Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. Here are key formulas to understand:

1. Basic Definitions

  • Sine (sin): $ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $
  • Cosine (cos): $ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $
  • Tangent (tan): $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{Opposite}}{\text{Adjacent}} $
Right_Triangle

2. Fundamental Identities

  • Pythagorean Identity: $ \sin^2(\theta) + \cos^2(\theta) = 1 $
  • Reciprocal Identities:
    • $ \csc(\theta) = \frac{1}{\sin(\theta)} $
    • $ \sec(\theta) = \frac{1}{\cos(\theta)} $
    • $ \cot(\theta) = \frac{1}{\tan(\theta)} $

3. Angle Addition Formulas

  • $ \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) $
  • $ \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) $
  • $ \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} $
Unit_Circle

4. Double Angle Formulas

  • $ \sin(2\theta) = 2\sin(\theta)\cos(\theta) $
  • $ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) $
  • $ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} $

5. Key Relationships

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

For deeper exploration, check our Trigonometry Tutorial or Trigonometry Examples section. Happy learning! 😊