Applications of Trigonometry - Theory & Concepts

Applications of Trigonometry

Trigonometry is indispensable in fields like surveying, navigation, construction, and engineering. It provides the mathematical framework to calculate unknown distances and angles based on a few known measurements, bridging the gap between theoretical geometry and real-world physical space.

Solving Right Triangles

When dealing with right triangles (where one angle is exactly $90^\circ$ ), we primarily rely on the Pythagorean Theorem ( $a^2 + b^2 = c^2$ ) and the fundamental trigonometric ratios (SOH CAH TOA).

Angle of Elevation and Depression

Angle of Elevation: The angle measured upward from the horizontal line of sight to an object situated above the observer.
Angle of Depression: The angle measured downward from the horizontal line of sight to an object situated below the observer.

Note: Because horizontal lines of sight at different elevations are parallel to each other, the angle of elevation from observer A up to object B is geometrically equal to the angle of depression from object B down to observer A (they form alternate interior angles).

Explore angles of elevation and depression in the interactive scene below — adjust height and distance to solve the resulting right triangle.

Angle of Elevation & Depression

Explore real-world right triangle applications by adjusting heights and observer distances.

Height (

h

)50 m

10 m55 m100 m

Distance (

d

)80 m

20 m85 m150 m

Triangle Results

Angle (

\theta

)

32.01°

0.5586 rad

Line of Sight (

L

)

94.34 m

Hypotenuse

Step-by-Step Math

1. Find the Angle:

\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{d} = \frac{50}{80} = 0.6250

\theta = \arctan(0.6250) \approx 32.01^\circ

2. Find Line of Sight (L):

L = \sqrt{h^2 + d^2} = \sqrt{50^2 + 80^2}

L = \sqrt{8900} \approx 94.34\text{ m}

Angle of Elevation (from Ground): 32.0°

Angle of Depression (from Top): 32.0°

Key Takeaways

Right Triangles: Use SOH CAH TOA and the Pythagorean theorem exclusively for right triangles.
Visualization: Always sketch and meticulously label a diagram for word problems. A good drawing is half the solution.

Bearings and Navigation

In navigation, direction is often given as a bearing.

Bearing Notation

Compass Bearing: Measured from North or South towards East or West.
- Example: $\text{N } 30^\circ \text{ E}$ means start facing North, then turn $30^\circ$ towards East.
- Example: $\text{S } 45^\circ \text{ W}$ means start facing South, then turn $45^\circ$ towards West.
True Bearing (Azimuth): Measured clockwise from North ( $0^\circ$ $0^{\circ}$ to $360^\circ$ $36 0^{\circ}$ ).
- Example: $135^\circ$ is Southeast.

Key Takeaways

Bearings: Always establish a clear North-South-East-West coordinate cross at your starting point. Pay close attention to the base reference (N or S) and the direction of the turn (E or W).

Solving Oblique Triangles

An oblique triangle is any triangle that is not a right triangle. To solve these, we use the Law of Sines, Law of Cosines, and occasionally the Law of Tangents.

Law of Sines

Use when you know ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or SSA (Side-Side-Angle, the Ambiguous Case).

Law of Sines

Relates the lengths of the sides of a triangle to the sines of its angles.

Variables

Symbol	Description	Unit
$a, b, c$	Lengths of the sides	m
$A, B, C$	Angles opposite to the respective sides	$^\circ$

Interactively solve oblique triangles using the Law of Sines below — select your known information and observe the solution steps.

The Ambiguous Case (SSA)

When given two sides and an angle opposite one of them (SSA), zero, one, or two triangles may exist.

If side opposite given angle $\lt$ height ( $h = b \sin A$ ), 0 triangles.
If side opposite = height, 1 right triangle.
If side opposite $\gt$ height but $\lt$ other side, 2 triangles (one acute, one obtuse).
If side opposite $\ge$ other side, 1 triangle.

Law of Cosines

Use when you know SAS (Side-Angle-Side) or SSS (Side-Side-Side).

Standard Form

Law of Cosines (Standard Form)

Relates the lengths of the sides of a triangle to the cosine of one of its angles.

Variables

Symbol	Description	Unit
$a, b, c$	Lengths of the sides	m
$C$	Angle opposite side c	$^\circ$

Visualize and solve oblique triangles with the Law of Cosines interactively — drag side lengths and angles to see the calculations update.

Law of Cosines

Alternative Forms

Law of Cosines (Alternative Forms)

Other forms for solving sides a or b.

Variables

Symbol	Description	Unit
$a, b, c$	Lengths of the sides	m
$A, B$	Angles opposite the respective sides	$^\circ$

Law of Cosines

Solved for Angle

Law of Cosines (Angle Form)

Rearranged to solve directly for the cosine of an angle.

Variables

Symbol	Description	Unit
$a, b, c$	Lengths of the sides	m
$C$	Angle opposite side c	$^\circ$

Law of Tangents

A lesser-known law, historically useful for calculations using logarithmic tables, relates the tangents of sums and differences of angles to the sums and differences of side lengths.

Law of Tangents

Relates the tangents of the sum and differences of two angles to their opposite sides.

