angle of elevation

Question

The angle of elevation from a point on level ground to the top of a 50-ft tree is 25°. How much closer to the tree does a point on the ground need to be for the measure of the angle of elevation to be twice the measure of the original triangle?

Answer 1

Let’s dive into this classic trigonometry scenario just like a friendly tutoring session! The goal is to give every learner clarity, confidence, and a spark of curiosity.

The Problem Recap: Detective Math in the Field

Imagine you’re standing in a sunny field staring up at a towering 50-foot tree. From where you stand, the angle of elevation—the tilt from your line of sight on the ground to the top of the tree—is 25°. The question is: How much closer should you move so the angle you see doubles?

Step 1: Visualizing with a Triangle

Whenever you see a problem about heights and angles, there’s a right triangle hiding in there! In this case:

- The height of the tree is the “opposite” side.

- Your distance from the tree is the “adjacent” side.

- The line from your eyes to the top of the tree is the “hypotenuse.”

- The angle between the ground and your gaze is the angle of elevation.

When height, distance, and angles appear in questions, trigonometry is the perfect tool to break the problem open. Tutors often emphasize this because it’s so useful and reliable.

Step 2: Why Trigonometry Works Here

Trigonometry helps us use two known measurements (like an angle and a distance) to uncover a third unknown side or angle. Since we have a right triangle with an angle and a couple of sides, trig ratios give us the answer.

Step 3: Choosing the Tangent Function

Which trig ratio to use? Tangent is the right fit because it relates the vertical height (“opposite”) and horizontal distance (“adjacent”) without needing the hypotenuse.

The formula:

$$

\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

$$

For the tree:

- Opposite side = 50 ft (height)

- Adjacent side = distance from tree

Step 4: Set Up the Two Distances

Let:

- $$x_1$$ = distance from tree at 25°

- $$x_2$$ = distance from tree at 50° (twice the angle)

Write these relationships:

$$

\tan(25^\circ) = \frac{50}{x_1}

$$

$$

\tan(50^\circ) = \frac{50}{x_2}

$$

### Step 5: Calculate the Distances

1. Original distance:

$$

x_1 = \frac{50}{\tan(25^\circ)} \approx \frac{50}{0.4663} \approx 107.3 \text{ ft}

$$

2. New distance:

$$

x_2 = \frac{50}{\tan(50^\circ)} \approx \frac{50}{1.1918} \approx 41.95 \text{ ft}

$$

3. Distance moved closer:

$$

107.3 - 41.95 \approx 65.35 \text{ ft}

$$

### Visualize the Triangle

Picture or draw:

```

/|

/ |

50 / | x (distance from tree)

/ |

/____|

angle

```

As the angle goes from 25° to 50°, the “adjacent” side shrinks, meaning you get closer.

Step 6: How to Spot Problems Like This

Look for keywords like “angle of elevation,” “height,” or “distance along the ground.” If two sides (height and ground distance) and an angle are part of the info, tangent is usually your go-to ratio. Drawing the triangle helps you visualize which sides are opposite and adjacent.

Why a Tutors Makes a Difference

A tutor helps:

- Spot the right triangle and identify sides, even if the problem looks complex.

- Provide visual aids and memory tools (like SOH-CAH-TOA).

- Build your confidence by walking you through the thinking and calculations step-by-step.

- Make abstract trig formulas something real and easy to understand.

***

Quick Reference Table

| Angle | Distance from Tree | Calculation |

|-------|--------------------|-------------------|

| 25° | ~107.3 ft | $$50 / \tan 25^\circ$$ |

| 50° | ~41.95 ft | $$50 / \tan 50^\circ$$ |

| Move | ~65.35 ft closer | $$107.3 - 41.95$$ |

One-Minute Summary

For any word problem involving right triangles, angles, and two known sides, turn to trigonometry to find the unknown. If the hypotenuse isn’t mentioned but height and ground distance are, tangent is your best friend. Always start by drawing a triangle to visualize the problem. A helpful tutor will make this process enjoyable, clear, and rewarding.

Thanks for your curiosity about trig—it’s a superpower worth mastering! Each problem solved builds skills that you’ll use again and again.

Sources

[1] [PDF] Word Problems Angles of Elevation and Depression https://coachgriffin.weebly.com/uploads/2/4/6/5/24656764/day_3_-_word_problem_hw_-_key.pdf

[2] Angle of Elevation and Depression Word Problems Trigonometry ... https://www.youtube.com/watch?v=uyKvSe6Ltgs

[3] Trigonometry - Word Problems - MathBitsNotebook(Geo) https://mathbitsnotebook.com/Geometry/Trigonometry/TGElevDepress.html

[4] angle of elevation and depression word problem - trigonometry https://www.youtube.com/watch?v=UHGD6sDZIg4

[5] How to Find Angles of Elevation in a Word Problem Using ... https://study.com/skill/learn/how-to-find-angles-of-elevation-in-a-word-problem-using-trigonometry-explanation.html

[6] Angle of Elevation - Formula | Angle of Depression - Cuemath https://www.cuemath.com/trigonometry/angle-of-elevation/

[7] Word Problems- Angle of Depression and Elevation - Andymath.com https://andymath.com/angle-of-depression-and-elevation/

[8] [Grade 9 Trigonometry] Angle of elevation worded question - Reddit https://www.reddit.com/r/HomeworkHelp/comments/139ik4p/grade_9_trigonometry_angle_of_elevation_worded/