Trigonometry: Angles and Triangles

Trigonometry

Angles and Triangles

• Angle measurements: Degrees, radians, and conversions between them
• Triangles: Right-angled triangles, oblique triangles, and Pythagorean triples
• Angle relationships: Complementary, supplementary, and corresponding angles

Trigonometric Ratios

• Sine (sin): Opposite side over hypotenuse
• Cosine (cos): Adjacent side over hypotenuse
• Tangent (tan): Opposite side over adjacent side
• Cotangent (cot): Adjacent side over opposite side
• Secant (sec): Hypotenuse over adjacent side
• Cosecant (csc): Hypotenuse over opposite side

Trigonometric Identities

• Pythagorean identity: sin^2(A) + cos^2(A) = 1
• Sum and difference formulas: sin(A+B), cos(A+B), sin(A-B), cos(A-B)
• Double and half angle formulas: sin(2A), cos(2A), sin(A/2), cos(A/2)

Graphs of Trigonometric Functions

• Sine and cosine waves: Period, amplitude, and phase shift
• Tangent and cotangent graphs: Period and asymptotes

Applications of Trigonometry

• Triangles and waves: Solving triangles, modeling periodic phenomena
• Analytic geometry: Coordinate geometry and trigonometric functions
• Physics and engineering: Describing motion, forces, and energies

Trigonometry

Angles and Triangles

• Angle measurements can be expressed in degrees or radians, with 1 radian equal to 180/π degrees.
• Right-angled triangles have one right angle (90°), and oblique triangles have no right angles.
• Pythagorean triples are sets of integers that satisfy the Pythagorean theorem, such as 3-4-5.

Trigonometric Ratios

• Sine (sin) is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
• Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle.
• Cotangent (cot) is the ratio of the adjacent side to the opposite side in a right-angled triangle.
• Secant (sec) is the ratio of the hypotenuse to the adjacent side in a right-angled triangle.
• Cosecant (csc) is the ratio of the hypotenuse to the opposite side in a right-angled triangle.

Trigonometric Identities

• The Pythagorean identity states that sin²(A) + cos²(A) = 1 for any angle A.
• The sum and difference formulas for sine and cosine allow for the calculation of trigonometric ratios for sums and differences of angles.
• The double and half angle formulas for sine and cosine allow for the calculation of trigonometric ratios for double and half angles.

Graphs of Trigonometric Functions

• Sine and cosine waves have a period of 360° (or 2π radians), and can be shifted horizontally or vertically.
• The amplitude of a sine or cosine wave is the maximum displacement from the mean position.
• The phase shift of a sine or cosine wave is the horizontal displacement from the origin.
• Tangent and cotangent graphs have vertical asymptotes at certain angles.

Applications of Trigonometry

• Trigonometry is used to solve triangles, including finding unknown sides and angles.
• Trigonometry is used to model periodic phenomena, such as sound waves and light waves.
• Trigonometry is used in analytic geometry to describe coordinates and geometric shapes.
• Trigonometry is used in physics and engineering to describe motion, forces, and energies.

Numbers

• Natural numbers are also known as counting numbers, and they start from 1 (e.g. 1, 2, 3,...)
• Whole numbers include natural numbers and zero (e.g. 0, 1, 2, 3,...)
• Integers include whole numbers and negative numbers (e.g. ..., -3, -2, -1, 0, 1, 2, 3,...)
• Rational numbers can be expressed as a fraction (e.g. 1/2, 3/4), and they can be either positive or negative
• Irrational numbers cannot be expressed as a fraction (e.g. π, e), and they are non-repeating and non-terminating
• Real numbers include both rational and irrational numbers, making them a comprehensive set of numbers
• Complex numbers include real numbers and imaginary numbers (e.g. 3 + 4i), and they are used to represent quantities with both magnitude and direction

Algebra

• Variables are letters or symbols that represent unknown values, and they are used to solve equations and inequalities
• Constants are numbers that do not change, and they are used to perform mathematical operations
• Expressions are combinations of variables, constants, and mathematical operations, and they can be simplified or evaluated
• Equations are statements that express the equality of two expressions, and they can be solved using algebraic methods
• Inequalities are statements that compare two expressions using greater than, less than, or equal to, and they can be solved using algebraic methods

Geometry

• Points are locations in space, represented by coordinates (x, y), and they can be used to define shapes and figures
• Lines are sets of points that extend infinitely in two directions, and they can be used to define shapes and figures
• Angles are formed by two lines or rays sharing a common endpoint, and they can be acute, obtuse, or right angles
• Triangles are polygons with three sides and three angles, and they can be classified into different types (e.g. equilateral, isosceles, scalene)
• Quadrilaterals are polygons with four sides and four angles, and they can be classified into different types (e.g. rectangles, squares, trapezoids)

Measurement

• Units are standard amounts used to measure quantities, such as inches, meters, and grams
• Conversion involves changing from one unit to another, such as from inches to feet or from pounds to kilograms
• Perimeter is the distance around a shape, and it can be calculated using different formulas depending on the shape
• Area is the size of a two-dimensional shape, and it can be calculated using different formulas depending on the shape
• Volume is the amount of space inside a three-dimensional shape, and it can be calculated using different formulas depending on the shape

Calculus

• Limits involve the behavior of a function as the input approaches a certain value, and they are used to define the basic operations of calculus
• Derivatives are rates of change of functions, and they are used to find the slope of a tangent line to a curve at a point
• Integrals are accumulations of quantities over a defined interval, and they are used to find the area under a curve or the volume of a solid
• Applications of calculus involve optimization, motion, and related rates, and they are used to solve real-world problems in fields such as physics, engineering, and economics

Statistics

• Data is a collection of information, and it can be qualitative or quantitative
• Graphs are visual representations of data, and they can be used to display and analyze data
• Mean is the average value of a dataset, and it can be calculated using different methods depending on the dataset
• Median is the middle value of a dataset when in order, and it can be used to describe the central tendency of a dataset
• Mode is the most frequently occurring value in a dataset, and it can be used to describe the central tendency of a dataset
• Probability is the chance of an event occurring, and it can be calculated using different methods depending on the event and the dataset