Dimensions
What is Trigonometry?
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. The word comes from Greek words trigonon (triangle) and metron (measure).
It is used extensively in engineering, physics, navigation, architecture, and many other fields to calculate distances, heights, and angles that would be difficult to measure directly.
Right Triangle Formulas
Pythagorean Theorem
c² = a² + b²
Where c is the hypotenuse, and a and b are the other two sides
Trigonometric Ratios
sin(α) = opposite/hypotenuse
cos(α) = adjacent/hypotenuse
tan(α) = opposite/adjacent
Inverse Trigonometric Functions
α = arcsin(opposite/hypotenuse)
α = arccos(adjacent/hypotenuse)
α = arctan(opposite/adjacent)
Area of a Right Triangle
Area = (base × height) ÷ 2
Non-Right Triangle Formulas
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
Relates the sides of a triangle to the sines of the opposite angles
Law of Cosines
c² = a² + b² - 2ab·cos(C)
b² = a² + c² - 2ac·cos(B)
a² = b² + c² - 2bc·cos(A)
Area Formulas
Area = (1/2) × base × height
Area = (1/2) × ab × sin(C)
Area = √(s(s-a)(s-b)(s-c))
Heron's formula, where s = (a+b+c)/2
Sum of Angles
A + B + C = 180°
Applications in Engineering
Structural Engineering: Trigonometry is used to analyze forces in trusses, beams, and supports, ensuring buildings and bridges remain stable.
Electrical Engineering: Sine and cosine functions are used to analyze alternating current circuits and electromagnetic waves.
Mechanical Engineering: Used for analyzing components like cams, cranks, and pistons, which operate through circular or rotational motion.
Civil Engineering: Surveyors use trigonometry to determine distances, elevations, and land areas for construction projects.
Interesting Facts
Historical Origins
The earliest known use of trigonometry dates back to the Egyptians and Babylonians around 1500 BCE, who used it for astronomy, navigation, and building pyramids. The Greek astronomer Hipparchus (190-120 BCE) is often credited as the father of trigonometry.
The Unit Circle
The unit circle (a circle with radius 1) connects trigonometry with geometry. Any point (x,y) on the unit circle can be written as (cos θ, sin θ), where θ is the angle formed with the positive x-axis.
Euler's Identity
One of the most beautiful equations in mathematics connects trigonometry with complex numbers: eiπ + 1 = 0. This equation relates five fundamental mathematical constants (e, i, π, 1, and 0) and uses trigonometric functions in its derivation.
Periodic Nature
Trigonometric functions repeat their values in regular intervals (periods). This property makes them extremely useful for modeling natural phenomena that repeat, such as sound waves, light waves, ocean tides, and seasonal variations.