Calculate Right Triangle

Trigonometry: Complete calculation from one side and one angle

Trigonometric Calculator

Trigonometry in Right Triangles

Trigonometry enables calculation of all parameters from one side length and one angle α. With sine, cosine, and tangent, all missing values are determined.

Enter Known Parameters
Choose the known side
Value of the known side
Acute angle α (0° < α < 90°)
Calculated Properties
Leg a:
Leg b:
Hypotenuse c:
Height h:
Area A:
Perimeter P:
Segment p:
Segment q:
Angle α:
Angle β:

Trigonometry Visualization

Right triangle with trigonometry

Right triangle with angle α and trigonometric relationships.
Sine, cosine, and tangent connect angles with side ratios.

Trigonometric Relationships
sin(α) = a/c
Opposite / Hypotenuse
cos(α) = b/c
Adjacent / Hypotenuse
tan(α) = a/b
Opposite / Adjacent

Trigonometry in Right Triangles

Trigonometry connects angles with side ratios in right triangles:

  • Sine (sin): Ratio of opposite side to hypotenuse
  • Cosine (cos): Ratio of adjacent side to hypotenuse
  • Tangent (tan): Ratio of opposite side to adjacent side
  • Applications: Navigation, physics, engineering
  • Inverse functions: arcsin, arccos, arctan for angle calculation
  • Unit circle: Foundation of trigonometry

Angles and Their Meaning

Angle α is the key to trigonometric calculation:

Angle α (Alpha)
  • Acute angle between 0° and 90°
  • Located at vertex A of the triangle
  • Determines all trigonometric ratios
  • Complementary to angle β (α + β = 90°)
Calculation Strategy
  • With one side and α → all other sides calculable
  • Trigonometric functions as tools
  • Systematic solution approach
  • Verification possible through Pythagoras

The Trigonometric Functions

Sine, cosine, and tangent are the fundamental functions of trigonometry:

Sine (sin)
\[\sin(\alpha) = \frac{a}{c}\]

Opposite side divided by hypotenuse

Cosine (cos)
\[\cos(\alpha) = \frac{b}{c}\]

Adjacent side divided by hypotenuse

Tangent (tan)
\[\tan(\alpha) = \frac{a}{b}\]

Opposite side divided by adjacent side

Practical Applications of Trigonometry

Trigonometry is indispensable in many fields:

Navigation & Surveying
  • Height measurement of mountains and buildings
  • GPS navigation and triangulation
  • Maritime and aviation navigation
  • Cartography and land surveying
Technology & Physics
  • Mechanical engineering: slopes and inclinations
  • Optics: refraction angles and reflection
  • AC electrical engineering and oscillations
  • Computer graphics and animation

Trigonometric Formulas for Right Triangles

Basic Functions
\[\sin(\alpha) = \frac{a}{c}\] \[\cos(\alpha) = \frac{b}{c}\] \[\tan(\alpha) = \frac{a}{b}\]

Ratios in right triangles

Inverse Functions
\[\alpha = \arcsin\left(\frac{a}{c}\right)\] \[\alpha = \arccos\left(\frac{b}{c}\right)\] \[\alpha = \arctan\left(\frac{a}{b}\right)\]

Angles from side ratios

Side Calculations
Hypotenuse c:
\[c = \frac{a}{\sin(\alpha)} = \frac{b}{\cos(\alpha)}\]
Leg a:
\[a = c \sin(\alpha) = b \tan(\alpha)\]
Leg b:
\[b = c \cos(\alpha) = \frac{a}{\tan(\alpha)}\]
Additional Parameters
Area: \(A = \frac{a \cdot b}{2}\)
Perimeter: \(P = a + b + c\)
Height: \(h = \frac{a \cdot b}{c}\)
Complementary angle: \(\beta = 90° - \alpha\)
Important Trigonometric Identities
Pythagoras: \(\sin^2(\alpha) + \cos^2(\alpha) = 1\)
Tangent: \(\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}\)
Complementary: \(\sin(\alpha) = \cos(90° - \alpha)\)
Complementary: \(\cos(\alpha) = \sin(90° - \alpha)\)
Notation and Symbols
  • α (Alpha): Acute angle at point A
  • β (Beta): Acute angle at point B
  • γ (Gamma): Right angle (90°)
  • a: Leg (opposite to α)
  • b: Leg (adjacent to α)
  • c: Hypotenuse (longest side)
  • h: Height to hypotenuse
  • °: Degrees (angle measure)

Calculation Example

Given
Leg a = 3 Angle α = 36.87°

Classic 3-4-5 triangle

1. Calculate Hypotenuse
\[c = \frac{a}{\sin(\alpha)} = \frac{3}{\sin(36.87°)} = \frac{3}{0.6} = 5\]

sin(36.87°) ≈ 0.6 (exactly 3/5)

