Applications of Trigonometry

Learning Objectives

  • Check whether the known measurements are sufficient, then select a method for a right or oblique triangle.
  • Classify acute SSA cases with the altitude threshold and handle right or obtuse SSA cases with their side-length condition.
  • Verify every candidate against the original measurements and triangle angle and side constraints.

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.

Minimum Data Before Solving

A right triangle requires its known 90°90° angle plus two independent measurements, with at least one side (for example, two sides or one side and one acute angle). An oblique triangle requires ASA/AAS, SAS, SSS, or SSA data. Angle-only measurements do not establish a triangle's scale, and a lone right angle is not enough to solve one.

Right and Oblique Triangle Solution Selector

First confirm that enough independent measurements are known. Then choose a right-triangle ratio or oblique-triangle law, resolve SSA solution count, and verify each reported triangle against the inputs.

Before selecting a method, the input must contain enough independent measurements. For a right triangle, the known 90° angle plus two independent measurements are sufficient when at least one is a side (for example, two sides or one side and one acute angle). For an oblique triangle, sufficient patterns are ASA/AAS, SAS, SSS, and SSA. Angle-only data, including a lone right angle, are insufficient. Failed SSS inequalities and failed post-solve triangle checks stop for corrected input. The SSA branch uses local construction labels: A is the known angle, a is the known adjacent side, and b is the known opposite side. Thus sin(B) = a sin(A)/b. First classify the known A. For acute A only, compute h = a sin(A): b < h gives 0 triangles; b = h gives 1 right triangle; h < b < a gives 2 triangles; and b ≥ a gives 1 non-degenerate triangle (the second candidate at equality is degenerate). For right or obtuse A, do not use those acute-angle altitude bands: b > a gives one triangle, solved from the Law of Sines; b ≤ a gives none. In the 2-solution case, test B and 180° − B and retain only non-degenerate triangles. These labels are scoped to this branch; elsewhere the lesson uses conventional lowercase side names opposite their matching uppercase angles. The comparisons assume degree measure. Convert radians before comparing A with 90 degrees or using a 180-degree angle sum.

