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We are given a right triangle with angles \( \angle A = 28^\circ \), \( \angle B = 61^\circ \), and the distance from \( A \) to \( D \) (the adjacent side to \( \angle A \)) is 151 units. We need to find the height \( h \), which is opposite to \( \angle C \).

To solve for \( h \), we can use trigonometric ratios. Specifically, the sine function relates the opposite side to the hypotenuse in a right triangle:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

From the diagram, \( \sin(28^\circ) \) will give us the ratio of \( h \) to the hypotenuse \( AD \), but we are first interested in finding the full hypotenuse using the known adjacent side.

Using the tangent function:

\[
\tan(28^\circ) = \frac{h}{151}
\]

Now, solving for \( h \):

\[
h = 151 \times \tan(28^\circ)
\]

Let's calculate the value of \( h \).The height \( h \) is approximately \( 80.3 \) units when rounded to the nearest tenth.

Would you like further clarification or details? Here are some related questions:

1. How can we calculate the length of the hypotenuse \( AD \) using trigonometry?
2. What trigonometric function could be used if we wanted to find the side adjacent to \( \angle B \)?
3. How would the solution change if the angle \( A \) were different?
4. How do we know when to use sine, cosine, or tangent in right triangles?
5. Could we also calculate the height using \( \sin(61^\circ) \)?

**Tip:** Always check whether you are using the correct trigonometric ratio by identifying the side relative to the angle given.

Find h to the nearest tenth, given a right triangle with angles of 28°, 61°, and a base length of 151 units.

Learn how to find the height of a right triangle using trigonometric ratios, given a 28° angle and a base length of 151 units. This problem covers essential trigonometry concepts and is suitable for high school students.

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