Math12 lesson8
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Trigonometric equations involve trigonometric functions and can be solved using algebraic techniques like isolating variables, collecting like terms, and factoring. There are two types of trigonometric equations: identical equations that are true for all values of the variable, and conditional equations that are true for some but not all values. To solve trigonometric equations, one should use identities to obtain a single function, substitute values to solve for an angle, or use the quadratic formula if a single function is quadratic.
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moments couples and force couple systems by ahmad khan
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The document discusses various topics related to differential equations including:
1. Linear differential equations of nth order with constant coefficients and the characteristic equation. The general solution depends on whether the roots of the characteristic equation are real/complex, distinct/multiple.
2. Finding the complementary function and particular integral to get the general solution for a second order differential equation with constant coefficients.
3. Solving simultaneous linear differential equations using the D-operator method and Laplace transform method.
4. Changing dependent and independent variables to solve second order differential equations numerically using an approximation technique similar to the trapezoid rule.
Similar to Math12 lesson8 (11)
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
Activity 1 (Directional Derivative and Gradient with minimum 3 applications)....
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13 sequences and series
The document defines sequences and their different types, including arithmetic and geometric sequences. It provides examples of finding the nth term or general term of sequences given initial terms. It also discusses how to find the common difference of arithmetic sequences and common ratio of geometric sequences. The document explains how to calculate arithmetic means, harmonic means, and geometric means of sequences. It provides information on finding the sums of finite and infinite arithmetic, harmonic, and geometric sequences.
Math10 (mwf)course calendar 1 st qtr 2011 12
The document provides a course calendar and summary score sheet for a differential equations class over two quarters. It lists the dates, topics to be covered from the textbook, assigned problem sets and projects, and spaces to record scores on quizzes, problem sets, and final grades. The course covers key topics in differential equations over 31 class days each quarter, including definitions, families of curves, exact and linear equations, integrating factors, undetermined coefficients, variation of parameters, and Laplace transforms. It assesses students with three long quizzes and records their ongoing and total scores on problem sets and projects.
8 polynomial functions
The document discusses polynomial functions. It defines a polynomial function and distinguishes it from other function types. It lists the objectives of understanding degree, using theorems to determine factors and roots, and approximating graphs of polynomial functions. Key points are that every polynomial equation has at least one real or complex root based on the Fundamental Theorem of Algebra, and that a polynomial of degree n will have n roots.
7 functions
This document defines functions and relations. It discusses identifying the domain and range of functions and relations, evaluating functions, and performing operations on functions such as addition, subtraction, multiplication, division, and composition. It also covers graphing functions, including piecewise functions, absolute value functions, greatest and least integer functions. Key examples are provided to illustrate how to identify domains and ranges, evaluate functions, perform operations on functions, and graph different types of functions.
Malaysia and indonesian
Malaysian and Indonesian literature share a common linguistic ancestor before the arrival of Islam in the 14th century. Early Malay literature was influenced by Indian epics, while 19th century works included religious texts, romances, and tales. Indonesian literature can be divided into periods such as the Pujangga Lama (Old Poets) and AngkatanBalaiPustaka (Colonial Office for Popular Literature generation). Major genres include syair (narrative poetry), pantun (quatrains), and hikayat (stories and chronicles). By the 1920s, fiction and Western-style works replaced earlier forms. Thai traditional literature focused on Buddhism and Hinduism through various emotive verse forms, exempl
Math12 lesson11
This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions have the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and have the form f(x) = logbx. Their graphs stretch downwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula and rules for simplifying expressions.
- Exponential and logarithmic equations can be solved
Math12 lesson10
Spherical trigonometry concerns triangles on the surface of a sphere, with formulas to solve for unknown sides and angles of spherical triangles based on relationships between opposite, adjacent, and included parts. Formulas are provided for right, oblique, and general spherical triangles, along with the six cases for solving oblique triangles based on given parts. Examples demonstrate applying the formulas to solve various spherical triangle problems.
Math12 lesson9
The document derives the Law of Sines and Law of Cosines, which relate the angles and sides of triangles. It discusses using these laws to solve oblique triangles given certain information like two angles and a side, two sides and the angle opposite one of the sides, two sides and the angle between them, or all three sides. It also covers finding the area of a triangle given two sides and the included angle, or using Heron's Formula with all three sides. Application problems demonstrate using these concepts to solve real-world geometry problems.
Math12 lesson7
The document defines inverse trigonometric functions. It states that trigonometric functions are not one-to-one but their domains can be restricted to make them one-to-one, allowing inverses. The inverses of sinx, cosx, tanx, cotx, secx, and cscx are then defined. Examples are given of finding exact values of inverse trigonometric functions.
Systems of equations lesson 5
The document provides information about solving systems of linear and nonlinear equations:
- It discusses the three types of solutions to systems of linear equations in two and three variables - one solution, no solution, and infinitely many solutions. Methods for solving include substitution, elimination, and graphing.
- Systems of nonlinear equations like quadratics are also introduced.
- Several examples of application problems involving systems of equations with two or three variables are presented and solved.
Math12 lesson 6
This document provides an overview of trigonometric identities and examples of their applications. It discusses basic identities, verifying identities, and advanced identities involving sums, differences, doubles angles, and half angles. Examples are provided for applying product-to-sum and sum-to-product identities. The objectives are to review and apply various trigonometric identities to simplify expressions and evaluate functions.
