Height and Distance: Complete Guide for Students

Height and distance is a fundamental topic in trigonometry that addresses the quantitative relationships between the vertical height of objects and their horizontal separation, utilizing right triangle geometry and primary trigonometric ratios to navigate solutions where direct measurement is impractical or impossible.

Geometric Structure of Height and Distance Problems

In every height and distance scenario, the arrangement of the observer, the object, and the reference axis forms a right-angled triangle wherein the vertical segment represents the height, the horizontal segment denotes the distance, and the direct line from the observer’s eye to the object constitutes the line of sight, acting as the hypotenuse.

Let $AB$ represent a vertical object of unknown height, $P$ a point on the horizontal ground at a known distance from the foot $B$ of the object, and let $AP$ be the line of sight inclined at angle $\theta$ with respect to the horizontal. In $\triangle ABP$, $\angle ABP = \theta$, $BP$ is the known distance, $AB$ is the required height, and $AP$ is the hypotenuse.

Mathematical Formulation Using Right Triangle Trigonometry

The fundamental trigonometric ratios used in height and distance problems are defined for a right triangle as follows:

$\sin\theta = \dfrac{\text{Opposite Side}}{\text{Hypotenuse}}$

$\cos\theta = \dfrac{\text{Adjacent Side}}{\text{Hypotenuse}}$

$\tan\theta = \dfrac{\text{Opposite Side}}{\text{Adjacent Side}}$

Within the context of height and distance, the opposite side often represents the unknown or known height, the adjacent side the known or unknown horizontal distance, and the hypotenuse the line of sight. The ratio selected depends upon which quantities are given and which is to be determined.

Interpretation of Angle of Elevation and Angle of Depression

The angle of elevation is defined as the angle formed between the horizontal and the line of sight when the observer looks upward toward an object. Mathematically, if an observer at $P$ looks at the top of object $A$ vertically above ground $B$, the angle $\theta = \angle BPA$ is the angle of elevation.

The angle of depression is defined as the angle formed between the horizontal and the line of sight when the observer looks downward at an object situated below the horizontal level of the observer's eye. If the observer is at point $A$ (above ground), the line joining $A$ parallel to the ground is the horizontal, and the line $AP$ to point $P$ lower than $A$ makes angle $\theta = \angle PAH$ (where $H$ is the projection on the horizontal), then $\theta$ is the angle of depression.

Mathematical Representation in Terms of Trigonometric Ratios

To solve for unknown height when distance and angle of elevation are known, use:

$\tan\theta = \dfrac{\text{Height}}{\text{Distance}}$

More explicitly, for the right triangle with vertical side $AB$ (height), base $BP$ (distance), and angle of elevation $\theta$ at point $P$:

$\tan\theta = \dfrac{AB}{BP}$

From which, $AB = BP \cdot \tan\theta$.

Algebraic Derivation Using Pythagoras’ Theorem

The Pythagorean theorem applies to all right-angled triangles, stating that the sum of the squares of the non-hypotenuse sides equals the square of the hypotenuse:

$\left(\text{Hypotenuse}\right)^2 = (\text{Base})^2 + (\text{Height})^2$

Given hypotenuse $AP$ (line of sight), base $BP$ (distance), and height $AB$:

$AP^2 = BP^2 + AB^2$

This relationship allows determination of any one side when the other two are known; for instance, if $AB$ and $BP$ are known, $AP = \sqrt{AB^2 + BP^2}$.

Analysis of Multiple Observation Scenarios

When observations are made from two distinct points at varying distances from the base of the object, and the respective angles of elevation are known, the height or unknown distance can be determined by constructing two independent trigonometric relations and solving the resulting system. Let points $P_1$ and $P_2$ be at distances $d_1$ and $d_2$ from foot $B$ of object $AB$ with angles of elevation $\theta_1$ and $\theta_2$. Then:

$\tan\theta_1 = \dfrac{AB}{d_1}$

$\tan\theta_2 = \dfrac{AB}{d_2}$

If $d_2 = d_1 + x$, $AB$ can be isolated by equating both forms and solving for $x$ or $AB$ as required.

