Here are the key steps to solve this problem:
1) Use the half-angle formula for cosine:
cos(2α) = 1 - 2sin^2(α)
2) Given: sec(α) = 2
Use the identity: sec^2(α) = 1 + tan^2(α)
Solve for tan(α): tan(α) = -√3
3) Substitute tan(α) = -√3 into the half-angle formula for sine:
sin(α) = ±√(1-cos^2(α)/2)
sin(α) = ±1/2
4) Given that tan(α) < 0
This document contains derivations of trigonometric identities. It simplifies expressions involving trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant by rewriting them in terms of other trigonometric functions or common denominators. It also uses factoring and trigonometric angle addition and subtraction formulas to derive new identities from existing ones. The document derives over a dozen new trigonometric identities through algebraic manipulation and rearrangement of trigonometric expressions.
This document contains solutions to trigonometric equations on various intervals. It first determines that 4π/5 is not a solution to 2sin(2θ)=0. It then finds that 4π/5 and 7π/6 are solutions to 2cos(3θ)-5=0 on the interval [0,2π]. Similarly, it finds all solutions to equations involving tangent, cosine, sine and combinations of trigonometric functions on specified intervals through repetitive use of trigonometric identities and interval arithmetic.
The document provides a trigonometry diagnostic exam with 4 problems:
1) Find trig functions if sinθ = 3/5
2) Find trig functions if secM = 6/5
3) Find 6 trig functions of angle P
4) Solve a trig expression given sin, tan, cos values
The problems require finding trig functions based on a given value, expressing answers in simplest form. Students have 10 minutes to complete the problems.
1) The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for trig functions, inverse trig functions, and laws of sines, cosines, and tangents.
2) Tables give values of trig functions for angles on the unit circle, along with properties like domain, range, and periodicity.
3) The cheat sheet is a reference for definitions, formulas, and properties of trigonometric functions.
This document introduces mathematical concepts in QBASIC programming language. It discusses arithmetic calculations like addition, subtraction, multiplication, division and modulus operator. It explains how decimal number system works and how to obtain individual digits of a number using modulus and integer division operators in a while loop. It also demonstrates different patterns that can be generated using for loops and techniques to reverse, find sum and product of digits in a number. Finally, it describes some inbuilt string functions in QBASIC for operations like extracting characters, reversing, checking palindromes and counting vowels/consonants in a string.
This document provides formulas and definitions for trigonometric functions including the definitions of sine, cosine, and tangent using right triangles and the unit circle. It also includes information on domains, ranges, periods, identities, inverse trig functions, complex numbers, conic sections, and formulas for working with angles in degrees and radians. Key aspects covered are the definitions of trig functions, trig identities, inverse trig functions, and formulas for circles, ellipses, hyperbolas, and parabolas.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
The document discusses Fourier series and periodic functions. It provides:
- Definitions of Fourier series and periodic functions.
- Examples of periodic functions including trigonometric and other functions.
- Euler's formulae for calculating the coefficients of a Fourier series.
- Integration properties used to solve Fourier series problems.
- Two examples of determining the Fourier series for given periodic functions and using it to deduce mathematical results.
This document contains derivations of trigonometric identities. It simplifies expressions involving trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant by rewriting them in terms of other trigonometric functions or common denominators. It also uses factoring and trigonometric angle addition and subtraction formulas to derive new identities from existing ones. The document derives over a dozen new trigonometric identities through algebraic manipulation and rearrangement of trigonometric expressions.
This document contains solutions to trigonometric equations on various intervals. It first determines that 4π/5 is not a solution to 2sin(2θ)=0. It then finds that 4π/5 and 7π/6 are solutions to 2cos(3θ)-5=0 on the interval [0,2π]. Similarly, it finds all solutions to equations involving tangent, cosine, sine and combinations of trigonometric functions on specified intervals through repetitive use of trigonometric identities and interval arithmetic.
