https://file.aiquickdraw.com/mathbot/file/1726604897375sjkso43k.jpg

Let's break down the two problems from the image:

### Problem 1: Angles of a Right Triangle
We are given the lengths of three sticks: 0.47 m, 0.62 m, and 0.78 m, forming a right triangle. We need to find the angles of the triangle.

The angles of a right triangle can be found using trigonometric ratios. Since it's a right triangle, one angle is \(90^\circ\). For the other two angles, we can use the sine, cosine, or tangent functions.

1. Let the sides be as follows:
   - \(a = 0.47\) m
   - \(b = 0.62\) m
   - \(c = 0.78\) m (hypotenuse)

2. The angles \(\theta_1\) and \(\theta_2\) can be found using:
   \[
   \theta_1 = \arcsin\left(\frac{a}{c}\right), \quad \theta_2 = \arcsin\left(\frac{b}{c}\right)
   \]
   or alternatively using the tangent function:
   \[
   \theta_1 = \arctan\left(\frac{a}{b}\right), \quad \theta_2 = 90^\circ - \theta_1
   \]

### Problem 2: Ladder Problem
We are given a ladder of length 2.5 meters leaning against a wall at an angle of 32 degrees with the wall. The height above the floor where the ladder touches the wall is \(h\), and we need to calculate \(h\).

Using the cosine function for the right triangle formed by the ladder, wall, and ground:
\[
\cos(\theta) = \frac{h}{\text{hypotenuse}} = \frac{h}{2.5}
\]
\[
h = 2.5 \times \cos(32^\circ)
\]

Let's calculate both problems.### Problem 1: Angles of the Right Triangle
The three angles of the triangle are:
- \( 90^\circ \) (the right angle),
- \( 37^\circ \) (rounded from 37.16),
- \( 53^\circ \) (rounded from 52.84).

So, the correct answer is:
- **\(90^\circ, 53^\circ, 37^\circ\)**.

### Problem 2: Ladder Problem
The height \( h \) above the floor where the ladder makes contact with the wall is approximately:
- **2.12 meters**.

So, the correct answer for the second problem is **2.1 m**.

---

Do you want further details or have any questions? Here are five related questions to explore:

1. How would you calculate the angles of a triangle if only two sides were given?
2. What other trigonometric functions could be used to solve right triangle problems?
3. How does changing the angle of the ladder affect the height?
4. What is the relationship between the sine and cosine of complementary angles?
5. How can you calculate the area of a right triangle using trigonometry?

**Tip:** Always ensure you are working with the correct trigonometric ratios when solving right triangles to avoid errors.

Three sticks are arranged to form a right triangle. If the lengths of the three sticks are 0.47 m, 0.62 m, and 0.78 m, what are the three angles of the triangle? A 2.5 m ladder leans against a wall and makes an angle with the wall of 32 degrees. What is the height above the floor where the ladder makes contact with the wall?

Learn how to calculate the angles of a right triangle with sides 0.47 m, 0.62 m, and 0.78 m using trigonometric functions. Also, find the height of a ladder leaning at a 32-degree angle. Explore trigonometric formulas and theorems, suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation