Trigonometric Identities and Properties Notes

Trigonometric Identities: Complementary Angle Relationships

• The following identities relate trigonometric functions of complementary angles (angles that add up to 90° or \frac{\pi}{2} radians):

  • \cos(\theta) = \sin(90^\circ - \theta)
  • \cot(\theta) = \tan(90^\circ - \theta)
  • \csc(\theta) = \sec(90^\circ - \theta)
  • \sin(\theta) = \cos(90^\circ - \theta)
  • \tan(\theta) = \cot(90^\circ - \theta)
  • \sec(\theta) = \csc(90^\circ - \theta)

• Working with radians, where 90^\circ is equivalent to \frac{\pi}{2} radians:

  • \cos(x) = \sin(\frac{\pi}{2} - x)
  • \cot(x) = \tan(\frac{\pi}{2} - x)
  • \csc(x) = \sec(\frac{\pi}{2} - x)
  • \sin(x) = \cos(\frac{\pi}{2} - x)
  • \tan(x) = \cot(\frac{\pi}{2} - x)
  • \sec(x) = \csc(\frac{\pi}{2} - x)

Even and Odd Trigonometric Functions

• Cosine and secant are even functions:

  • \cos(-x) = \cos(x)
  • \sec(-x) = \sec(x)

• Sine, tangent, cosecant, and cotangent are odd functions:

  • \sin(-x) = -\sin(x)
  • \csc(-x) = -\csc(x)
  • \tan(-x) = -\tan(x)
  • \cot(-x) = -\cot(x)

Reciprocal Trigonometric Identities

• \sin(x) = \frac{1}{\csc(x)}
• \csc(x) = \frac{1}{\sin(x)}
• \cos(x) = \frac{1}{\sec(x)}
• \sec(x) = \frac{1}{\cos(x)}
• \tan(x) = \frac{1}{\cot(x)}
• \cot(x) = \frac{1}{\tan(x)}

Quotient Identities

• \tan(x) = \frac{\sin(x)}{\cos(x)}
• \cot(x) = \frac{\cos(x)}{\sin(x)}

Pythagorean Identities

• \cos^2(x) + \sin^2(x) = 1
• 1 + \tan^2(x) = \sec^2(x)
• \cot^2(x) + 1 = \csc^2(x)