Geometry Trig Unit: Word Problems on Angles of Elevation and Depression with step-by-step solutions.

You've uploaded a worksheet with four trigonometry word problems involving angles of elevation and depression, and you're asking to solve and explain the solution.

Let’s go through each problem one by one, verify the given solutions, and provide clear explanations.

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## ✔ Problem 1:
> A woman is standing on the ground at a point 78 ft from the base of a building. The angle of elevation to the top of the building is 57°. To the nearest foot, how high is the building?

📐 Diagram:

Right triangle:
- Adjacent side (ground distance) = 78 ft
- Angle of elevation = 57°
- Opposite side = height of building = `h`

🔍 Solution:

We use tangent, since we have opposite and adjacent:

```
tan(57°) = h / 78
→ h = 78 × tan(57°)
```

Using calculator:
`tan(57°) ≈ 1.53986`
→ `h ≈ 78 × 1.53986 ≈ 120.109`

✔ Rounded to nearest foot: `h ≈ 120 ft`

✔️ Your answer is correct.

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## ✔ Problem 2:
> The sun shines on a flagpole, causing a shadow to be cast on the ground. The distance from the base of the pole to the tip of the shadow is 49 feet. At that time of day, the sun’s rays make an angle of 38° with the ground. How tall is the flagpole?

📐 Diagram:

Right triangle:
- Adjacent = shadow length = 49 ft
- Angle = 38° (angle between sun ray and ground)
- Opposite = height of flagpole = `h`

🔍 Solution:

Again, use tangent:

```
tan(38°) = h / 49
→ h = 49 × tan(38°)
```

`tan(38°) ≈ 0.78129`
→ `h ≈ 49 × 0.78129 ≈ 38.283`

✔ Rounded to nearest tenth: `h ≈ 38.3 ft`

✔️ Your answer is correct.

---

## ✔ Problem 3:
> An observer on a cliff 1000 dm above sea level sights two ships due east. The angles of depression of the ships are 47° and 32°. Find, to the nearest decimeter, the distance between the two ships.

📐 Diagram:

Observer at top of cliff (height = 1000 dm). Two lines of sight down to ships at angles of depression 47° and 32°. Since the observer is looking *down*, the angles of depression equal the angles of elevation from the ships up to the observer (alternate interior angles).

So we form two right triangles sharing the same vertical leg (1000 dm), but different horizontal legs (`x` and `y`) to each ship.

Let:
- `x` = horizontal distance to nearer ship (angle 47°)
- `y` = horizontal distance to farther ship (angle 32°)

Then, distance between ships = `y - x`

🔍 Solution:

For ship at 47°:

```
tan(47°) = 1000 / x
→ x = 1000 / tan(47°)
```

For ship at 32°:

```
tan(32°) = 1000 / y
→ y = 1000 / tan(32°)
```

Calculate:

- `tan(47°) ≈ 1.07237` → `x ≈ 1000 / 1.07237 ≈ 932.5 dm`
- `tan(32°) ≈ 0.62487` → `y ≈ 1000 / 0.62487 ≈ 1600.5 dm`

Distance between ships = `y - x ≈ 1600.5 - 932.5 = 668.0 dm`

✔ Rounded to nearest decimeter: `668 dm`

✔️ Your answer is correct.

*(Note: You wrote “ans: 668 dm” — perfect!)*

---

## ✔ Problem 4:
> A 200 ft high television transmitting tower is to be supported by guy wires running from the ground to the top of the tower. The wires make an angle of 63° with the ground.

Part (a): How long is each wire?

📐 Diagram:

Right triangle:
- Opposite side = tower height = 200 ft
- Angle with ground = 63°
- Hypotenuse = wire length = `w`

🔍 Solution:

Use sine (opposite/hypotenuse):

```
sin(63°) = 200 / w
→ w = 200 / sin(63°)
```

`sin(63°) ≈ 0.89101`
→ `w ≈ 200 / 0.89101 ≈ 224.46 ft`

✔ Rounded to nearest tenth: `w ≈ 224.5 ft`

✔️ Your answer is correct.

---

Part (b): How far from the base of the tower must they meet the ground?

📐 Diagram:

Same triangle:
- Adjacent side = distance from base = `b`
- Angle = 63°
- Opposite = 200 ft

🔍 Solution:

Use tangent (opposite/adjacent):

```
tan(63°) = 200 / b
→ b = 200 / tan(63°)
```

`tan(63°) ≈ 1.96261`
→ `b ≈ 200 / 1.96261 ≈ 101.90 ft`

✔ Rounded to nearest tenth: `b ≈ 101.9 ft`

✔️ Your answer is correct.

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## ✔ Final Summary:

| Problem | Answer | Correct? |
|--------|--------|--------|
| 1 | 120 ft | ✔ Yes |
| 2 | 38.3 ft | ✔ Yes |
| 3 | 668 dm | ✔ Yes |
| 4a | 224.5 ft | ✔ Yes |
| 4b | 101.9 ft | ✔ Yes |

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## 🧠 Key Concepts Used:
- Angle of Elevation/Depression: Angles measured from horizontal.
- Trigonometric Ratios:
- `tan(θ) = opposite / adjacent` → used when you know or want a side next to the angle.
- `sin(θ) = opposite / hypotenuse` → used when you know or want the hypotenuse.
- `cos(θ) = adjacent / hypotenuse` → not needed here.
- Angles of Depression: Equal to angles of elevation from the object’s perspective (due to parallel lines & transversal).

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## 💡 Tip for Future Problems:
Always sketch the triangle, label known values, and ask:
> “Which side do I know? Which side do I need? What ratio connects them?”

Then pick sine, cosine, or tangent accordingly.

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✔ All your answers are mathematically correct and well-explained in your work! Great job!