Right and Oblique Triangle Solution SelectorFirst confirm that enough independent measurements are known. Then choose a right-triangle ratio or oblique-triangle law, resolve SSA solution count, and verify each reported triangle against the inputs.. Before selecting a method, the input must contain enough independent measurements. For a right triangle, the known 90° angle plus two independent measurements are sufficient when at least one is a side (for example, two sides or one side and one acute angle). For an oblique triangle, sufficient patterns are ASA/AAS, SAS, SSS, and SSA. Angle-only data, including a lone right angle, are insufficient. Failed SSS inequalities and failed post-solve triangle checks stop for corrected input. The SSA branch uses local construction labels: A is the known angle, a is the known adjacent side, and b is the known opposite side. Thus sin(B) = a sin(A)/b. First classify the known A. For acute A only, compute h = a sin(A): b < h gives 0 triangles; b = h gives 1 right triangle; h < b < a gives 2 triangles; and b ≥ a gives 1 non-degenerate triangle (the second candidate at equality is degenerate). For right or obtuse A, do not use those acute-angle altitude bands: b > a gives one triangle, solved from the Law of Sines; b ≤ a gives none. In the 2-solution case, test B and 180° − B and retain only non-degenerate triangles. These labels are scoped to this branch; elsewhere the lesson uses conventional lowercase side names opposite their matching uppercase angles. The comparisons assume degree measure. Convert radians before comparing A with 90 degrees or using a 180-degree angle sum.. Start → Sketch the geometry; check angle range, positive lengths, and units; Sketch the geometry; check angle range, positive lengths, and units → Enough independent triangle data?; Enough independent triangle data? — Yes → Is a 90° angle known?; Enough independent triangle data? — No → Stop: request sufficient data; Is a 90° angle known? — Yes → Choose a reference angle; label O, A, and H; Choose a reference angle; label O, A, and H → Use SOH-CAH-TOA or Pythagorean theorem; Use SOH-CAH-TOA or Pythagorean theorem → Compute the remaining angle and side; Is a 90° angle known? — No → Classify the complete side-angle pattern; Classify the complete side-angle pattern → ASA or AAS?; ASA or AAS? — Yes → Find the third angle; apply Law of Sines to opposite pairs; Find the third angle; apply Law of Sines to opposite pairs → Compute the remaining angle and side; ASA or AAS? — No → SAS or SSS?; SAS or SSS? — Yes → Is the known pattern SSS?; Is the known pattern SSS? — Yes → Do the SSS lengths satisfy triangle inequality?; Is the known pattern SSS? — No → Use Law of Cosines for SAS data; Do the SSS lengths satisfy triangle inequality? — Yes → Use Law of Cosines for valid SSS data; Do the SSS lengths satisfy triangle inequality? — No → Stop: correct the SSS side lengths; Use Law of Cosines for SAS data → Compute the remaining angle and side; Use Law of Cosines for valid SSS data → Compute the remaining angle and side; SAS or SSS? — No → SSA with a known opposite pair?; SSA with a known opposite pair? — Yes → SSA rule: acute A uses h bands; right/obtuse A requires b > a; SSA with a known opposite pair? — No → Stop: correct or complete the side-angle pattern; SSA rule: acute A uses h bands; right/obtuse A requires b > a → How many non-degenerate SSA triangles fit?; How many non-degenerate SSA triangles fit? — 0 → Report that no triangle satisfies the data; How many non-degenerate SSA triangles fit? — 1 → Use sin(B) = a sin(A)/b; retain the unique valid B; How many non-degenerate SSA triangles fit? — 2 → Test B and 180° − B; retain both valid angles; Use sin(B) = a sin(A)/b; retain the unique valid B → Compute the remaining angle and side; Test B and 180° − B; retain both valid angles → Compute the remaining angle and side; Compute the remaining angle and side → Check positive lengths, triangle inequalities, and 180° angle sum; Check positive lengths, triangle inequalities, and 180° angle sum → Do calculated sides and angles form a valid triangle?; Do calculated sides and angles form a valid triangle? — Yes → Do calculated pairs reproduce the original data?; Do calculated sides and angles form a valid triangle? — No → Stop: correct the input or review the result; Do calculated pairs reproduce the original data? — Yes → Report the valid triangle solution(s); Do calculated pairs reproduce the original data? — No → Review rounding, side labels, and source measurements; Review rounding, side labels, and source measurements → Stop: correct the data or calculation

Start → Sketch the geometry; check angle range, positive lengths, and units; Sketch the geometry; check angle range, positive lengths, and units → Enough independent triangle data?; Enough independent triangle data? — Yes → Is a 90° angle known?; Enough independent triangle data? — No → Stop: request sufficient data; Is a 90° angle known? — Yes → Choose a reference angle; label O, A, and H; Choose a reference angle; label O, A, and H → Use SOH-CAH-TOA or Pythagorean theorem; Use SOH-CAH-TOA or Pythagorean theorem → Compute the remaining angle and side; Is a 90° angle known? — No → Classify the complete side-angle pattern; Classify the complete side-angle pattern → ASA or AAS?; ASA or AAS? — Yes → Find the third angle; apply Law of Sines to opposite pairs; Find the third angle; apply Law of Sines to opposite pairs → Compute the remaining angle and side; ASA or AAS? — No → SAS or SSS?; SAS or SSS? — Yes → Is the known pattern SSS?; Is the known pattern SSS? — Yes → Do the SSS lengths satisfy triangle inequality?; Is the known pattern SSS? — No → Use Law of Cosines for SAS data; Do the SSS lengths satisfy triangle inequality? — Yes → Use Law of Cosines for valid SSS data; Do the SSS lengths satisfy triangle inequality? — No → Stop: correct the SSS side lengths; Use Law of Cosines for SAS data → Compute the remaining angle and side; Use Law of Cosines for valid SSS data → Compute the remaining angle and side; SAS or SSS? — No → SSA with a known opposite pair?; SSA with a known opposite pair? — Yes → SSA rule: acute A uses h bands; right/obtuse A requires b > a; SSA with a known opposite pair? — No → Stop: correct or complete the side-angle pattern; SSA rule: acute A uses h bands; right/obtuse A requires b > a → How many non-degenerate SSA triangles fit?; How many non-degenerate SSA triangles fit? — 0 → Report that no triangle satisfies the data; How many non-degenerate SSA triangles fit? — 1 → Use sin(B) = a sin(A)/b; retain the unique valid B; How many non-degenerate SSA triangles fit? — 2 → Test B and 180° − B; retain both valid angles; Use sin(B) = a sin(A)/b; retain the unique valid B → Compute the remaining angle and side; Test B and 180° − B; retain both valid angles → Compute the remaining angle and side; Compute the remaining angle and side → Check positive lengths, triangle inequalities, and 180° angle sum; Check positive lengths, triangle inequalities, and 180° angle sum → Do calculated sides and angles form a valid triangle?; Do calculated sides and angles form a valid triangle? — Yes → Do calculated pairs reproduce the original data?; Do calculated sides and angles form a valid triangle? — No → Stop: correct the input or review the result; Do calculated pairs reproduce the original data? — Yes → Report the valid triangle solution(s); Do calculated pairs reproduce the original data? — No → Review rounding, side labels, and source measurements; Review rounding, side labels, and source measurements → Stop: correct the data or calculation

