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Pythagorean Trigonometric Identities

Sep 9, 2025

Overview

This lecture covers the three Pythagorean trigonometric identities, demonstrates how to use them to find unknown trig values, and explains how quadrant location affects sign selection.

The Pythagorean Identities

  • The three Pythagorean identities are:
    • sin²θ + cos²θ = 1
    • 1 + tan²θ = sec²θ
    • 1 + cot²θ = csc²θ
  • These identities relate the squares of trigonometric functions of the same angle.

Using sin²θ + cos²θ = 1

  • If sin θ = 4/5 and 0° < θ < 90°, find cos θ:
    • (4/5)² + cos²θ = 1
    • 16/25 + cos²θ = 25/25
    • cos²θ = (25 - 16)/25 = 9/25
    • cos θ = Âą3/5
    • Since θ is in quadrant I, cos θ is positive: cos θ = 3/5

Determining Sign Based on Quadrant

  • The sign of a trig function depends on the quadrant:
    • Quadrant I: sin, cos, tan all positive
    • Quadrant II: sin positive, cos and tan negative
    • Quadrant III: tan positive, sin and cos negative
    • Quadrant IV: cos positive, sin and tan negative

Example: Finding sin θ When cos θ Is Given

  • If cos θ = 8/17 and θ is between 3π/2 and 2π (quadrant IV):
    • sin²θ + (8/17)² = 1
    • sin²θ + 64/289 = 289/289
    • sin²θ = (289 - 64)/289 = 225/289
    • sin θ = Âą15/17
    • In quadrant IV, sin is negative: sin θ = -15/17

Example: Finding cos θ Given sin θ and Tangent Sign

  • If sin θ = 2/5 and tan θ < 0:
    • Sin positive ⇒ quadrant I or II
    • Tan negative ⇒ quadrant II or IV
    • Both true only in quadrant II or IV, but sin positive only in quadrant II
    • cos²θ = 1 - (2/5)² = 1 - 4/25 = 21/25
    • cos θ = ±√21/5
    • In quadrant II, cos is negative: cos θ = -√21/5

Key Terms & Definitions

  • Pythagorean Identity — Equation expressing the relationship between squares of trigonometric functions.
  • Quadrant — One of four regions divided by x- and y-axes; affects trig function signs.
  • Sine (sin) — Opposite/hypotenuse in a right triangle.
  • Cosine (cos) — Adjacent/hypotenuse in a right triangle.
  • Tangent (tan) — Opposite/adjacent in a right triangle.

Action Items / Next Steps

  • Practice finding unknown trig values using Pythagorean identities and determining signs based on quadrants.
  • Review quadrant sign rules for trigonometric functions.

Full transcript