1. **State the problem:** Given that $\tan \theta = \frac{3}{4}$ for an acute angle $\theta$, find integers $a$ and $b$ such that $\cos \theta = \frac{a}{b}$. 2. **Recall the definitions and formula:** - $\tan \theta = \frac{\sin \theta}{\cos \theta}$ - For an acute angle, $\sin \theta$, $\cos \theta$, and $\tan \theta$ are positive. - Using the Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$ 3. **Express $\sin \theta$ and $\cos \theta$ in terms of $a$ and $b$: ** Since $\tan \theta = \frac{3}{4}$, we can think of a right triangle where the opposite side is 3 and adjacent side is 4. 4. **Calculate the hypotenuse:** $$\text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 5. **Find $\cos \theta$:** $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}$$ 6. **Identify $a$ and $b$:** $$a = 4, \quad b = 5$$ **Final answer:** $$\cos \theta = \frac{4}{5}$$