The document provides a trigonometry diagnostic exam with 4 problems:
1) Find trig functions if sinθ = 3/5
2) Find trig functions if secM = 6/5
3) Find 6 trig functions of angle P
4) Solve a trig expression given sin, tan, cos values
The problems require finding trig functions based on a given value, expressing answers in simplest form. Students have 10 minutes to complete the problems.
1) The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for trig functions, inverse trig functions, and laws of sines, cosines, and tangents.
2) Tables give values of trig functions for angles on the unit circle, along with properties like domain, range, and periodicity.
3) The cheat sheet is a reference for definitions, formulas, and properties of trigonometric functions.
This document introduces mathematical concepts in QBASIC programming language. It discusses arithmetic calculations like addition, subtraction, multiplication, division and modulus operator. It explains how decimal number system works and how to obtain individual digits of a number using modulus and integer division operators in a while loop. It also demonstrates different patterns that can be generated using for loops and techniques to reverse, find sum and product of digits in a number. Finally, it describes some inbuilt string functions in QBASIC for operations like extracting characters, reversing, checking palindromes and counting vowels/consonants in a string.
This document provides formulas and definitions for trigonometric functions including the definitions of sine, cosine, and tangent using right triangles and the unit circle. It also includes information on domains, ranges, periods, identities, inverse trig functions, complex numbers, conic sections, and formulas for working with angles in degrees and radians. Key aspects covered are the definitions of trig functions, trig identities, inverse trig functions, and formulas for circles, ellipses, hyperbolas, and parabolas.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
The document discusses Fourier series and periodic functions. It provides:
- Definitions of Fourier series and periodic functions.
- Examples of periodic functions including trigonometric and other functions.
- Euler's formulae for calculating the coefficients of a Fourier series.
- Integration properties used to solve Fourier series problems.
- Two examples of determining the Fourier series for given periodic functions and using it to deduce mathematical results.
This document summarizes key concepts in singular perturbation problems. It provides an example problem of determining concentration profiles in a catalyst pellet where the reaction rate is fast compared to diffusion. Singular perturbation problems require rescaling to identify dominant terms as the perturbation parameter approaches zero. Solutions in different regions must then be matched asymptotically. The example demonstrates obtaining outer and inner solutions, and matching them to solve the full problem.
Mathematics assignment sample from assignmentsupport.com essay writing services https://writeessayuk.com/
The document proves the product rule for derivatives. It begins by writing the derivative of fg as the limit definition. It then subtracts and adds fg(x) to rewrite this in a form where the limit can be split into two pieces. Taking the limits individually and factoring terms provides the product rule, where the derivative of fg is f'g + fg'.
System dynamics 3rd edition palm solutions manualSextonMales
System dynamics 3rd edition palm solutions manual
Full download: https://goo.gl/7Z6QZ3
People also search:
system dynamics palm 3rd edition pdf
system dynamics palm 3rd edition solutions pdf
system dynamics palm 3rd edition free pdf
system dynamics palm pdf
system dynamics palm 3rd edition ebook
system dynamics 3rd edition ogata pdf
system dynamics palm 2nd edition solution manual
system dynamics palm 3rd edition academia
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
This document contains 5 limit calculation problems:
1) Finding the limit as x approaches infinity of the fraction x5-2x4+3x2-2 over 3x5-2x+1.
2) Finding the limit as x approaches 2 of the difference of two fractions.
3) Finding the limit as x approaches 1 of the difference of two fractions.
4) Finding the limit as x approaches 4 of the fraction x-4 over 1-√x-3.
5) Finding the limit as x approaches 1 of the fraction x2-5x+4 over x3-1.
This document contains 5 limit calculation problems:
1) Finding the limit as x approaches infinity of the fraction x5-2x4+3x2-2 over 3x5-2x+1.
2) Finding the limit as x approaches 2 of the difference of two fractions.
3) Finding the limit as x approaches 1 of the difference of two fractions.
4) Finding the limit as x approaches 4 of the fraction x-4 over 1-√x-3.
5) Finding the limit as x approaches 1 of the fraction x2-5x+4 over x3-1.
