Notes on Vectors and Trigonometry Foundations

Introduction to Vectors and Trigonometry

• This material covers Chapter 5 of Psychopathetics, focusing on the foundational concepts and applications of vectors and trigonometry.

• The lesson is divided into two primary sections: the algebraic handling of vectors and the geometric and analytical applications of trigonometric functions.

Fundamentals of Vectors

• Conceptual Definition: A vector is a mathematical object that has a specific starting position and moves in a designated direction. It is conceptually similar to a ray studied in geometry.

• Note: While vectors are represented geometrically in pictures as rays, this course focuses on treating them algebraically as numbers.

• Dimensions: Vectors can exist in multiple dimensions, including two, three, four, and five dimensions. While higher dimensions (three through five) cannot be easily drawn on a 2D surface, the algebraic rules remain consistent across all dimensions.

Vector Arithmetic and Operations

• Addition and Subtraction Rule: To add or subtract vectors, they must be the same size. You perform the operation on the corresponding parts (components) of the vectors.

  • Addition Example: Given vectors $u = (-1, 2, 3)$ and $v = (2, -6, 0)$.

  • $u + v = (-1 + 2, 2 + (-6), 3 + 0)$

  • The resulting vector is $(1, -4, 3)$.

• Scalar Multiplication: A scalar is a coefficient placed in front of a vector. To multiply a vector by a scalar, apply the distributive property by multiplying each term in the vector by the scalar.

  • Example: Multiply vector $v = (2, -6, 0)$ by the scalar $4$.

  • $4 \times (2, -6, 0) = (4 \times 2, 4 \times -6, 4 \times 0)$

  • The resulting vector is $(8, -24, 0)$.

• Complex Combined Operations: Problems can combine scalar multiplication and subtraction.

  • Example Problem: Solve $-3u - 2v$ where $u = (-1, 2, 3)$ and $v = (2, -6, 0)$.

  • Step 1 (Distributing across u): $-3 \times (-1, 2, 3) = (3, -6, -9)$.

  • Step 2 (Distributing across v): Distribute the $-2$ into vector $v$: $-2 \times (2, -6, 0) = (-4, 12, 0)$. Alternatively, distribute the positive $2$ and subtract the result later.

  • Step 3 (Summation): Add the components together: $(3 + (-4), -6 + 12, -9 + 0)$.

  • Final Answer: $(-1, 6, -9)$.

Basic Trigonometry and SOHCAHTOA

• Scope: The trigonometry covered here deals primarily with right-angled triangles.

• Triangle Variables: In a standard right triangle, we identify:

  • Angles: $\text{Angle } a$ and $\text{Angle } b$.

  • Sides: $x$, $y$, and $r$ (where $r$ typically represents the hypotenuse).

• Primary Functions: There are three main trig functions: Sine ($\text{sine}$), Cosine ($\text{cosine}$), and Tangent ($\text{tangent}$).

  • Note: While Secant, Cosecant, and Cotangent exist, they are not the focus of this introductory material.

• SOHCAHTOA Mnemonic: A crucial acronym to remember the ratios for each function:

  • SOH: $\sin = \frac{\text{Opposite}}{\text{Hypotenuse}}$

  • CAH: $\cos = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

  • TOA: $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$

• Alternative Mnemonics:

  • "Some Old Horse Caught Another Horse Taking Oats Away."

  • "Some armed hippie caught another hippie tripping on acid."

Identifying Opposite, Adjacent, and Hypotenuse Sides

• The Hypotenuse: The side labeled $r$ is always the hypotenuse because it is the longest side and is directly across from the right angle.

• Relative Sides: The "Opposite" and "Adjacent" labels change depending on which angle is being analyzed.

  • Using Angle $a$:

    • The Opposite side is across from the angle ($y$).

    • The Adjacent side is beside the angle ($x$).

    • $\sin(a) = \frac{y}{r}$, $\cos(a) = \frac{x}{r}$, $\tan(a) = \frac{y}{x}$.

  • Using Angle $b$:

    • The Opposite side is now across from $b$ ($x$).

    • The Adjacent side is now beside $b$ ($y$).

    • $\sin(b) = \frac{x}{r}$, $\cos(b) = \frac{y}{r}$, $\tan(b) = \frac{x}{y}$.

Solving Right Triangles for Sides and Angles

• Calculator Setup: Ensure the calculator is in Degree Mode. To do this, go to "Mode" and select "Degree" instead of "Radian."

• Example 1: Finding an Unknown Side: A triangle has an angle of $35^{\circ}$, a hypotenuse of $4$, and an unknown adjacent side $x$.

  • Setup: $\cos(35^{\circ}) = \frac{x}{4}$.

  • Algebra: Multiply both sides by $4$ to isolate $x$.

  • Calculation: $4 \times \cos(35^{\circ}) \approx 3.28$.

  • Check: The result ($3.28$) is less than the hypotenuse ($4$), which is a logical requirement.

• Example 2: Unknown in the Denominator: A triangle has an angle of $65^{\circ}$, an opposite side of $1\,cm$, and an unknown hypotenuse $x$.

  • Setup: $\sin(65^{\circ}) = \frac{1}{x}$.

  • Algebra: Multiply by $x$, then divide by $\sin(65^{\circ})$.

  • Shortcut: The unknown $x = \frac{1}{\sin(65^{\circ})}$.