  • Start: terminator
  • Sketch the geometry; check angle range, positive lengths, and units: process
  • Enough independent triangle data?: decision
  • Is a 90° angle known?: decision
  • Choose a reference angle; label O, A, and H: process
  • Use SOH-CAH-TOA or Pythagorean theorem: process
  • Classify the complete side-angle pattern: process
  • ASA or AAS?: decision
  • Find the third angle; apply Law of Sines to opposite pairs: process
  • SAS or SSS?: decision
  • Is the known pattern SSS?: decision
  • Do the SSS lengths satisfy triangle inequality?: decision
  • Use Law of Cosines for SAS data: process
  • Use Law of Cosines for valid SSS data: process
  • SSA with a known opposite pair?: decision
  • Stop: request sufficient data: terminator
  • Stop: correct or complete the side-angle pattern: terminator
  • Stop: correct the SSS side lengths: terminator
  • SSA rule: acute A uses h bands; right/obtuse A requires b > a: process
  • How many non-degenerate SSA triangles fit?: decision
  • Use sin(B) = a sin(A)/b; retain the unique valid B: process
  • Test B and 180° − B; retain both valid angles: process
  • Compute the remaining angle and side: process
  • Check positive lengths, triangle inequalities, and 180° angle sum: process
  • Do calculated sides and angles form a valid triangle?: decision
  • Do calculated pairs reproduce the original data?: decision
  • Report the valid triangle solution(s): terminator
  • Report that no triangle satisfies the data: terminator
  • Stop: correct the input or review the result: terminator
  • Review rounding, side labels, and source measurements: process
  • Stop: correct the data or calculation: terminator

Solving Right Triangles

When dealing with right triangles (where one angle is exactly 90∘90^\circ), we primarily rely on the Pythagorean Theorem (a2+b2=c2a^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).

Right-Triangle Side Roles Follow the Reference Angle
Read opposite and adjacent from the chosen acute angle. The two legs exchange roles when the reference angle moves; the hypotenuse remains opposite 90 degrees.

Right-Triangle Interactive Exploration

Adjust the height and horizontal distance to observe how they determine the line-of-sight angle and length.

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Right-Triangle Solver Exploration

Adjust the two legs to observe how the hypotenuse, acute angles, and triangle area respond.

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

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.

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

The Ambiguous Case (SSA)

For this SSA construction, let AA be the known angle with 0°<A<180°0° < A < 180°, aa the known side adjacent to AA, and bb the known side opposite AA. These are local construction labels; the conventional side-angle pairing used in the Law of Sines formula above remains unchanged elsewhere.

  • For acute AA, compute h=asin⁡Ah = a \sin A: b<hb < h gives no triangle; b=hb = h gives one right triangle; h<b<ah < b < a gives two triangles; and b≥ab \ge a gives one non-degenerate triangle.
  • For 90°≤A<180°90° \le A < 180°, do not use the acute-angle altitude bands: one triangle exists only when b>ab > a; otherwise no triangle exists.
  • For a candidate angle BB, use sin⁡B=(a/b)sin⁡A\sin B = (a/b)\sin A. In the two-solution case, test both BB and 180°−B180°-B; retain each only when the angles form a non-degenerate triangle.
Acute SSA Cases: How the Altitude Sets the Side-Length Threshold
These four bands apply only when angle A is acute. The local construction uses adjacent side a, opposite side b, and h = a sin(A); b measures from the fixed endpoint to candidate B on the ray. At b = a, one candidate collapses at A and is degenerate, leaving one non-degenerate triangle. For right or obtuse A, compare b with a instead: b > a gives one triangle and b ≤ a gives none. The sketches are schematic and use degree measure.