Multiple Choice Questions of Successive Differentiation (Calculus) for B.Sc. 1st Semester (Panjab University ) Mathematics students.There are 2o questions with answer keys.
pedagogy of mathematics part ii (numbers and sequence - ex 2.7), numbers and sequences, Std X samacheer Kalvi, Geometric progression, definition of geometric progression, general form of geometric progression, general term of geometric progression,
This document describes a novel steganographic method for hiding data in JPEG images. It proposes improvements to the existing matrix encoding (F5) technique. Specifically, it introduces overlapped matrix encoding, modified matrix encoding using 2 or 3 coefficient flips instead of 1, and an insert-remove approach. Experimental results show this method achieves higher data hiding capacity while decreasing detectability compared to the original F5 technique according to steganalysis using 274 features.
This document provides formulas and examples for calculating the distance and midpoint between two points.
The distance formula is used to find the distance between two points (x1, y1) and (x2, y2). The midpoint formula finds the midpoint between two points by taking the average of the x-coordinates and y-coordinates.
Several examples demonstrate using the distance and midpoint formulas to find distances, midpoints, and missing endpoint coordinates. The key steps are to work inside parentheses, do exponents before addition, and plug points into the formulas.
The document discusses coordinate geometry and determining the position of a point P that divides a line segment AB based on a ratio m:n. It provides examples of finding the coordinates of points that divide line segments in different ratios. It also covers topics related to the gradient of a line, parallel and perpendicular lines, and finding the midpoint and length of a line segment.
This document discusses the technique of substitution for evaluating indefinite integrals. It defines the differential du in terms of a function u and its derivative, and shows how to rewrite integrals in terms of u and du. Examples are provided of using substitution to evaluate integrals involving roots, exponents, logarithms, and trigonometric functions. The document also addresses cases where the original variable remains after substitution and provides a practice problem to apply the technique.
The document provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. It asks to:
1) Solve quadratic equations by factoring.
2) Solve quadratic equations by completing the square.
3) Solve quadratic equations using the quadratic formula.
4) Determine the value of p that makes a given quadratic equation have equal roots.
This document contains information about basic math solutions for the third semester intended for self-study. It includes sample solutions to exercises on elementary group theory, including determining if a set forms a group based on properties like closure, identity, and inverses. It also contains multiplication tables and worked examples of checking properties for specific sets and binary operations.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
The document discusses addition and subtraction formulas for trigonometric functions. It provides examples of using these formulas to find the exact trigonometric values of angles that are not on the unit circle, such as sin(75°) and cos(265°). It then works through examples of using the addition formulas to find sin(105°) and cos(5π/12), providing step-by-step workings. The document also proves several trigonometric identities using the addition formulas, such as tan(θ + π/4) = (tanθ + 1)/(1 - tanθ).
1. The document provides steps to solve 6 trigonometric equations algebraically.
2. For the first equation, the solutions are θ = 0 and θ = π.
3. The second equation has solutions of θ = π/3 and θ = 5π/3.
4. The sixth equation has the solution cos(x) = -1/3.
This document provides fully worked solutions to exam questions from Form 4 mathematics chapters on standard form, quadratic expressions and equations, sets, mathematical reasoning, the straight line, and statistics. The solutions include:
1) Detailed working to obtain the answers for multiple choice and structured questions.
2) Explanations of mathematical concepts and reasoning such as determining gradients, interpreting graphs, and identifying argument forms.
3) Step-by-step derivations to find equations of lines from given points and gradients.
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for domain, range, period, identities, inverses, and laws involving trig functions like the Law of Sines, Cosines, and Tangents. Key formulas include definitions of sine, cosine, tangent and their inverses, as well as the Pythagorean, double angle, and sum and difference identities.
This document summarizes key concepts in singular perturbation problems. It provides an example problem of determining concentration profiles in a catalyst pellet where the reaction rate is fast compared to diffusion. Singular perturbation problems require rescaling to identify dominant terms as the perturbation parameter approaches zero. Solutions in different regions must then be matched asymptotically. The example demonstrates obtaining outer and inner solutions, and matching them to solve the full problem.
