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

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