Fundamental Trigonometric Identities

Fundamental Trigonometric Identities are key tools in understanding relationships between trigonometric functions. These identities simplify expressions and solve equations, making them essential for mastering concepts in AP Pre-Calculus and building a strong math foundation.

1. Reciprocal Identities

   • The reciprocal identities express each trigonometric function in terms of its reciprocal.
   • Key identities include:
     • ( \sin(\theta) = \frac{1}{\csc(\theta)} )
     • ( \cos(\theta) = \frac{1}{\sec(\theta)} )
     • ( \tan(\theta) = \frac{1}{\cot(\theta)} )
   • These identities are essential for simplifying expressions and solving equations.

2. Pythagorean Identities

   • Derived from the Pythagorean theorem, these identities relate the squares of sine and cosine.
   • The fundamental identity is:
     • ( \sin^2(\theta) + \cos^2(\theta) = 1 )
   • Other forms include:
     • ( 1 + \tan^2(\theta) = \sec^2(\theta) )
     • ( 1 + \cot^2(\theta) = \csc^2(\theta) )

3. Quotient Identities

   • These identities express tangent and cotangent in terms of sine and cosine.
   • Key identities include:
     • ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
     • ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} )
   • Useful for converting between different trigonometric functions.

4. Even-Odd Identities

   • These identities describe the symmetry properties of trigonometric functions.
   • Key identities include:
     • ( \sin(-\theta) = -\sin(\theta) ) (odd function)
     • ( \cos(-\theta) = \cos(\theta) ) (even function)
   • Important for simplifying expressions involving negative angles.

5. Cofunction Identities

   • These identities relate the trigonometric functions of complementary angles.
   • Key identities include:
     • ( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) )
     • ( \cos(\frac{\pi}{2} - \theta) = \sin(\theta) )
   • Useful for solving problems involving angles that add up to ( \frac{\pi}{2} ).

6. Sum and Difference Identities for Sine and Cosine

   • These identities allow the calculation of sine and cosine for the sum or difference of two angles.
   • Key identities include:
     • ( \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) )
   • Essential for simplifying complex trigonometric expressions.

7. Double Angle Identities

   • These identities express trigonometric functions of double angles in terms of single angles.
   • Key identities include:
     • ( \sin(2\theta) = 2\sin(\theta)\cos(\theta) )
     • ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) )
   • Useful for solving equations and simplifying expressions involving double angles.

8. Half Angle Identities

   • These identities express trigonometric functions of half angles in terms of single angles.
   • Key identities include:
     • ( \sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}} )
     • ( \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} )
   • Important for solving problems involving half angles.

9. Product-to-Sum Identities

   • These identities convert products of sine and cosine into sums.
   • Key identities include:
     • ( \sin(a)\sin(b) = \frac{1}{2}[\cos(a-b) - \cos(a+b)] )
     • ( \cos(a)\cos(b) = \frac{1}{2}[\cos(a-b) + \cos(a+b)] )
   • Useful for simplifying integrals and solving equations.

10. Sum-to-Product Identities

    • These identities convert sums of sine and cosine into products.
    • Key identities include:
      • ( \sin(a) + \sin(b) = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) )
      • ( \cos(a) + \cos(b) = 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) )
    • Helpful for simplifying expressions and solving trigonometric equations.