Mathematics assignment sample from assignmentsupport.com essay writing services https://writeessayuk.com/
The document proves the product rule for derivatives. It begins by writing the derivative of fg as the limit definition. It then subtracts and adds fg(x) to rewrite this in a form where the limit can be split into two pieces. Taking the limits individually and factoring terms provides the product rule, where the derivative of fg is f'g + fg'.
System dynamics 3rd edition palm solutions manualSextonMales
System dynamics 3rd edition palm solutions manual
Full download: https://goo.gl/7Z6QZ3
People also search:
system dynamics palm 3rd edition pdf
system dynamics palm 3rd edition solutions pdf
system dynamics palm 3rd edition free pdf
system dynamics palm pdf
system dynamics palm 3rd edition ebook
system dynamics 3rd edition ogata pdf
system dynamics palm 2nd edition solution manual
system dynamics palm 3rd edition academia
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
This document contains 5 limit calculation problems:
1) Finding the limit as x approaches infinity of the fraction x5-2x4+3x2-2 over 3x5-2x+1.
2) Finding the limit as x approaches 2 of the difference of two fractions.
3) Finding the limit as x approaches 1 of the difference of two fractions.
4) Finding the limit as x approaches 4 of the fraction x-4 over 1-√x-3.
5) Finding the limit as x approaches 1 of the fraction x2-5x+4 over x3-1.
This document contains 5 limit calculation problems:
1) Finding the limit as x approaches infinity of the fraction x5-2x4+3x2-2 over 3x5-2x+1.
2) Finding the limit as x approaches 2 of the difference of two fractions.
3) Finding the limit as x approaches 1 of the difference of two fractions.
4) Finding the limit as x approaches 4 of the fraction x-4 over 1-√x-3.
5) Finding the limit as x approaches 1 of the fraction x2-5x+4 over x3-1.
Multiple Choice Questions of Successive Differentiation (Calculus) for B.Sc. 1st Semester (Panjab University ) Mathematics students.There are 2o questions with answer keys.
pedagogy of mathematics part ii (numbers and sequence - ex 2.7), numbers and sequences, Std X samacheer Kalvi, Geometric progression, definition of geometric progression, general form of geometric progression, general term of geometric progression,
This document describes a novel steganographic method for hiding data in JPEG images. It proposes improvements to the existing matrix encoding (F5) technique. Specifically, it introduces overlapped matrix encoding, modified matrix encoding using 2 or 3 coefficient flips instead of 1, and an insert-remove approach. Experimental results show this method achieves higher data hiding capacity while decreasing detectability compared to the original F5 technique according to steganalysis using 274 features.
This document provides formulas and examples for calculating the distance and midpoint between two points.
The distance formula is used to find the distance between two points (x1, y1) and (x2, y2). The midpoint formula finds the midpoint between two points by taking the average of the x-coordinates and y-coordinates.
Several examples demonstrate using the distance and midpoint formulas to find distances, midpoints, and missing endpoint coordinates. The key steps are to work inside parentheses, do exponents before addition, and plug points into the formulas.
The document discusses coordinate geometry and determining the position of a point P that divides a line segment AB based on a ratio m:n. It provides examples of finding the coordinates of points that divide line segments in different ratios. It also covers topics related to the gradient of a line, parallel and perpendicular lines, and finding the midpoint and length of a line segment.
This document discusses the technique of substitution for evaluating indefinite integrals. It defines the differential du in terms of a function u and its derivative, and shows how to rewrite integrals in terms of u and du. Examples are provided of using substitution to evaluate integrals involving roots, exponents, logarithms, and trigonometric functions. The document also addresses cases where the original variable remains after substitution and provides a practice problem to apply the technique.
The document provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. It asks to:
1) Solve quadratic equations by factoring.
2) Solve quadratic equations by completing the square.
3) Solve quadratic equations using the quadratic formula.
4) Determine the value of p that makes a given quadratic equation have equal roots.
