Fundamental Trigonometric Identities to Know for AP Pre-Calculus

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.