https://r2.erweima.ai/mathbot/image/1725759190810sfpskypz.jpg

Let's go over the image to analyze the math problem step by step.

From what I can see, two problems are written, labeled as 12 and 13. Let's review them:

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### Problem 12:

1. The equation seems to be starting with:
   \[
   \frac{\text{tan}\, A}{4} + \sin \, A = \frac{4}{5} \quad \text{and} \quad \pi \angle A = \frac{3\pi}{2}
   \]
2. This looks like a trigonometric problem, possibly involving finding the value of \( A \) using the tangent and sine functions.
3. You have written several steps involving finding the value of \( A \) by solving for tangent and sine using square root identities:
   - Something like \( y = 16 - x^2 \), and solving for \( y = 25 \).
4. You eventually seem to have reached a final value for \( \tan A \), but the notation appears a little unclear. It looks like you're trying to find values like \( \tan \theta \).

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### Problem 13:

1. The equation starts with:
   \[
   \text{tan}\, A = \frac{1}{4} \quad \text{and} \quad \tan^2 \, A
   \]
2. You are trying to solve for \( A \) again using trigonometric properties.
3. I can see that you used trigonometric identities such as \( \sin^2 A + \cos^2 A = 1 \) and seem to have reached a point involving finding square roots of fractions.
4. There is a step where you write:
   \[
   \text{tan}\, A = \frac{4}{3}
   \]
5. The overall steps appear correct, but the notations can be made a little clearer.

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### Feedback:

- The use of square roots and identities looks good, but some steps might benefit from clearer labeling of variables.
- Make sure that each transformation follows logically, and double-check calculations involving square roots or division to ensure there are no errors.

Would you like to walk through any specific part or verify the steps further?

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Here are 5 related questions to help you understand better:

1. How do you determine the angle \( A \) from trigonometric equations like \( \sin A \) or \( \tan A \)?
2. How can you simplify square roots when solving trigonometric identities?
3. What is the relationship between sine, cosine, and tangent in a right-angled triangle?
4. How do trigonometric identities like \( \sin^2 A + \cos^2 A = 1 \) help solve problems?
5. How can we verify the solutions obtained for \( A \) using a calculator?

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### Tip:
When solving trigonometric problems, double-check the range of angles involved (like \( A \)) to ensure that the solution makes sense in the given context (e.g., \( \tan \) and \( \sin \) can give multiple values depending on the quadrant).

Explore detailed solutions for trigonometric equations involving tangent and sine identities. Learn how to apply Pythagorean identities and trigonometric properties to solve for angles. Suitable for high school students in Grades 10-12.

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