{3}{5}$$

$$\boxed{\sin A = \frac45,\qquad \cos A = \frac35}$$

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### (b)

**Step 1 — Use the fundamental identity.**
$$\sin^2 A + \cos^2 A = 1$$

**Step 2 — Make $\cos A$ the subject.**
$$\cos^2 A = 1 - \sin^2 A \ \Rightarrow\ \cos A = \sqrt{1 - \sin^2 A}$$

**Step 3 — Write $\tan A$ using the quotient relation.**
$$\tan A = \frac{\sin A}{\cos A}$$

**Step 4 — Substitute the value of $\cos A$.**
$$\tan A = \frac{\sin A}{\sqrt{1 - \sin^2 A}}$$

$$\boxed{\cos A = \sqrt{1-\sin^2 A},\qquad \tan A = \frac{\sin A}{\sqrt{1-\sin^2 A}}}$$