  • Calculation: $x \approx 1.10\,cm$.

• Example 3: Solving for an Angle: Use inverse trigonometric functions ($\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) when the angle is unknown.

  • Setup: A triangle has an opposite side of $5$ and an adjacent side of $2$. Find angle $x$.

  • Ratio: $\tan(x) = \frac{5}{2}$.

  • Calculation: $x = \tan^{-1}(\frac{5}{2}) \approx 68.2^{\circ}$.

Real-World Applications: Sonar Case Study

• Scenario: A destroyer ship detects a submarine using sonar. The submarine is at a depth of $200\,m$. The angle from the destroyer to the sub is $30^{\circ}$.

• Question: How far apart are the vessels on the surface ($x$)?

• Diagram Interpretation: The depth ($200\,m$) is the opposite side relative to the angle. The surface distance ($x$) is the adjacent side.

• Calculation:

  • $\tan(30^{\circ}) = \frac{200}{x}$

  • $x = \frac{200}{\tan(30^{\circ})}$

  • $x \approx 346.41\,m$.

Angles of Elevation and Depression

• Angle of Depression: The angle formed looking downward from a height (e.g., a person on a mountain looking down).

• Angle of Elevation: The angle formed looking upward from the ground (e.g., an ant looking up at the mountain).

• Relationship: The angle of depression is equal to the angle of elevation. This is because they are alternate interior angles formed by a transversal cutting through two parallel lines (the horizontal eye-line and the ground).

Graphing Trigonometric Functions without a Calculator

• Periodic Function: Trig functions are periodic, meaning the graph repeats its shape indefinitely after completing one "period."

• Special Degree Points for Sine ($\sin(x)$):

  • $\sin(0^{\circ}) = 0$

  • $\sin(90^{\circ}) = 1$

  • $\sin(180^{\circ}) = 0$

  • $\sin(270^{\circ}) = -1$

  • $\sin(360^{\circ}) = 0$

• Special Degree Points for Cosine ($\cos(x)$):

  • $\cos(0^{\circ}) = 1$

  • $\cos(90^{\circ}) = 0$

  • $\cos(180^{\circ}) = -1$

  • $\cos(270^{\circ}) = 0$

  • $\cos(360^{\circ}) = 1$

Degrees to Radians Conversion

• When graphing, mathematicians often use radians instead of degrees. Significant conversions include:

  • $0^{\circ} = 0$

  • $90^{\circ} = \frac{\pi}{2}$

  • $180^{\circ} = \pi$

  • $270^{\circ} = \frac{3\pi}{2}$

  • $360^{\circ} = 2\pi$

Characteristics of the Sine and Cosine Parent Functions

• Period: Both $\sin(x)$ and $\cos(x)$ have a standard period of $2\pi$.

• Range: Both functions fluctuate between a maximum y-value of $1$ and a minimum y-value of $-1$.

• The Sine Pattern: Starts at the origin $(0,0)$, goes up to the peak, returns to the middle, goes down to the trough, and returns to the middle middle.

• The Cosine Pattern: Starts at the peak $(0,1)$, goes to the middle, goes down to the trough, returns to the middle, and returns to the peak.

• Comparison: Cosine is essentially the same shape as Sine, just shifted horizontally.

Transformations: Amplitude and Period Changes

• Amplitude ($A$): This is the coefficient in front of the function (e.g., $y = A\sin(x)$).

  • It determines how far up and down the graph moves from the center line.

  • A whole number (e.g., $y = 2\sin(x)$) stretches the graph vertically.

  • A fraction (e.g., $y = \frac{1}{2}\cos(x)$) shrinks the graph vertically.

• Period Change ($b$): This is the coefficient in front of the $x$ variable (e.g., $y = \sin(bx)$).

  • It changes the length of one full cycle.

  • Formula: $\text{New Period} = \frac{2\pi}{b}$.

  • Example 1: $y = \sin(2x)$. Period is $\frac{2\pi}{2} = \pi$. The full graph completes twice within the original $0$ to $2\pi$ range.

  • Example 2: $y = 5\cos(\frac{1}{3}x)$.

    • Amplitude is $5$.

    • Period is $\frac{2\pi}{\frac{1}{3}} = 6\pi$. Within the original $0$ to $2\pi$ range, you would only see one-third of the total graph.

Reflections of Trigonometric Functions

• A negative sign in front of the function reflects the graph across the horizontal axis.

• Negative Sine ($y = -\sin(x)$): Still starts in the middle, but goes down first instead of up.

• Negative Cosine ($y = -\cos(x)$): Starts at the bottom (minimum value) instead of the top peak.

Mixed Transformation Example

• Function: $y = -4\sin(5x)$

• Components Analysis:

  • Amplitude: $4$ (the graph goes from $4$ to $-4$).

  • Period: Use the formula $\frac{2\pi}{5}$. This is the value of the final mark on the x-axis for one cycle.

  • Reflection: Because of the negative sign, start at $(0,0)$ and move downward toward $-4$ first.

• Graphing Steps:

  1. Mark the vertical axis up to $4$ and down to $-4$.

  2. Mark the end of the period at $\frac{2\pi}{5}$.

  3. Create three subdivisions between $0$ and $\frac{2\pi}{5}$.

  4. Plot points following the inverted sine pattern: middle, down, middle, up, middle.