📌 What is Trigonometry?
Trigonometry (or "trig") is a branch of mathematics that studies the relationships between the angles and sides of triangles. It’s especially useful in geometry, physics, architecture, and any field involving waves or periodic motion.
🎯 Angles and Their Measurement
- Degrees (°): A full circle is 360°
- Radians (rad): A full circle is 2π radians
- Conversion:
π radians = 180°
Trigonometry commonly uses both degrees and radians. You'll need to be comfortable switching between them.
📐 Right Triangle Trigonometry
In a right triangle (one angle = 90°), the basic trig functions are defined using ratios:
- Sine:
sin(θ) = opposite / hypotenuse - Cosine:
cos(θ) = adjacent / hypotenuse - Tangent:
tan(θ) = opposite / adjacent
Mnemonic: SOH-CAH-TOA
🧭 The Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It helps define trig functions for all angles.
Each point on the unit circle corresponds to:
(cos(θ), sin(θ))— coordinates at angle θ- Common angles: 0°, 30°, 45°, 60°, 90°, etc. (and their radian equivalents)
📊 Trigonometric Functions
There are six primary trig functions:
- sin(θ) – sine
- cos(θ) – cosine
- tan(θ) – tangent
- csc(θ) – cosecant (1 / sin)
- sec(θ) – secant (1 / cos)
- cot(θ) – cotangent (1 / tan)
These functions repeat in cycles and are periodic.
📈 Graphs of Trig Functions
Trigonometric functions create wave-like graphs. Key characteristics:
- Sine & Cosine: smooth, periodic curves (period = 2π)
- Tangent: has asymptotes, period = π
- Amplitude: height from midline to peak
- Phase Shift: horizontal movement of the graph
These are used in physics (waves, sound), engineering, and even art.
📘 Trigonometric Identities
These are formulas that are always true and help simplify expressions and solve equations.
🧮 Fundamental Identities:
sin²(θ) + cos²(θ) = 11 + tan²(θ) = sec²(θ)1 + cot²(θ) = csc²(θ)
📦 Angle Addition Formulas:
sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)
📐 Inverse Trig Functions
Used to find the angle when given a ratio.
sin⁻¹(x)→ gives angle whose sine is xcos⁻¹(x)→ gives angle whose cosine is xtan⁻¹(x)→ gives angle whose tangent is x
🏗️ Applications of Trigonometry
- Architecture and construction
- Physics (motion, waves, oscillations)
- Engineering (electrical, mechanical)
- Navigation, astronomy, and satellite communication
- Computer graphics and game design