We are given a right triangle ABC, with hypotenuse $AC = 13 \, \text{cm}$ and one leg $BC = 5 \, \text{cm}$. We are asked to find the approximate difference between the angles $\angle C$ and $\angle A$.

Step 1: Use the Pythagorean Theorem to find the length of the other leg $AB$.

The Pythagorean theorem states: $AC^2 = AB^2 + BC^2$ Substitute the known values: $13^2 = AB^2 + 5^2$ $169 = AB^2 + 25$ $AB^2 = 169 - 25 = 144$ $AB = \sqrt{144} = 12 \, \text{cm}$

Step 2: Find the angles $\angle A$ and $\angle C$.

We can use trigonometry to find the angles. Since $\triangle ABC$ is a right triangle, we know that $\angle B = 90^\circ$. We can use the sine, cosine, or tangent functions.

Angle $\angle C$:

To find $\angle C$, use the sine function: $\sin(C) = \frac{BC}{AC} = \frac{5}{13}$ Now, solve for $C$: $C = \sin^{-1}\left(\frac{5}{13}\right) \approx 22.6^\circ$

Angle $\angle A$:

Since the sum of the angles in a triangle is $180^\circ$, we know: $\angle A = 90^\circ - \angle C = 90^\circ - 22.6^\circ = 67.4^\circ$

Step 3: Find the difference between $\angle A$ and $\angle C$.

The difference is: $\Delta \theta = \angle A - \angle C = 67.4^\circ - 22.6^\circ = 44.8^\circ$

Answer:

The approximate difference between $\angle A$ and $\angle C$ is 44.8°.

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Would you like a more detailed explanation of any step? Or have any further questions?

Related Questions:

1. How would the triangle change if $BC$ were 7 cm instead of 5 cm?
2. Can the Pythagorean theorem be used in non-right triangles?
3. How do you calculate the other angles in a triangle if one angle is known?
4. What are the trigonometric functions and their applications in solving triangles?
5. What are the exact formulas for finding the angles in right triangles?

Tip:

When solving for angles in right triangles, using the sine, cosine, or tangent functions is often the most efficient way to find the angles, especially when you know the lengths of the sides.