Given \(\sin A = \frac{3}{4}\), we can calculate \(\cos A\) and \(\tan A\) using the Pythagorean identity and the definition of the tangent function.

### Step 1: Calculate \(\cos A\)

We know from the Pythagorean identity:
\[
\sin^2 A + \cos^2 A = 1
\]
Substitute \(\sin A = \frac{3}{4}\) into the equation:
\[
\left(\frac{3}{4}\right)^2 + \cos^2 A = 1
\]
\[
\frac{9}{16} + \cos^2 A = 1
\]
Subtract \(\frac{9}{16}\) from both sides:
\[
\cos^2 A = 1 - \frac{9}{16}
\]
\[
\cos^2 A = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}
\]
Taking the square root of both sides:
\[
\cos A = \pm \sqrt{\frac{7}{16}} = \pm \frac{\sqrt{7}}{4}
\]
So, \(\cos A = \pm \frac{\sqrt{7}}{4}\). The sign depends on the quadrant in which angle \(A\) is located.

### Step 2: Calculate \(\tan A\)

We know that:
\[
\tan A = \frac{\sin A}{\cos A}
\]
Substitute the values we have:
\[
\tan A = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} = \frac{3\sqrt{7}}{7}
\]
So, \(\tan A = \frac{3\sqrt{7}}{7}\).

### Final Answers:
- \(\cos A = \pm \frac{\sqrt{7}}{4}\)
- \(\tan A = \frac{3\sqrt{7}}{7}\)

Would you like more details on any step or have any other questions?

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Here are some related questions you might find interesting:

1. How do you determine the sign of \(\cos A\) based on the quadrant?
2. What is the value of \(\sin A\) if \(\cos A = \frac{5}{13}\)?
3. How can you use the unit circle to find \(\cos A\) and \(\tan A\)?
4. What is the relationship between \(\sin A\), \(\cos A\), and \(\tan A\) in different quadrants?
5. How do you calculate \(\sin 2A\) given \(\sin A\) and \(\cos A\)?
6. Can you determine \(\sec A\) if you know \(\sin A\) and \(\cos A\)?
7. What is the geometric interpretation of \(\sin A\), \(\cos A\), and \(\tan A\)?
8. How do you find \(\cot A\) if \(\sin A\) and \(\cos A\) are known?

**Tip:** When dealing with trigonometric identities, always keep in mind the quadrant to determine the correct signs of the trigonometric functions.

Learn how to calculate cos A and tan A given sin A = 3/4 using trigonometric identities and definitions. Get step-by-step solutions with explanations and explore related trigonometric concepts.

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