Given the values of $\sin \theta$ and $\cos \theta$ for the acute angle $\theta$:

$\sin \theta = \frac{4}{7}, \quad \cos \theta = \frac{\sqrt{33}}{7}$

we need to find the exact values of the four remaining trigonometric functions: $\tan \theta$, $\cot \theta$, $\sec \theta$, and $\csc \theta$.

Step-by-Step Solution

1. Tangent ($\tan \theta$):

   $\tan \theta = \frac{\sin \theta}{\cos \theta}$ Substituting the given values: $\tan \theta = \frac{\frac{4}{7}}{\frac{\sqrt{33}}{7}} = \frac{4}{\sqrt{33}}$

2. Cotangent ($\cot \theta$):

   $\cot \theta = \frac{\cos \theta}{\sin \theta}$ Substituting the given values: $\cot \theta = \frac{\frac{\sqrt{33}}{7}}{\frac{4}{7}} = \frac{\sqrt{33}}{4}$

3. Secant ($\sec \theta$):

   $\sec \theta = \frac{1}{\cos \theta}$ Substituting the value of $\cos \theta$: $\sec \theta = \frac{1}{\frac{\sqrt{33}}{7}} = \frac{7}{\sqrt{33}}$

   To rationalize the denominator: $\sec \theta = \frac{7}{\sqrt{33}} \times \frac{\sqrt{33}}{\sqrt{33}} = \frac{7\sqrt{33}}{33}$

4. Cosecant ($\csc \theta$):

   $\csc \theta = \frac{1}{\sin \theta}$ Substituting the value of $\sin \theta$: $\csc \theta = \frac{1}{\frac{4}{7}} = \frac{7}{4}$

Summary of Results

• $\sin \theta = \frac{4}{7}$
• $\cos \theta = \frac{\sqrt{33}}{7}$
• $\tan \theta = \frac{4}{\sqrt{33}}$
• $\cot \theta = \frac{\sqrt{33}}{4}$
• $\sec \theta = \frac{7\sqrt{33}}{33}$
• $\csc \theta = \frac{7}{4}$

Would you like to see more details on any of these steps, or do you have any questions?

Follow-up Questions:

1. Would you like to learn more about the process of rationalizing denominators?
2. Are you interested in how these trigonometric identities relate to the unit circle?
3. Do you want to understand how to use these identities to solve other types of problems?
4. Would you like examples of other trigonometric functions given different values?
5. Do you need a refresher on the Pythagorean identity involving $\sin$ and $\cos$?

Tip:

When working with trigonometric functions, always remember that the square of $\sin \theta$ plus the square of $\cos \theta$ is equal to 1 for any angle $\theta$. This is a useful identity for checking your work: $\sin^2 \theta + \cos^2 \theta = 1$.