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.

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

Oblique-Triangle Interactive Exploration

Adjust the known parts in AAS, SAS, or SSS mode to see the Law of Sines or Law of Cosines determine the remaining sides and angles.

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Law of Cosines

Alternative Forms

Law of Cosines (Alternative Forms)

Other forms for solving sides a or b.

a2=b2+c2−2bccos⁡Ab2=a2+c2−2accos⁡B\begin{aligned} a^2 &= b^2 + c^2 - 2bc \cos A \\ b^2 &= a^2 + c^2 - 2ac \cos B \end{aligned}

Law of Cosines

Solved for Angle

Law of Cosines (Angle Form)

Rearranged to solve directly for the cosine of an angle.

cos⁡C=a2+b2−c22ab\cos C = \frac{a^2 + b^2 - c^2}{2ab}

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.

a−ba+b=tan⁡[12(A−B)]tan⁡[12(A+B)]\frac{a - b}{a + b} = \frac{\tan\left[\frac{1}{2}(A - B)\right]}{\tan\left[\frac{1}{2}(A + B)\right]}

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=a+b+c2s = \frac{a+b+c}{2} be the semi-perimeter, and rr 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.

cot⁡(A2)=s−arcot⁡(B2)=s−brcot⁡(C2)=s−cr\begin{aligned} \cot\left(\frac{A}{2}\right) &= \frac{s - a}{r} \\ \cot\left(\frac{B}{2}\right) &= \frac{s - b}{r} \\ \cot\left(\frac{C}{2}\right) &= \frac{s - c}{r} \end{aligned}

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.

a+bc=cos⁡[12(A−B)]sin⁡(12C)a−bc=sin⁡[12(A−B)]cos⁡(12C)\begin{aligned} \frac{a + b}{c} &= \frac{\cos\left[\frac{1}{2}(A - B)\right]}{\sin\left(\frac{1}{2}C\right)} \\ \frac{a - b}{c} &= \frac{\sin\left[\frac{1}{2}(A - B)\right]}{\cos\left(\frac{1}{2}C\right)} \end{aligned}

Area of a Triangle

SAS Area Formula

Triangle Area (SAS)

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

Area=12absin⁡C\text{Area} = \frac{1}{2}ab \sin C

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.

Area=s(s−a)(s−b)(s−c)\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

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.

Area=(s−a)(s−b)(s−c)(s−d)\text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)}

Area of a Triangle

where s=a+b+c+d2s = \frac{a+b+c+d}{2}. Notice that if side d=0d=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.

Vectors and Physics Applications

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

Vector Resolution

Any 2D vector V⃗\vec{V} with magnitude VV and angle θ\theta (measured from the positive x-axis) can be resolved into horizontal (VxV_x) and vertical (VyV_y) components:

Vector Components

Resolves a vector into orthogonal horizontal and vertical components.

Vx=Vcos⁡θVy=Vsin⁡θ\begin{aligned} V_x &= V \cos \theta \\ V_y &= V \sin \theta \end{aligned}

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.

V=Vx2+Vy2θ=arctan⁡(VyVx)\begin{aligned} V &= \sqrt{V_x^2 + V_y^2} \\ \theta &= \arctan \left(\frac{V_y}{V_x}\right) \end{aligned}

Vector Resolution

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

Key Takeaways
  • Right triangles: Use SOH CAH TOA or the Pythagorean theorem after labeling sides from the chosen reference angle; sketch the measured geometry first.
  • Bearings: Establish the north-south-east-west reference and record the starting direction and turn convention.
  • Law of Sines: Use ASA/AAS directly and treat SSA as a zero-, one-, or two-solution case that requires candidate validation.
  • Law of Cosines and Heron's formula: Use the cosine law for SAS/SSS and Heron's formula for area from three valid side lengths.
  • Vector components: Resolve a vector into perpendicular components using its stated reference angle and preserve component signs.

References