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To calculate the height \( h \) of the triangle, we can use trigonometry and the law of sines. The given triangle has the following key information:

- The side opposite the \( 105^\circ \) angle (hypotenuse of the right triangle) is 28 cm.
- Two angles are provided: \( 105^\circ \) and \( 45^\circ \).

### Steps to calculate height \( h \):

1. **Find the missing angle**:
   The third angle can be found because the sum of all angles in a triangle is \( 180^\circ \). Since we already have \( 105^\circ \) and \( 45^\circ \):

   \[
   \text{Third angle} = 180^\circ - 105^\circ - 45^\circ = 30^\circ
   \]

2. **Apply the Law of Sines**:
   Using the Law of Sines:

   \[
   \frac{28}{\sin 30^\circ} = \frac{h}{\sin 45^\circ}
   \]

3. **Solve for height \( h \)**:
   We know \( \sin 30^\circ = 0.5 \) and \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), so the equation becomes:

   \[
   \frac{28}{0.5} = \frac{h}{\frac{\sqrt{2}}{2}}
   \]

   Simplify the left-hand side:

   \[
   56 = \frac{h \cdot 2}{\sqrt{2}}
   \]

   Now, solve for \( h \):

   \[
   56 = \frac{2h}{\sqrt{2}} \quad \Rightarrow \quad 56 \cdot \sqrt{2} = 2h \quad \Rightarrow \quad h = \frac{56 \cdot \sqrt{2}}{2}
   \]

   Simplifying further:

   \[
   h = 28\sqrt{2}
   \]

Thus, the height \( h \) is \( 28\sqrt{2} \, \text{cm} \).

Would you like further explanation or details?

### Related Questions:
1. What is the Law of Sines and how is it applied in triangles?
2. How would you calculate the height if one of the angles was not provided?
3. Can the Law of Cosines be used to solve this problem as well?
4. How does the \( 45^\circ \) angle affect the trigonometric calculations for height?
5. What are the differences in solving for height in a right triangle vs. an oblique triangle?

**Tip:** Remember that the Law of Sines is especially useful in non-right triangles when two angles and one side are known.

Work out the height, h, of a triangle where one angle is 105°, another angle is 45°, and the hypotenuse is 28 cm. Give your answer in its simplest form.

Learn how to calculate the height of a triangle using trigonometric concepts such as the Law of Sines. Given a hypotenuse of 28 cm, an angle of 105°, and a right triangle angle of 45°, this problem provides a step-by-step guide for solving for the height.

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