Welcome to Week 1! This week, we tackled two essential mathematical pillars: Right Angle Trigonometry and a General Strategy for Factoring. Whether you are solving for the height of a tree using a shadow or breaking down complex polynomials, having a systematic approach is key.

Below, you will find a summary of the concepts, the downloadable notes from Thursday and Friday’s classes, and a review of this week’s quiz.

Part 1: Right Angle Trigonometry

When working with right triangles, we use specific tools to find missing side lengths and angle measures. Always remember that the Pythagorean Theorem ($a^2 + b^2 = c^2$) only applies to right triangles where $c$ is the hypotenuse.

To relate angles to sides, we use the trigonometric functions. The mnemonic SOH CAH TOA is the best way to remember these ratios:

  • SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Key Concept: Angle of Elevation vs. Depression
It is crucial to distinguish between these two angles when solving word problems:

  • Angle of Elevation: The angle measured from the horizontal looking up at an object (like looking at the top of a flagpole).
  • Angle of Depression: The angle measured from the horizontal looking down on an object (like a bird on a roof looking down at a worm).

Part 2: A General Strategy for Factoring

Factoring is one of the most important skills in algebra. Instead of guessing, use this checklist based on the number of terms in the expression. Always factor out the Greatest Common Factor (GCF) first!

  1. Two Terms: Check for special formulas.
    • Difference of Squares: $x^2 - y^2 = (x-y)(x+y)$
    • Sum of Cubes: $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$
    • Difference of Cubes: $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$
  2. Three Terms: Use Reverse FOIL or the AC Method (Splitting the Middle Term).
    • If the coefficient in front of $x^2$ is 1: Find two numbers that multiply to the constant and add to the middle coefficient.
    • If the coefficient is $>1$: Multiply the first coefficient by the constant to find factors that split the middle term, then factor by grouping.
  3. Four Terms: Factor by Grouping. Group the first two terms and the last two terms, factor out the GCF from each pair, and look for the common binomial.

Class Downloads

Attached you will find the detailed worksheets and handwritten notes from our sessions:

Week 1 Quiz Review

Below is the study guide based on this week's quiz questions. Use these to test your understanding.

  1. What are the three different skills needed to find the parts of a triangle?
    You need to utilize the 180-degree rule (all angles sum to 180), the Pythagorean Theorem ($a^2+b^2=c^2$), and the Trigonometric Functions (Sin, Cos, and Tan).
  2. What is the interesting word used to remember the trig functions?
    SOH CAH TOA.
  3. What is the difference between angle of elevation and angle of depression?
    With an angle of elevation, you are looking up at the object from a horizontal line. With an angle of depression, you are looking down on the object from a horizontal line.
  4. If you had to explain to a friend how to factor, what would you tell them?
    First, always look for a GCF. Then, look at the number of terms:
    • If there is no number in front of the $x^2$ (coefficient is 1), find two numbers that multiply to the constant term and add to the middle term. Format: $(x+?)(x+?)$.
    • If there is a number in front of the $x^2$, multiply that leading number by the constant term. Find factors that multiply to this new number and add to the middle term. Split the middle term into two pieces using these numbers, creating a 4-term polynomial. Finally, factor by grouping.

Good luck with your studies this week! Remember to show your work and check your answers to ensure they make sense physically (no negative lengths for triangles!).