Variables

Symbol	Description	Unit
$a, b$	Lengths of two sides	m
$A, B$	Angles opposite sides a and b	$^\circ$

Law of Cotangents

Similar to the Law of Sines, Cosines, and Tangents, the Law of Cotangents relates the sides of a triangle to the cotangents of its half-angles. It is particularly useful when dealing with the inscribed circle (incircle) of a triangle.

Let $s = \frac{a+b+c}{2}$ be the semi-perimeter, and $r$ be the radius of the incircle.

Law of Cotangents

Relates the sides of a triangle to the cotangents of its half-angles and the inscribed circle radius.

Variables

Symbol	Description	Unit
$a, b, c$	Lengths of the sides	m
$A, B, C$	Angles of the triangle	$^\circ$
$s$	Semi-perimeter (\frac{a+b+c}{2})	m
$r$	Radius of the inscribed circle	m

Mollweide's Equations

Mollweide's equations relate all six parts of a triangle (three sides and three angles) simultaneously. Because they use every part of the triangle, they are historically used to check solutions for oblique triangles. If you solve a triangle and the values do not satisfy Mollweide's equations, a mistake was made.

Mollweide's Equations

Two equations incorporating all six parts of a triangle to verify calculated oblique triangle solutions.

Variables

Symbol	Description	Unit
$a, b, c$	Lengths of the sides	m
$A, B, C$	Angles of the triangle	$^\circ$

Area of a Triangle

SAS Area Formula

Triangle Area (SAS)

Calculates the area of a triangle given two sides and the included angle.

Variables

Symbol	Description	Unit
$\text{Area}$	Area of the triangle	$m^{2}$
$a, b$	Lengths of two sides	m
$C$	Included angle	$^\circ$

Area of a Triangle

Heron's Formula (SSS)

Heron's Formula

Calculates the area of a triangle when only the three side lengths are known.

Variables

Symbol	Description	Unit
$\text{Area}$	Area of the triangle	$m^{2}$
$a, b, c$	Lengths of the three sides	m
$s$	Semi-perimeter (s = \frac{a+b+c}{2})	m

Area of a Triangle

Brahmagupta's Formula (Cyclic Quadrilaterals)

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle. Its area can be found using a generalization of Heron's formula:

Brahmagupta's Formula

Calculates the area of a cyclic quadrilateral.

Variables

Symbol	Description	Unit
$\text{Area}$	Area of the cyclic quadrilateral	$m^{2}$
$a, b, c, d$	Lengths of the four sides	m
$s$	Semi-perimeter (s = \frac{a+b+c+d}{2})	m

Area of a Triangle

where $s = \frac{a+b+c+d}{2}$ . Notice that if side $d=0$ , the quadrilateral becomes a triangle, and Brahmagupta's formula reduces exactly to Heron's formula.

Triangulation and Trilateration

In civil engineering and surveying, the principles of oblique triangles are applied through two primary methods to determine coordinates of points over large areas:

Triangulation: This method relies heavily on the Law of Sines. Surveyors measure a single baseline distance between two points accurately, and then measure only angles to other points. By creating a network of triangles and using the Law of Sines, they can calculate all other distances without needing to measure them physically.
Trilateration: This method relies heavily on the Law of Cosines. Instead of measuring angles, surveyors measure the distances between points (often using electronic distance measuring or GPS). Once three sides (SSS) of a triangle are known, the internal angles can be found using the Law of Cosines.

Modern GPS systems fundamentally use a 3D form of trilateration (measuring distances from satellites) to determine your exact position on Earth.

Key Takeaways

Law of Sines: Ideal for solving triangles when you know ASA or AAS. Exercise extreme caution with SSA, as it may result in zero, one, or two valid triangles (the Ambiguous Case).
Law of Cosines: The go-to tool for SAS or SSS scenarios. It acts as a generalized form of the Pythagorean theorem for oblique triangles.
Heron's Formula: Perfect for determining the area of a triangle when only the three side lengths (SSS) are known.

Vectors and Physics Applications

Trigonometry is heavily used in physics to decompose vectors into orthogonal components.

Vector Resolution

Any 2D vector $\vec{V}$ with magnitude $V$ and angle $\theta$ (measured from the positive x-axis) can be resolved into horizontal ( $V_x$ ) and vertical ( $V_y$ ) components:

Vector Components

Resolves a vector into orthogonal horizontal and vertical components.

Variables

Symbol	Description	Unit
$V_{x}$	Horizontal component	-
$V_{y}$	Vertical component	-
$V$	Magnitude of the vector	-
$\theta$	Directional angle	$^\circ$

Vector Resolution

Conversely, if the components are known, the magnitude and direction are found using:

Vector Magnitude and Direction

Determines the magnitude and angle of a vector from its orthogonal components.

Variables

Symbol	Description	Unit
$V$	Magnitude of the vector	-
$\theta$	Directional angle	$^\circ$
$V_{x}, V_{y}$	Horizontal and vertical components	-

Vector Resolution

This principle is universally applied in statics (resolving forces), kinematics (projectile motion), and electrical engineering (AC circuits).

Key Takeaways

Vector Components: Use cosine for the adjacent component and sine for the opposite component to break down forces and velocities.

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