## Math 9 Chapter 1 Lesson 5: Practical applications of trigonometric ratios of acute angles. Outdoor practice

## 1. Summary of theory

### 1.1. Determine the height

**a) Duties: **Determine the height of a tower without going to the top of the tower

**b) Preparation: **Protractor, tape measure, pocket calculator

**c) Implementation instructions:**

**– **Place the angle meter vertically away from the tower \(a\), the height of the gauge is \(b\).

– Rotate the angle bar so that when looking along this bar we can see the top of the tower. Read the measure of the angle on the trigonometer as \(\alpha\).

– Use the calculator \(tan\alpha\). Then we have the height of the tower: \(b+a.tan\alpha\)

### 1.2. Determine the distance

**a) Duties: **Determine the width of a river section that is measured only at one bank of the river

**b) Preparation: **Ekemeter, trigonometry, tape measure, calculator or trigonometric table

**c) Implementation instructions:**

-Choose a point across the river close to the river bank set as B, take a point on this side of the river close to the bank.

-Draw a line on this side of the river so that it is perpendicular to AB. take a point C on the perpendicular line just drawn

– The segment \(AC=a\) uses a protractor to measure\(\widehat{ACB}=\alpha\) . Then the width of the river section is the value of: \(a.tan\alpha\)

## 2. Illustrated exercise

### 2.1. Form 1: Determine the height of the object

**Question 1. **Calculate the height of a green tree knowing that a person 1.7m tall standing looking up at the top of the tree, the direction of view is 35 degrees with the ground and the distance from that person to the tree is 20m.

**Solution guide**

We see the problem as the picture above \(\widehat{ABC}=90^{\circ}\)

Then the tree height to be calculated is the following: \(CF=CB+BF=AB.tan35^{\circ}+AE=20.tan35^{\circ}+1.7\simeq 15.7 (m)\ )

**Verse 2. **A tree is struck by lightning in the middle of the trunk causing the trunk to fall to the ground, making an angle with the ground of \(40^{\circ}\). Know that the tree trunk is still standing 3m high.

Calculate the initial height of the tree

**Solution guide**

We consider the problem as shown with \(\widehat{ABC}=90^{\circ}\)

Then the initial tree length is: \(BC+AC=BC+\frac{BC}{sinA}=3.(1+\frac{1}{sin40^{\circ}})\simeq 7.67 (m)\)

**Verse 3.** A 6m long double ladder is used to climb a roof. Know that when climbing each foot of the ladder makes an angle of 60 degrees with the ground. Calculate the height of that house

**Solution guide**

We see the topic as the picture above

Then we have regular \(\Delta ABC\) and \(CD=AC.sin60^{\circ}=6.\frac{\sqrt{3}}{2}=3\sqrt{3}\)

### 2.2. Form 2: Determine the distance of the object

A lighthouse observer 88 feet above sea level looks at the distant ship at an angle \(0{}^\circ 42’\). What is the distance from the ship to the foot of the lighthouse in nautical miles?

**Solution guide**

The height of the lighthouse is the right angle opposite the angle \(0{}^\circ 42’\), the distance from the ship to the foot of the lighthouse is the side adjacent to the acute angle.

So the distance from the boat to the foot of the lighthouse is:

\(80.\cot 0{}^\circ 42’\approx 6547.76\,(feet)\approx 1.24\) (nautical miles).

## 3. Practice

### 3.1. Essay exercises

**Question 1. **Ladder AB 6m long leaning against the wall makes an angle \(63{}^\circ \) with the ground. What is the height of the ladder reached above the ground?

**Verse 2.** Make a flag zipper: Find the length of the flag’s zipper, knowing that the flagpole’s shadow (lighted by sunlight) is 9m long and the angle of view of the sun is \(41{}^\circ 38’\).

**Verse 3. **The Toronto Observatory, Ontario, Canada is 533m high. At a certain time during the day, the sun shines with a shadow of 1000m long. What is the angle made by the sun’s rays and the ground at that time?

### 3.2. Multiple choice exercises

**Question 1. **A lamp post is 5m high. At one point the sun’s rays make an angle of 60 degrees with the ground. How long is the shadow of that lamp post on the ground?

A. \(\frac{5}{\sqrt{2}}\)

B. \(\frac{5}{\sqrt{3}}\)

C. \(\frac{5}{2}\)

D. \(\frac{10}{\sqrt{2}}\)

**Verse 2. **A building at a time when the sun’s rays make an angle of 50 degrees with the ground, the shadow of the building on the ground is 7m long.

The height of the building is:

A. \(\simeq 4.5\)

B. \(\simeq 5.36\)

C. \(\simeq 5.87\)

D. \(\simeq 8.34\)

**Verse 3. **Let ABC be a right triangle at B with BC=20, D is a point on side AB such that \(\widehat{BCD}=50^{\circ}, \widehat{DCA}=15^{\circ}\)

AD length is:

A. \(\simeq 20.78\)

B. \(\simeq 2.805\)

C. \(\simeq 19.05\)

D. \(\simeq 21\)

**Verse 4. **A boat crosses a river. Due to the water flowing, the direction of the boat is at an angle of 30 degrees compared to the direction of going straight across the other shore. Assume that the speed of the boat is 3m/s and that the boat travels in 3 minutes. What is the length of the river?

A. \(270\sqrt{3}\)

B. \(540\)

C. \(270\sqrt{2}\)

D. \(540\sqrt{3}\)

**Question 5.** A person uses a long ladder to climb a building. Know that the shadow of the building when the sun’s rays make an angle of 45 degrees with the ground is 3m and the distance from the bottom of the ladder to the house is 1.5m. Calculate the angle formed by the ladder and the ground

A. \(48^{\circ}14{}’\)

B. \(72^{\circ}14{}’\)

C. \(26^{\circ}14{}’\)

D. \(63^{\circ}26{}’\)

## 4. Conclusion

Through this lesson, students should meet the following requirements:

- Master the relations for sides and angles in right triangles.
- Apply knowledge to solve problems about right triangles.

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