Inequalities lesson 4
The document discusses inequalities and their solutions. It defines absolute and conditional inequalities and explains how to represent solutions using interval, set, and graphical notation. Methods are presented for solving linear, polynomial, rational, and absolute value inequalities. Key steps include determining intervals where an expression is positive or negative, identifying valid intervals based on inequality signs, and expressing the solution in interval notation. Examples are provided throughout to demonstrate these techniques.
Quadratic equations lesson 3
The document discusses various methods for solving quadratic equations, including factoring, square root method, completing the square, and the quadratic formula. It also covers solving other types of equations that are quadratic in form, such as radical equations, through transformations. The objectives are to solve quadratic, radical, and other equations that are quadratic in form and to find sums and products of roots, the quadratic equation given roots, and solve application problems involving these equation types.
Week 1 2 linear equations (2)
The document provides an overview of lessons on linear equations for a math class over 3 days. Key points covered include:
- Day 1 focuses on defining linear equations, properties of equality, and solving linear and rational equations.
- Day 2 covers solving equations involving radicals, absolute values, and literal equations.
- Day 3 involves word problems including number, geometry, digit, investment, age, mixture, uniform motion, work, and clock problems that can be modeled and solved using linear equations.
Kong zi
Kong Zi was a philosopher who proposed a socio-political-moral philosophy based on benevolence, righteousness, propriety, wisdom, and sincerity. He aimed to create an ideal, harmonious society with well-ordered structure and citizens with mutual obligations. Kong Zi transmitted ancient wisdom through instruction, rather than proposing new ideas or religion. His teachings were recorded in Lun Yu as instructive sayings between Kong Zi and his disciples. Key concepts in his philosophy included Ren (benevolence or goodness), Zhong (rectitude), Yi (duty), Li (ritual propriety), and Zhi (wisdom and virtue).
Week3 ratio
This document discusses different types of variation that can be used to model relationships between quantities: direct variation occurs when one quantity is directly proportional to another; inverse variation occurs when one quantity is inversely proportional to another; joint variation occurs when one quantity is proportional to the product of two or more other quantities; and combined variation occurs when one quantity is directly proportional to some quantities and inversely proportional to others. Examples are provided to illustrate direct, inverse, joint and combined variation and how to write equations to describe each type of variation. Classwork and homework exercises are assigned for students to practice applying these concepts.
Math12 lesson 4
This document discusses graphs of trigonometric functions. It aims to teach students how to develop properties of trig functions, find the amplitude, period, phase shift and vertical shift from equations, and graph trig functions. Examples are provided to illustrate finding these characteristics from trig function equations and using them to graph the functions.
Educational system ii
The educational system in the Philippines took over 3 and a half centuries to develop after the Spanish colonized the islands. The Spanish friars first introduced parochial schools modeled after those used in Europe during the Dark Ages. In 1591, Governor Dasmarinas ordered that taxes from land grants be used to support schools, though the friars taught in native dialects rather than Spanish. The early schools run by friars were simple primary schools focused on religious instruction, not graded systems, and were governed by the friars with little oversight. Formal schools began emerging in the 18th and 19th centuries focused on subjects like artillery, navigation, commerce, and the arts.
Math12 lesson 3
1) This document provides lesson objectives and examples for solving right triangles using trigonometric functions, the Pythagorean theorem, and angle relationships. It defines trigonometric ratios, angle of elevation/depression, bearing, and course.
2) Examples are provided to solve right triangles, find missing angles and sides, and solve real-world problems involving width of a stream, height of a flagpole, camera angle of depression, and height of a tower.
3) Additional examples solve problems involving slant distance to a sunken ship, plane bearings and courses between locations, and references are provided for further reading.
Math12 lesson 2
This document discusses trigonometric functions of right triangles. It defines the six trigonometric functions as ratios of the sides of a right triangle. It also discusses reciprocal functions, the Pythagorean theorem, functions of complementary angles, and the trigonometric functions of specific angles like 45°, 30° and 60° degrees. Examples are provided to evaluate trigonometric functions and to solve right triangles.
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Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
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Math12 lesson8
- 2. DEFINITION:TRIGONOMETRIC EQUATIONS Equations that involve the trigonometric functions are called trigonometric equations. Solving a trigonometric equation means the same thing as solving an algebraic equation, like finding the values of the variable that satisfy the equation. To solve trigonometric equation, we use the same techniques used in algebraic equations such as isolating the variable, collecting like terms and factoring. In addition to these techniques, we can also simplify the equation by substituting trigonometric identities.
- 4. GUIDELINES IN SOLVING TRIGONOMETRIC EQUATIONS If the equation contains one function of a single angle, use algebraic technique to solve for the angle. Solve a quadratic equation containing a single function of the same angle by factoring. Otherwise, use quadratic formula. If the equation contains several functions of the same angle, substitute trigonometric identities to obtain a single function. If the equation contains several angles, substitute trigonometric identities to obtain a function of a single angle.
- 5. EXAMPLE: Find the exact solutions to each equation for the interval . II. Find the exact solutions to each equation for the interval .
- 6. Application: If a projectile is fired with velocity V0 at an angle ,then its range, the horizontal distance it travels (in feet ), is modeled by the function . If V0 =2200 ft/sec, what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 feet away?
Editor's Notes
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