This document contains information about basic math solutions for the third semester intended for self-study. It includes sample solutions to exercises on elementary group theory, including determining if a set forms a group based on properties like closure, identity, and inverses. It also contains multiplication tables and worked examples of checking properties for specific sets and binary operations.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
The document discusses addition and subtraction formulas for trigonometric functions. It provides examples of using these formulas to find the exact trigonometric values of angles that are not on the unit circle, such as sin(75°) and cos(265°). It then works through examples of using the addition formulas to find sin(105°) and cos(5π/12), providing step-by-step workings. The document also proves several trigonometric identities using the addition formulas, such as tan(θ + π/4) = (tanθ + 1)/(1 - tanθ).
1. The document provides steps to solve 6 trigonometric equations algebraically.
2. For the first equation, the solutions are θ = 0 and θ = π.
3. The second equation has solutions of θ = π/3 and θ = 5π/3.
4. The sixth equation has the solution cos(x) = -1/3.
This document provides fully worked solutions to exam questions from Form 4 mathematics chapters on standard form, quadratic expressions and equations, sets, mathematical reasoning, the straight line, and statistics. The solutions include:
1) Detailed working to obtain the answers for multiple choice and structured questions.
2) Explanations of mathematical concepts and reasoning such as determining gradients, interpreting graphs, and identifying argument forms.
3) Step-by-step derivations to find equations of lines from given points and gradients.
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for domain, range, period, identities, inverses, and laws involving trig functions like the Law of Sines, Cosines, and Tangents. Key formulas include definitions of sine, cosine, tangent and their inverses, as well as the Pythagorean, double angle, and sum and difference identities.
The document discusses solving polynomial equations. It begins by explaining quadratic equations, including how to solve them by factoring or using the quadratic formula. It then introduces polynomial equations of higher degree and methods for determining their real or complex roots, including the fundamental theorem of algebra. Examples are provided to illustrate solving quadratic and polynomial equations using these various methods.
The document discusses calculating the numerical value of algebraic expressions by substituting values for variables and performing the indicated operations. It provides examples of substituting values into single-variable expressions like P(x) and multi-variable expressions like Q(y,z) to find the numerical value. It also covers adding, subtracting, and multiplying polynomials, demonstrating how to combine like terms and distribute coefficients.
The document provides a review of trigonometry concepts related to right triangles, including:
- The definitions of the trigonometric functions sine, cosine, and tangent using the ratios of sides in a right triangle (SOH CAH TOA)
- Finding all trig values of an angle given one ratio
- Special right triangles and their connections to the unit circle
- The Pythagorean theorem and Pythagorean triples
- Angles of elevation and depression and using trig functions to find unknown side lengths
- Using the inverse trig functions (arcsine, arccosine, arctangent) to find an angle given a side ratio
This document contains suggested solutions to tutorial problems about mathematics of games.
[1] It calculates the expected value of a $1 field bet in craps to be -5.56%, meaning the bettor expects to lose $0.0556 on average per $1 bet.
[2] It determines that for a casino where a total roll of 3 results in no outcome for Don't Pass bettors, the house edge on a Don't Pass bet would be -4.1%, so it would be better to bet Pass.
[3] It proves that the binomial coefficients in the expansion of (a + b)n are in ascending then descending order when n is even, and
This document provides exercises to calculate the domain of various mathematical functions. It lists 21 functions and their corresponding domains. The domains range from subsets of the real numbers to empty sets, with intervals including things like -2 to 2, -infinity to -2 union 2 to infinity, and 1 to 2. The goal is to determine the valid inputs that each function can accept to produce a real output.
1. The document describes techniques for integrating trigonometric functions using trigonometric substitution and identities involving sine, cosine, tangent, and secant.
2. Trigonometric substitution involves redefining the variable in terms of a trigonometric function, unlike traditional substitution which defines a new variable.
3. The techniques are demonstrated through examples such as finding antiderivatives of √9-x^2/x^2 and √x^2+4/x^2.
Fjfjdjejrkdidjxjsjrkkdjdkkjjkkfkckckdkükememememdjdjdjjdnrnene3irjrjrjüjejsjdndnenejejdjdjdjdjdjsjsjejrururirjrjrjddjdjjdjjrjrjrjrjriejrjrjridududjrjrjejejeididjdjdje**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit simpler. By managing notifications and removing distractions like Facebook, you can focus on what's important in your life.**Introduction:**
If you want to focus better and get more things done, there are two simple solutions to consider. First, turn off notifications on your phone, especially from news apps and emails. Second, try deleting Facebook from your smartphone – it can help you avoid wasting time.
