Precal Project

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Math Summative

luonut Maham Saif - T L Kennedy SS (2352)

6/19 Phoenix Math Workshop

luonut Erick Hofacker

Calculus III

luonut Melanie Chua

Chapter 1

luonut Jolie Tang

Chapter 8

Cross Product

Trig

Trigonometry is needed for calculating the cross product when only the magnitudes and the angle are known

The cross product formula uses the sine of the angle between vectors

The direction of the resulting vector follows the right-hand rule: Point your fingers in the direction of A, curl them towards B, and your thumb points in the direction of A × B

The magnitude of the cross product is |A| * |B| * sin(θ), where θ is the angle between A and B

The cross product is anticommutative: A × B = - (B × A)

For two vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), the cross product is given by

A × B = (Ay * Bz - Az * By, Az * Bx - Ax * Bz, Ax * By - Ay * Bx)

Unlike the dot product, the cross product results in a vector that is perpendicular to the plane containing the original vectors

The cross product (also known as the vector product) is an operation that combines two vectors to produce a new vector

Dot Product

Properties

A · B is positive if the angle between A and B is acute, negative if it's obtuse

If the dot product is zero (A · B = 0), the vectors are orthogonal (perpendicular)

The dot product is commutative: A · B = B · A

Calculating

A · B = |A| * |B| * cos(θ), where θ is the angle between A and B

A · B = Ax * Bx + Ay * By

It measures the "projection" of one vector onto another, taking into account their magnitudes and the angle between them

The dot product (also known as the scalar product) is an operation that combines two vectors to produce a scalar (a single number)

Vectors

Scalar Multiplication of Vectors

Multiplying a vector A by scalar 'c' results in a new vector with magnitude |c| * |A| and the same direction if 'c' is positive

Scalar multiplication changes the magnitude of a vector but not its direction

Adding and Subtracting Vectors

The result is the vector that connects the starting point of the first vector to the endpoint of the last vector

Vectors can be added geometrically by placing them head to tail

Trigonometry is used to calculate angles and magnitudes in vector calculations

Vectors can be represented using trigonometric notation (magnitude and direction) or Cartesian notation (components)

Vectors are often represented as arrows pointing in a certain direction with a specific length

They are used to represent various physical quantities like displacement, force, and velocity

Vectors are mathematical quantities that have both magnitude and direction

Parametric Equations

Converting to Cartesian Form

Solve one equation for 't' and substitute into the other equation

You can convert parametric equations to Cartesian form by eliminating the parameter 't'

Parametric equations often involve trigonometric functions

Instead of the usual x and y coordinates, you have equations for x and y separately in terms of a parameter, often denoted as 't'

Parametric equations are a way to express the coordinates of a point in terms of one or more parameters

Chapter 7

Polar Equations and Graphs

Limaçons

Shape depends on the values of 'a' and 'b', creating loops and cusps

Polar equations of the form r = a ± b * cos(θ)

Graphing Polar Equations

Identifying symmetrical patterns and repeating loops

Plotting points by calculating r for various θ values

Common Polar Equations

Spiral: r = a * θ or r = a * e^(bθ), resulting in spiraling patterns

Lemniscate: r² = a² * cos(2θ), creating an infinity symbol

Cardioid: r = a + b * cos(θ) or r = a + b * sin(θ), forming heart-like shapes

Circle: r = a, where 'a' is the radius

They allow us to describe complex shapes and patterns not easily represented in Cartesian coordinates

Polar equations express relationships between the distance (r) from the origin and the angle (θ) from the positive x-axis

Polar Coordinates

Problem Solving

Analyzing curves with rotational symmetry

Calculating areas and distances using polar coordinates

Solving problems involving circular or radial symmetry

Limitations

Take care with negative angles and ambiguous notation

Not suitable for every situation, particularly when Cartesian coordinates are more intuitive

Introduction

x = r * cos(θ), y = r * sin(θ)

Polar coordinates (r, θ) represent points using distance from the origin (r) and angle (θ) from the positive x-axis

Cartesian coordinates (x, y) represent points in a plane using horizontal and vertical distances

Area of a Triangle

Using Trigonometry for Area

Helpful when angles and two sides are provided

Area = 0.5 * a * b * sin(C)

