Heights and distances
Heights and Distances
Heights and distances problems are a practical application of trigonometry that involve calculating the heights of objects or distances between points when it is not practical to measure them directly. These problems often involve right-angled triangles and the use of trigonometric ratios: sine (sin), cosine (cos), and tangent (tan).
Understanding the Basics
In a right-angled triangle, the side opposite the right angle is called the hypotenuse, the side opposite the angle of interest is called the opposite side, and the side adjacent to the angle of interest is called the adjacent side.
The basic trigonometric ratios are defined as follows:
- Sine (sin) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Trigonometric Ratios
| Ratio | Definition | Formula |
|---|---|---|
| Sine (sin) | Opposite over Hypotenuse | $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ |
| Cosine (cos) | Adjacent over Hypotenuse | $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ |
| Tangent (tan) | Opposite over Adjacent | $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ |
Solving Heights and Distances Problems
To solve problems involving heights and distances, follow these steps:
- Draw a diagram to represent the scenario.
- Identify the right-angled triangles in the diagram.
- Label the sides of the triangle(s) as opposite, adjacent, or hypotenuse relative to the angle of interest.
- Choose the appropriate trigonometric ratio based on the sides you need to relate.
- Set up an equation using the trigonometric ratio and solve for the unknown side or angle.
Examples
Example 1: Calculating the Height of a Tree
Suppose you want to find the height of a tree. You measure the angle of elevation from a point 30 meters away from the base of the tree and find it to be 45 degrees.
Solution:
- Draw the scenario as a right-angled triangle with the tree as the opposite side and the distance from the tree as the adjacent side.
- The angle of elevation is 45 degrees, so we use the tangent ratio.
- Set up the equation using the tangent of 45 degrees:
$$ \tan(45^\circ) = \frac{\text{Height of the tree}}{30} $$
- Since $\tan(45^\circ) = 1$, the equation simplifies to:
$$ 1 = \frac{\text{Height of the tree}}{30} $$
- Solving for the height of the tree gives:
$$ \text{Height of the tree} = 30 \text{ meters} $$
Example 2: Calculating the Distance to an Object
Imagine you are standing at the top of a cliff that is 50 meters high and you want to find the distance to a boat at sea level, directly out from the base of the cliff. You measure the angle of depression to be 30 degrees.
Solution:
- Draw the scenario as a right-angled triangle with the height of the cliff as the opposite side and the distance to the boat as the adjacent side.
- The angle of depression from the top of the cliff is the same as the angle of elevation from the boat to the top of the cliff, which is 30 degrees.
- Use the tangent ratio:
$$ \tan(30^\circ) = \frac{50}{\text{Distance to the boat}} $$
- Knowing that $\tan(30^\circ) = \frac{\sqrt{3}}{3}$, the equation becomes:
$$ \frac{\sqrt{3}}{3} = \frac{50}{\text{Distance to the boat}} $$
- Solving for the distance to the boat gives:
$$ \text{Distance to the boat} = \frac{50 \times 3}{\sqrt{3}} \approx 86.60 \text{ meters} $$
Important Points to Remember
- Always draw a diagram to visualize the problem.
- Label the sides of the triangle correctly relative to the angle of interest.
- Choose the right trigonometric ratio based on the information given and what you need to find.
- Remember that the angle of elevation is measured upwards from the horizontal, and the angle of depression is measured downwards from the horizontal.
- In problems involving angles of elevation or depression, the angle inside the triangle is often equal to the angle of elevation or depression.
By understanding these principles and practicing various problems, you can master the topic of heights and distances for your exams.