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Reciprocal Trigonometric Identities

Sep 9, 2025

Overview

This lecture covers the fundamental reciprocal trigonometric identities, how to use them, and evaluates specific trigonometric functions using the unit circle and reference angles.

Reciprocal Trigonometric Identities

  • The reciprocal of a number is 1 divided by that number (e.g., reciprocal of 7 is 1/7).
  • (\sin \theta) reciprocal is (\csc \theta = 1/\sin \theta).
  • (\sec \theta) reciprocal is (1/\cos \theta).
  • (\cot \theta = 1/\tan \theta); (\tan \theta = 1/\cot \theta).
  • (\cos \theta = 1/\sec \theta) (less commonly used).
  • Memorize reciprocals: secant, cosecant, and cotangent.

Example Evaluations Using Reciprocal Identities

  • Secant of (\pi/3): (\sec(\pi/3) = 1/\cos(\pi/3) = 2).
  • Cosecant of (4\pi/3): (\csc(4\pi/3) = 1/\sin(4\pi/3) = -2\sqrt{3}/3) after rationalizing.
  • Secant of 330°: (\sec(330^\circ) = 1/\cos(330^\circ) = 2/\sqrt{3} = 2\sqrt{3}/3) after rationalizing.
  • Cosecant of –225° (or 135°): Reference angle 45°, terminal point ((-\sqrt{2}/2, \sqrt{2}/2)), so (\csc(-225^\circ) = 2/\sqrt{2} = \sqrt{2}).
  • Secant of (-7\pi/6): Coterminal to (5\pi/6), reference angle (\pi/6), so (\sec(-7\pi/6) = -2\sqrt{3}/3).

Evaluating Functions at Special Angles

  • Secant (0^\circ): (\sec(0^\circ) = 1/\cos(0^\circ) = 1).
  • Secant (90^\circ): (\sec(90^\circ) = 1/\cos(90^\circ)) is undefined ((1/0)).
  • Cosecant (180^\circ): (\csc(180^\circ) = 1/\sin(180^\circ)) is undefined ((1/0)).
  • Cosecant (270^\circ): (\csc(270^\circ) = 1/\sin(270^\circ) = -1).

Using the Unit Circle and Reference Angles

  • Reference angles help find standard values in any quadrant by adjusting the sign of coordinates.
  • The unit circle points are re-used in different quadrants with changing signs.

Key Terms & Definitions

  • Reciprocal — A value obtained by dividing 1 by the original number.
  • Secant ((\sec)) — Reciprocal of cosine: (\sec \theta = 1/\cos \theta).
  • Cosecant ((\csc)) — Reciprocal of sine: (\csc \theta = 1/\sin \theta).
  • Cotangent ((\cot)) — Reciprocal of tangent: (\cot \theta = 1/\tan \theta).
  • Reference Angle — The positive acute angle formed by the terminal side of an angle and the x-axis.
  • Coterminal Angles — Angles that share the same terminal side.

Action Items / Next Steps

  • Memorize the three main reciprocal identities: secant, cosecant, cotangent.
  • Practice evaluating trig functions at special and negative angles using the unit circle.
  • Rationalize denominators in final answers when necessary.

Full transcript