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 $90^\circ$.