Chapter 6

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

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

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

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

Solving basic trigonometry equations
helps identify periodic solutions

Trig functions are periodic
and have multiple solutions

General solutions are also
found using parameter "n"

Identifying patterns and trends
by looking at the graph

Graphing Trig Functions and Solutions

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

Using calculator or tables to
find angle given ratios

Finding multiple solutions.

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

Solving for angles in real world
problems, like inclined planes

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

This is used when combining
or amplifying waveforms

tan(A ± B) is the formula

Is applied to problems
involving angle of elevation

Solving problems involving two angles

It helps when determining unknown angles
in geometric setups

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

This is utilized when solving trig
equations and or simplifying

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

This is utilized when the problem
involves halving

Calculating and unknowns in scenarios
with reduced angles

Simplifying trig expressions in
calculus problems

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

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

Simplifying integrals and trig expressions

Calculating areas in polar coordinates

Converting trigonometric equations for easier solving

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

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

Chapter 7

One angle is exactly 90 degrees

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

Sine Ratio (sinθ) = Opposite / Hypotenuse

Cosine Ratio (cosθ) = Adjacent / Hypotenuse

Tangent Ratio (tanθ) = Opposite / Adjacent

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

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

Navigation, such as determining distances from elevation changes

Engineering for designing structures and systems

Typically useful when solving triangles that aren't
right triangles

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

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

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

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

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

Astronomy for calculating distances to celestial bodies

a² = b² + c² - 2bc * cos(A)

b² = a² + c² - 2ac * cos(B)

c² = a² + b² - 2ab * cos(C)

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

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

Calculating distances between points on Earth's surface

Designing structures, determining forces and tensions in bridges

Projectile motion and collision analysis

Geometry and construction for calculating surface areas

Surveying and cartography for determining land areas

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

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

Helpful when angles and two sides are provided

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

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

x = r * cos(θ), y = r * sin(θ)

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

Take care with negative angles and ambiguous notation

Solving problems involving circular or radial symmetry

Calculating areas and distances using polar coordinates

Analyzing curves with rotational symmetry

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

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

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

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

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

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

Plotting points by calculating r for various θ values

Identifying symmetrical patterns and repeating loops

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

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

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

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 often involve trigonometric functions

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

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

Vectors are mathematical quantities that have both magnitude and direction

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

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

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

Trigonometry is used to calculate angles and magnitudes in vector calculations

Vectors can be added geometrically by placing them head to tail

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

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

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

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

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

A · B = Ax * Bx + Ay * By

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

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

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

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

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

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

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)

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

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

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 cross product formula uses the sine of the angle between vectors

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