If two sides and the angle between them are known, the area can be calculated using trigonometric ratios

Surveying and cartography for determining land areas

Geometry and construction for calculating surface areas

Law of Cosines

Projectile motion and collision analysis

Designing structures, determining forces and tensions in bridges

Calculating distances between points on Earth's surface

Ambiguous Case (SSS)

When given all three sides, multiple triangles might satisfy the given conditions

Occurs when the side lengths form a triangle that can be "folded" to create two different triangles

Formulas

c² = a² + b² - 2ab * cos(C)

b² = a² + c² - 2ac * cos(B)

a² = b² + c² - 2bc * cos(A)

Law of Sines

Astronomy for calculating distances to celestial bodies

Ambiguous Case (SSA)

When given two sides and a non-included angle, there can be two, one, or no possible triangles

Solving

Ambiguous Case: When given two sides and an angle opposite one of them, there might be multiple solutions

Given two sides and a non-included angle, use the Law of Sines to find the missing parts

Law of Sines Intro

Ratio of a side's length to the sine of the opposite angle is constant for all sides and angles

sin(A) / a = sin(B) / b = sin(C) / c

Typically useful when solving triangles that aren't right triangles

Right Triangle Trig

Real-world Applications

Engineering for designing structures and systems

Navigation, such as determining distances from elevation changes

Special Right Triangles

30-60-90 triangle: One angle is 30 degrees, the other is 60 degrees

45-45-90 triangle: Legs are congruent, angles are 45 degrees each

Trigonometric Ratios in Right Triangles

Tangent Ratio (tanθ) = Opposite / Adjacent

Cosine Ratio (cosθ) = Adjacent / Hypotenuse

Sine Ratio (sinθ) = Opposite / Hypotenuse

Right Triangles

The side opposite the right angle is the hypotenuse, while the other two sides are the legs

One angle is exactly 90 degrees

Chapter 6

Sum to Product and Product to Sum

Product to Sum Formulas

1/2 * [cos(A - B) - cos(A + B)]

sin(A) * sin(B) or cos(A) * cos(B)

Applications

Converting trigonometric equations for easier solving

Calculating areas in polar coordinates

Simplifying integrals and trig expressions

Sum to Product Formulas

2 * sin((A + B) / 2) * cos((A - B) / 2)

sin(A) + sin(B) or cos(A) + cos(B)

Double and Half Angle Formulas

Applications in Trig Equations and Calculations

Simplifying trig expressions in calculus problems

Calculating and unknowns in scenarios with reduced angles

Formulas for Half Angles

This is utilized when the problem involves halving

sin(A/2), cos(A/2), tan(A/2)

Derivation and Formulas for Double Angles

This is utilized when solving trig equations and or simplifying

sin(2A), cos(2A), tan(2A)

Sum and Difference Formulas

Applications in Trig Equations

It helps when determining unknown angles in geometric setups

Solving problems involving two angles

Tangent of Sum/Difference

Is applied to problems involving angle of elevation

tan(A ± B) is the formula

Sine and Cosine of Sum/Difference

This is used when combining or amplifying waveforms

sin(A ± B) and cos(A ± B) are the formulas

Inverse Trig Functions

Applications in Geometry and Physics

Solving for angles in real world problems, like inclined planes

Angle measures in right triangles which are used later in the course.

Evaluating Inverse Trig Functions

Finding multiple solutions.

Using calculator or tables to find angle given ratios

Definitions and Notations

Inverse sine (arcsin), Inverse cosine (arccos), Inverse tangent (arctan).

Trig Equations

Periodic Solutions

Graphing Trig Functions and Solutions

Identifying patterns and trends by looking at the graph

Multiple Solutions and General Solutions

General solutions are also found using parameter "n"

Trig functions are periodic and have multiple solutions

Solving Basic Trig Equations

Solving basic trigonometry equations helps identify periodic solutions

You must isolate the trig function by itself in order to solve the problem.

Trig Identities

Fundamental Identites

Quotient Identities: tan(θ) = sin(θ)/cos(θ), cot(θ) = cos(θ)/sin(θ)

Reciprocal Identities: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ)

Pythagorean Identity: sin²(θ) + cos²(θ) = 1