Some Applications of Trigonometry
Use trigonometry to calculate the height of a tall object. The experiment relies on measuring the **angle of elevation** and the **distance** to the object.
Help & Instructions
â–¼The height ($H$) of the object above eye level is found using the tangent ratio:
$$H = \text{Distance} \times \tan(\theta)$$Total Height = $H + \text{Eye Level}$
- **Set Measurements:** Adjust the sliders for the measured **Distance** and **Angle of Elevation**.
- **Calculate:** Click "Calculate Height" to find the height based on your inputs.
- **Verify:** The true height is hidden initially. Try to get your calculated height to match the true height!
Measure the Lighthouse 🗼
Calculation Results (Eye Level = 1.6m)
Opposite Side ($H$): 0.00 m
Calculated Total Height: 0.00 m
**Heights and Distances** problems are direct applications of trigonometry. The line of sight, the vertical object, and the horizontal ground form a **right-angled triangle**. We use the **tangent** ratio ($\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$) because it relates the angle of elevation to the two measurable sides.
The Mathematics Behind the Measurement
The total height ($H_{total}$) is calculated as:
$$H_{total} = \text{Eye Level} + \left(\text{Distance} \times \tan(\theta)\right)$$This relies on the fact that $\tan(\theta) = \frac{\text{Opposite Side (H)}}{\text{Adjacent Side (Distance)}}$.
This method (trigonometric leveling) is used by surveyors in:
- **Mapping and Surveying:** Determining elevations and heights of large structures (e.g., towers, mountains).
- **Construction:** Laying out buildings and checking verticality.
- **Navigation:** Calculating distances to visible landmarks.