Structural Explanation of Similar Triangles in Height and Distance

All height and distance problems are underpinned by properties of similar right-angled triangles. When two or more right triangles share a common angle, their sides are in proportion. If two objects cast shadows and the angles of elevation of the sun for both cases are identical, the triangles are similar, giving:

$\dfrac{\text{Height}_1}{\text{Shadow}_1} = \dfrac{\text{Height}_2}{\text{Shadow}_2}$

Condition and Limitation for Application of Trigonometric Ratios

These trigonometric methodologies presuppose that the horizontal distance is measured along a perfectly flat ground and that the vertical object is normal (perpendicular) to this ground. The geometry is strictly valid when the triangle so formed is right-angled and the angle of elevation or depression is accurately measured at ground level or a stated reference point. If the observer’s eye is at a positive height above the ground, this contribution must be explicitly added or subtracted in the corresponding triangle calculation.

Worked Example: Determination of Height with Angle of Elevation

Given: An observer stands at a distance of $20\ \mathrm{m}$ from a tower. The angle of elevation of the top of the tower from this point is $60^\circ$.

Substitution: Let the height of the tower be $h$ and the base distance be $20\ \mathrm{m}$. Then:

$\tan 60^\circ = \dfrac{h}{20}$

$\Rightarrow \sqrt{3} = \dfrac{h}{20}$

Simplification: Multiply both sides by $20$:

$h = 20 \sqrt{3}$

Final result: The height of the tower is $20\sqrt{3}\ \mathrm{m}$.

Worked Example: Finding Distance Using Angle of Depression

Given: The angle of depression from the top of a lighthouse $30\ \mathrm{m}$ high to a boat is $45^\circ$.

Substitution: Let the distance between the base of the lighthouse and the boat be $d$.

$\tan 45^\circ = \dfrac{30}{d}$

$1 = \dfrac{30}{d}$

Simplification: $d = 30$

Final result: The horizontal distance from the lighthouse to the boat is $30\ \mathrm{m}$.

Special Derivation: Height from Two Directly Opposite Angles of Elevation

Suppose from two points on the same horizontal line at distances $x$ and $y$ from the base of the tower, the respective angles of elevation to the top are complementary, that is, $\theta$ and $90^\circ-\theta$. Let the height of the tower be $h$.

From the first position:

$\tan\theta = \dfrac{h}{x}$

$\Rightarrow h = x\,\tan\theta$

From the second position:

$\tan(90^\circ-\theta) = \cot\theta = \dfrac{h}{y}$

$\Rightarrow h = y\,\cot\theta$

Since both expressions represent the same height, equate:

$x\,\tan\theta = y\,\cot\theta$

$\Rightarrow x\,\tan\theta = y\,\dfrac{1}{\tan\theta}$

$\Rightarrow x (\tan\theta)^2 = y$

$\Rightarrow (\tan\theta)^2 = \dfrac{y}{x}$

Take square roots of both sides:

$\tan\theta = \sqrt{\dfrac{y}{x}}$

Return to the original expression: $h = x \tan\theta$

Substitute for $\tan\theta$:

$h = x \sqrt{\dfrac{y}{x}} = \sqrt{xy}$

Result: The height of the tower is $\sqrt{xy}$, where $x$ and $y$ are the distances from the two observation points with complementary angles of elevation.

Principle of Proportionality in Similar Shadow Problems

Consider two objects casting shadows under identical solar elevation. If the heights are $h_1$ and $h_2$, and the lengths of their shadows are $s_1$ and $s_2$:

$\tan\theta = \dfrac{h_1}{s_1} = \dfrac{h_2}{s_2}$

Thus, $h_1 : s_1 = h_2 : s_2$ and an unknown shadow or height can be found using proportionality as described in the Height And Distance concept page.

Practical Application Constraints and Adjustments

If the observer’s eye level is not at ground height but at elevation $h_0$, then in computing the height of an object, one must add or subtract $h_0$ to obtain the total or visible height, respectively. This adjustment reflects the non-ideal initial reference and may appear, for example, in problems involving the observer’s line of sight beginning above ground level, as is routinely encountered in standardized examinations and practical surveying.

Additional relations involving cotangent differences and multiple observation points arise in more advanced problems. For instance, if the angles of elevation from two points at distances $d_1$ and $d_2$ from the base are $\alpha$ and $\beta$, the height $h$ can be expressed as $h = \dfrac{d_2\,\tan\alpha - d_1\,\tan\beta}{\tan\alpha - \tan\beta}$, derived by setting up and solving the system of tangent equations for $h$. Full derivation of such expressions must be performed by explicitly expressing each triangle’s tangent ratio, solving both equations for the unknown, and equating or eliminating as required.