2. Second Leg
\[b = c \cos(\alpha) = 5 \times \cos(36.87°) = 5 \times 0.8 = 4\]

cos(36.87°) ≈ 0.8 (exactly 4/5)

3. Verification with Pythagoras
\[3^2 + 4^2 = 9 + 16 = 25 = 5^2 \checkmark\]

The famous 3-4-5 triangle

4. Additional Parameters
  • Area: A = (3×4)/2 = 6
  • Perimeter: P = 3+4+5 = 12
  • Angle β: β = 90°-36.87° = 53.13°
  • Height: h = (3×4)/5 = 2.4

Trigonometry and Right Triangles

Trigonometry (from Greek "triangle measurement") is one of the most fundamental areas of mathematics. The calculation of right triangles from one side length and one angle demonstrates the practical power of this mathematical discipline and its universal applicability.

Fundamentals of Trigonometry

The trigonometric functions arose from the practical need to solve right triangles:

  • Historical development: Used in antiquity for astronomy and navigation
  • Unit circle relationships: Modern definition via the unit circle
  • Periodic functions: Extension beyond the right triangle
  • Analytical properties: Differentiable, integrable, expandable in series
  • Universal application: From physics to computer science

The Trigonometric Functions in Detail

Sine (sin)

The sine gives the ratio of the opposite side to the hypotenuse. Geometrically, it corresponds to the y-coordinate on the unit circle.

Cosine (cos)

The cosine gives the ratio of the adjacent side to the hypotenuse. Geometrically, it corresponds to the x-coordinate on the unit circle.

Tangent (tan)

The tangent gives the ratio of the opposite side to the adjacent side. It corresponds to the slope of the line from the origin to the point on the unit circle.

Relationships

The functions are connected by fundamental identities: tan(α) = sin(α)/cos(α), sin²(α) + cos²(α) = 1.

Practical Applications of Trigonometric Calculation

The calculation of right triangles from an angle and a side is ubiquitous in practice:

Surveying and Navigation
  • Height measurement with theodolites and angle measurers
  • Distance determination without direct access
  • GPS triangulation and satellite navigation
  • Map projection and geographic coordinates
Construction and Architecture
  • Roof slopes and gable angles
  • Stair slopes and ramp inclinations
  • Static calculations of supporting structures
  • Optimization of constructions
Physics and Technology
  • Force resolution in mechanics
  • Wave theory and vibration analysis
  • Optics: refraction and reflection angles
  • AC electrical engineering and phase shifts
Modern Technologies
  • Computer graphics and 3D modeling
  • Robotics and motion control
  • Signal processing and communication technology
  • Astronomy and space travel

Calculation Strategies and Methods

Depending on the given parameters, different solution approaches emerge:

Given: Leg and Angle

When a leg and an acute angle are known, the position of the leg (opposite or adjacent) determines the function to use.

Given: Hypotenuse and Angle

With a known hypotenuse, both legs are calculated directly via sine and cosine.

Verification Methods

Pythagorean theorem and angle sum (α + β = 90°) always provide verification options for the calculation.

Numerical Aspects

Special caution with very acute or very obtuse angles due to numerical instabilities.

Special Angles and Their Significance

Certain angles have special significance in trigonometry:

  • 30°, 45°, 60°: Yield exact values for trigonometric functions
  • 30°-60°-90° triangle: Side ratio 1:√3:2
  • 45°-45°-90° triangle: Side ratio 1:1:√2
  • 36.87°-53.13°-90° triangle: The famous 3-4-5 triangle
  • Practical angles: Commonly used inclinations in technology and construction

Historical Development and Cultural Significance

Ancient Origins

Babylonians and Egyptians used trigonometric relationships for astronomical calculations and architecture.

Greek Mathematics

Hipparchus and Ptolemy developed the first systematic chord and angle tables.

Islamic Mathematics

Al-Battani and others perfected trigonometric methods and created the term "sine".

Modern Development

Euler and others developed analytical trigonometry and its applications in analysis.

Trigonometry in Modern Mathematics

Trigonometry has developed far beyond right triangles:

  • Fourier analysis: Decomposition of periodic functions into sine and cosine components
  • Complex analysis: Euler's formula e^(iα) = cos(α) + i·sin(α)
  • Differential equations: Solutions with periodic functions
  • Probability theory: Random processes with periodic properties
  • Numerical mathematics: Algorithms for approximation methods
Summary

The trigonometric calculation of right triangles from one side and one angle demonstrates the elegant connection between geometric relationships and analytical functions. This method is not only mathematically elegant but also practically indispensable in numerous technical and scientific applications. The trigonometric functions form a bridge between elementary geometry and higher mathematics.

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