Trigonometric functions are the building blocks of triangle math. They relate angles to side lengths, helping us understand shapes and cycles. From sine waves to tangent lines, these functions pop up everywhere in math and science.

Right triangles are the key to unlocking trig's secrets. By looking at ratios of sides, we can figure out angles and lengths. This connects geometry to algebra, giving us powerful tools to solve real-world problems.

Trigonometric Functions Fundamentals

Trigonometric functions in triangles

• Right triangle definitions relate trig functions to triangle side ratios
  • Sine (sin) opposite side divided by hypotenuse measures vertical component
  • Cosine (cos) adjacent side divided by hypotenuse measures horizontal component
  • Tangent (tan) opposite side divided by adjacent side measures steepness
  • Cosecant (csc) hypotenuse divided by opposite side reciprocal of sine
  • Secant (sec) hypotenuse divided by adjacent side reciprocal of cosine
  • Cotangent (cot) adjacent side divided by opposite side reciprocal of tangent
• Unit circle definitions connect trig functions to coordinates
  • Sine (sin) y-coordinate of point on unit circle vertical position
  • Cosine (cos) x-coordinate of point on unit circle horizontal position
  • Tangent (tan) $y / x$ or $\sin \theta / \cos \theta$ slope of radial line
  • Cosecant (csc) $1 / \sin \theta$ reciprocal of y-coordinate
  • Secant (sec) $1 / \cos \theta$ reciprocal of x-coordinate
  • Cotangent (cot) $x / y$ or $\cos \theta / \sin \theta$ reciprocal of slope

Trigonometry and right triangles

• Ratios of triangle sides link trig functions to geometry
  • Sine relates opposite side to hypotenuse vertical component
  • Cosine relates adjacent side to hypotenuse horizontal component
  • Tangent relates opposite side to adjacent side steepness or slope
• Reciprocal relationships connect inverse functions
  • Cosecant reciprocal of sine $\csc \theta = 1 / \sin \theta$
  • Secant reciprocal of cosine $\sec \theta = 1 / \cos \theta$
  • Cotangent reciprocal of tangent $\cot \theta = 1 / \tan \theta$
• Pythagorean identities fundamental trig equations
  • $\sin^2 \theta + \cos^2 \theta = 1$ relates sine and cosine
  • $1 + \tan^2 \theta = \sec^2 \theta$ involves tangent and secant
  • $1 + \cot^2 \theta = \csc^2 \theta$ connects cotangent and cosecant

Domain and range of trigonometry

• Sine function periodic wave
  • Domain all real numbers continuous input
  • Range [-1, 1] output bounded between -1 and 1
• Cosine function shifted sine wave
  • Domain all real numbers continuous input
  • Range [-1, 1] output bounded between -1 and 1
• Tangent function periodic with vertical asymptotes
  • Domain all real numbers except $\pi/2 + n\pi$, n is integer undefined at 90°, 270°
  • Range all real numbers unbounded output
• Cosecant function reciprocal of sine
  • Domain all real numbers except $n\pi$, n is integer undefined at 0°, 180°
  • Range $(-\infty, -1]$ and $[1, \infty)$ output never between -1 and 1
• Secant function reciprocal of cosine
  • Domain all real numbers except $\pi/2 + n\pi$, n is integer undefined at 90°, 270°
  • Range $(-\infty, -1]$ and $[1, \infty)$ output never between -1 and 1
• Cotangent function reciprocal of tangent
  • Domain all real numbers except $n\pi$, n is integer undefined at 0°, 180°
  • Range all real numbers unbounded output

Signs of trigonometric functions

• Quadrant I (0° to 90°) positive quadrant
  • All trigonometric functions positive
• Quadrant II (90° to 180°) sine positive quadrant
  • Sine and cosecant positive vertical components positive
  • Cosine, tangent, secant, and cotangent negative
• Quadrant III (180° to 270°) tangent positive quadrant
  • Tangent and cotangent positive slope-related functions positive
  • Sine, cosine, cosecant, and secant negative
• Quadrant IV (270° to 360°) cosine positive quadrant
  • Cosine and secant positive horizontal components positive
  • Sine, tangent, cosecant, and cotangent negative
• Mnemonic device All Students Take Calculus
  • A All functions positive (Quadrant I)
  • S Sine and its reciprocal (cosecant) positive (Quadrant II)
  • T Tangent and its reciprocal (cotangent) positive (Quadrant III)
  • C Cosine and its reciprocal (secant) positive (Quadrant IV)