**Body:**
1. *Turn Off Notifications:*
To concentrate on your tasks, start by stopping notifications from bothering you. Turn off those pop-up messages from news apps and emails. This way, you won't be interrupted while working or studying.
2. *Delete Facebook:*
Consider removing Facebook from your phone. It might seem difficult at first, but it can help you stop spending too much time scrolling through posts. This small change can give you more free time.
**Conclusion:**
These ideas won't magically make you super productive, but they can make your life a bit
This document contains solutions to trigonometric functions and logarithmic equations. Key solutions include:
1) The inverse cosine of 1/2 is π/3 or 5π/3.
2) The tangent of 3π/4 is -1.
3) The natural log of x+1 does not equal 2lnx + 0. It is the natural log of the entire quantity x+1.
This document provides examples and practice problems for adding and subtracting integers. It begins with a basic skills quiz involving integer operations. It then provides examples of worked problems, a flow chart for adding and subtracting integers, and practice problems for students to complete. It concludes by providing space for students to create an incorrect integer problem for others to correct as additional practice.
The document contains 20 math word problems involving operations like addition, subtraction, multiplication and division of variables like a, b, c, x, y. The problems are summarized in 3 sentences or less with the key steps and solutions.
Questions and Solutions Basic Trigonometry.pdferbisyaputra
Unlock a deep understanding of mathematics with our Module and Summary! Clear definitions, comprehensive discussions, relevant example problems, and step-by-step solutions will guide you through mathematical concepts effortlessly. Learn with a systematic approach and discover the magic in every step of your learning journey. Mathematics doesn't have to be complicated—let's make it simple and enjoyable!
This document contains information about a mathematics workshop held in Barquisimeto, Venezuela in January 2021. It lists the participants Antonio Maria and Norneris Melendez. It then provides definitions and examples of algebraic language, including terms like literal, coefficient, degree, monomial, binomial, trinomial, polynomial. It discusses operations like addition, subtraction, multiplication, and factorization of polynomials. It also covers special products and factoring techniques for polynomials like factoring out a common monomial and using the difference of squares.
This document provides formulas and definitions for trigonometric functions including:
- Definitions of trig functions using right triangles and the unit circle
- Domains and ranges of the trig functions
- Periods of trig functions
- Trigonometric identities and formulas
- Inverse trig functions and their properties
- Formulas for conic sections including circles, ellipses, parabolas, and hyperbolas.
Math resources trigonometric_formulas class 11th and 12thDeepak Kumar
This document provides formulas and definitions for trigonometric functions including the definitions of sine, cosine, and tangent using right triangles and the unit circle. It also includes formulas for trigonometric identities, inverse trig functions, complex numbers including DeMoivre's theorem, and conic sections including circles, ellipses, hyperbolas, and parabolas.
The document summarizes key concepts from the first chapter of a Pre-Calculus textbook. It introduces interval notation and defines common inequality symbols like greater than, less than, greater than or equal to, and less than or equal to. It provides examples of writing inequalities using interval notation, such as x > 3 representing the interval (3, ∞).
This document provides information about sequences and series. It defines sequences as functions with positive integers as the domain. It distinguishes between infinite and finite sequences. Examples of sequences are provided and explicit formulas for finding terms are derived. Methods for finding the nth term of arithmetic and geometric sequences are described.
The document discusses tangent lines to functions. It provides examples of finding the equation of a tangent line with a given slope to specific functions. It also discusses finding the average and instantaneous velocity of an object given its position function.
