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To solve for \( d \) and \( h \) in the two right triangles, we'll use trigonometric ratios based on the given angles and side lengths.

### For the left triangle (with a 60° and 30° angle):
- We know the side adjacent to the 30° angle is 6.
- We are tasked to find \( d \) (the hypotenuse) of the triangle.

Using the cosine of the 30° angle:
\[
\cos(30^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{6}{d}
\]
\[
d = \frac{6}{\cos(30^\circ)} = \frac{6}{\frac{\sqrt{3}}{2}} = \frac{6 \times 2}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3}
\]
Thus, \( d = 4\sqrt{3} \).

### For the right triangle (with a 45° angle):
- The hypotenuse is 5, and we need to find \( h \), the length of one of the legs.

Since it is a 45°-45°-90° triangle, the legs are equal in length, and the ratio of the hypotenuse to a leg in this type of triangle is \( \sqrt{2} \).

\[
h = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2}
\]
Thus, \( h = \frac{5\sqrt{2}}{2} \).

### Final answers:
- \( d = 4\sqrt{3} \)
- \( h = \frac{5\sqrt{2}}{2} \)

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the hypotenuse using the sine function?
2. What are the trigonometric identities for 30° and 45° angles?
3. How does the Pythagorean theorem apply in a 30°-60°-90° triangle?
4. Can the values of \( d \) and \( h \) be verified using another method?
5. What is the relationship between the side lengths in a 45°-45°-90° triangle?

**Tip**: In right triangles, the sine, cosine, and tangent ratios are powerful tools for solving unknown sides when angles and one side are known.

For the right triangles below, find the exact values of the side lengths d and h. If necessary, write your responses in simplified radical form.

Learn how to find the exact side lengths of right triangles using trigonometric ratios and simplified radical form. This problem involves the cosine of 30 degrees and the unique properties of 45°-45°-90° triangles, suitable for students in Grades 9-11.

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