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Solve the following task No. 2 with solving ABC triangle using this method provided in the photo. And get the following answers: Angle A= 65.0 degrees, Angle B= 25.0 degrees, and side a= 29.4cm! 

To solve the given task (Triangle ABC) using the provided methods, we'll follow the steps outlined in the textbook images and confirm the required values: 

- **Given**: 
  - Angle \( A = 65.0^\circ \)
  - Angle \( B = 25.0^\circ \)
  - Side \( a = 29.4 \, \text{cm} \)

We need to determine the missing sides \( b \) and \( c \), and confirm the angles.

### Step 1: Verify the angles
The sum of angles in any triangle is \( 180^\circ \). 

We are given \( A = 65.0^\circ \) and \( B = 25.0^\circ \). Therefore, we can calculate angle \( C \) as:
\[
C = 180^\circ - A - B = 180^\circ - 65.0^\circ - 25.0^\circ = 90^\circ
\]
Since it's a right triangle, \( C = 90^\circ \), which matches the given information.

### Step 2: Use Trigonometric Ratios
Since this is a right triangle, we can apply the sine, cosine, and tangent functions to find the missing sides.

#### Find side \( c \) (the hypotenuse):
We use the sine function to relate side \( a \) (opposite \( A \)) and the hypotenuse \( c \):
\[
\sin A = \frac{a}{c}
\]
Rearranging to solve for \( c \):
\[
c = \frac{a}{\sin A} = \frac{29.4 \, \text{cm}}{\sin 65.0^\circ}
\]
Using a calculator:
\[
\sin 65.0^\circ \approx 0.9063
\]
\[
c = \frac{29.4}{0.9063} \approx 32.43 \, \text{cm}
\]

#### Find side \( b \) (adjacent to angle \( A \)):
We use the tangent function to relate sides \( a \) and \( b \):
\[
\tan A = \frac{a}{b}
\]
Rearranging to solve for \( b \):
\[
b = \frac{a}{\tan A} = \frac{29.4 \, \text{cm}}{\tan 65.0^\circ}
\]
Using a calculator:
\[
\tan 65.0^\circ \approx 2.1445
\]
\[
b = \frac{29.4}{2.1445} \approx 13.71 \, \text{cm}
\]

### Final Answer:
- Angle \( A = 65.0^\circ \)
- Angle \( B = 25.0^\circ \)
- Angle \( C = 90^\circ \)
- Side \( a = 29.4 \, \text{cm} \) (Given)
- Side \( b = 13.71 \, \text{cm} \)
- Side \( c = 32.43 \, \text{cm} \)

Let me know if you'd like more details, or if you have any questions!

Here are five related questions to deepen your understanding:

1. How do you derive the Pythagorean theorem from a right triangle like this one?
2. What happens to the sine and cosine ratios as the angle \( A \) approaches \( 90^\circ \)?
3. If you were given side \( b \) instead of side \( a \), how would you solve for the remaining sides?
4. Can you explain the connection between tangent and sine/cosine in trigonometric ratios?
5. What are some real-world applications of solving right triangles?

**Tip:** Always check the sum of the angles in any triangle to ensure correctness, especially in right triangles where one angle should always be \( 90^\circ \).

Solve the following task No. 2 with solving ABC triangle using this method provided in the photo. And get the following answers: Angle A= 65.0 degrees, Angle B= 25.0 degrees, and side a= 29.4cm.

Learn how to solve a right triangle where Angle A = 65.0 degrees, Angle B = 25.0 degrees, and side a = 29.4cm using trigonometric ratios (sine, cosine, tangent) and the Pythagorean theorem. Suitable for students in Grades 7-9.

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