Precalculus Second Semester Final Review

Unit 1: Trigonometric Identities

• Verifying Trig Identities:
  • Choose the side with more operations for simplification.
  • Convert expressions to sine and cosine.
  • Get a common denominator.
  • Combine and simplify expressions.
  • Factor and use identities like ( 1 - \cos^2 \theta = \sin^2 \theta ).
  • Example: Show ( \tan \times \sin ) as ( \sin^2 \theta / \cos^2 \theta \times \sin^2 \theta ).

Binomial Theorem

• Pascal’s Triangle: Use appropriate row for coefficients.
• Expand Binomials:
  • Raise the first term to descending powers.
  • Raise the second term to increasing powers.
  • Simplify each term separately.
  • Example: ((x^2 - 2y)^4) results in alternating signs due to negative base.

Vectors and Bearings

• Vector Components:
  • Use cosine and sine with reference angles to find components.
  • Combine vectors for resultant.
• True Ground Speed and Bearing:
  • Use Pythagorean theorem for magnitude.
  • Use tangent inverse for bearing adjustments.

Power Series and Interval of Convergence

• Infinite Geometric Series:
  • Check the common ratio ( |r| < 1 ) for convergence.
  • Sum of series: ( \frac{a}{1-r} ).

Multiple Choice Problem Solving

• Get Common Denominators:
  • Use identities to simplify expressions.
• Use Sum and Difference Identities.
• Double Angle Identities:
  • ( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) )
• Solving Trigonometric Equations:
  • Substitute known values and solve for the unknown.

Graphing and Converting Polar Coordinates

• Standard Position and Bearings:
  • Measure angles from North for bearings.
• Reference Angles: Found by comparing against the nearest x-axis.

Trigonometric Problem Solving

• Perpendicular and Parallel Vectors:
  • Use dot product for perpendicular conditions.
• Parametric and Slope-Intercept Forms:
  • Convert between forms using known formulas.

Polar and Rectangular Conversions

• Conversions:
  • Between polar (r, θ) and rectangular (x, y) using trigonometric relations.

Polar Form of Complex Numbers

• Multiplying and Dividing Complex Numbers in Polar Form:
  • Multiply/divide r-values, add/subtract angles.
• Powers and Roots:
  • Use De Moivre’s Theorem for powers.

Sequences and Series

• Factorials and Simplification:
  • Simplify expressions using factorial properties.
• Permutations vs Combinations:
  • Permutations for ordered, combinations for unordered.

Summation and Series

• Sigma Notation:
  • Expand series terms.
• Type of Series and Convergence:
  • Geometric series with |r| < 1 converges.

Limits

• Polynomials:
  • Ratio of leading coefficients for same degree polynomials.
• Exponential Limits:
  • Base between -1 and 1 tends toward zero.

This review covers a wide range of topics crucial for the precalculus final, including trigonometric identities, binomial theorem, vectors, polar coordinates, complex numbers, series and sequences, and limits.