Here are the problems from the slides with their solutions:
1. Find the slope of the line tangent to the graph of the function f(x) = x^2 - 5x + 8 at the point P(1,4).
Slope = -3
2. Find the equation of the tangent line to the curve f(x) = 2x^2 - 3 at the point P(1,-1) using point-slope form.
y - (-1) = 4(x - 1)
3. Find the equation of the tangent line to the curve f(x) = x + 6 at the point P(3,3) using point-slope form.
y
This document contains examples and explanations of limits involving various functions. Some key points covered include:
- Substitution can be used to evaluate limits, such as substituting 2 into -2x^3.
- Left and right hand limits must agree for the overall limit to exist.
- The limit of a piecewise function exists if the left and right limits are the same.
- Graphs can help verify limit calculations and show discontinuities.
- Special limits involving trigonometric and greatest integer functions are evaluated.
The document provides an introduction to evaluating limits, including:
1. The limit of a constant function is the constant.
2. Common limit laws can be used to evaluate limits of sums, differences, products, and quotients if the individual limits exist.
3. Special techniques may be needed to evaluate limits that involve indeterminate forms, such as 0/0, infinity/infinity, or limits approaching infinity. These include factoring, graphing, and rationalizing.
The document discusses recursive rules for defining sequences. It explains that a recursive rule defines subsequent terms of a sequence using previous terms, with one or more initial terms provided. Examples are worked through, such as finding the first five terms of the sequence where a1 = 3 and an = 2an-1 - 1, which are 3, 5, 9, 17, 33. Other sequences discussed include the Fibonacci sequence and examples of finding recursive rules to define other given sequences.
The document discusses two methods for expanding binomial expressions: Pascal's triangle and the binomial theorem. Pascal's triangle uses a recursive method to provide the coefficients for expanding binomials, but is only practical for smaller values of n. The binomial theorem provides an explicit formula for expanding binomials of the form (a + b)n using factorials and combinations. It works better than Pascal's triangle when n is large. Examples are provided to demonstrate expanding binomials like (3 - xy)4 and (x - 2)6 using both methods.
The document discusses using mathematical induction to prove the formula:
3 + 5 + 7 +...+ (2k + 1) = k(k + 2)
It provides the base case of p(1) and shows that it is true. It then assumes p(k) is true, and shows that p(k+1) follows by algebraic manipulations. This completes the induction proof.
The document discusses mathematical induction. It provides examples of deductive and inductive reasoning. It then explains the principle of mathematical induction, which involves proving that a statement is true for a base case, and assuming the statement is true for some value k to prove it is also true for k+1. The document provides a full example of using mathematical induction to prove that the sum of the first k odd positive integers is equal to k^2. It demonstrates proving the base case of 1 and the induction step clearly.
The document discusses geometric sequences and series. It examines partial sums of geometric sequences, which involve adding a finite number of terms. It also explores whether infinite series, or adding an infinite number of terms, can converge to a limiting value. It provides an example of someone getting closer to a wall on successive trips, with the total distance traveled converging even as the number of trips approaches infinity. It analyzes the behavior of geometric series based on whether the common ratio r is less than, greater than, or equal to 1.
The document discusses geometric sequences. A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. The common ratio is found by taking the quotient of any two consecutive terms. Explicit formulas are provided to calculate specific terms based on knowing the first term and common ratio. Examples are worked through, including finding a specific term for given sequences.
Here are the key steps:
- Find the formula for the nth term (an) of an arithmetic sequence
- Plug the values given into the formula to find a and d
- Use the formula for the sum of the first n terms (Sn) of an arithmetic sequence
- Set the formula equal to the total sum given and solve for n
The goal is to set up and solve the equation systematically rather than guessing and checking numbers. Documenting the work shows the logical steps and thought process. Keep exploring new approaches to solving problems more efficiently!
The document defines arithmetic sequences as sequences where the difference between consecutive terms is constant. It provides the formula for an arithmetic sequence as an = an-1 + d, where d is the common difference. It then gives several examples of arithmetic sequences and exercises identifying sequences as arithmetic and finding their common differences. It also explains how given any two terms of a sequence, the entire sequence is determined by finding the common difference d and using the formula an = a1 + (n-1)d.
This document discusses sequences and summation notation on day four. It references a bible verse about love and laying down one's life for others. It also contains instructions to be sure homework questions are addressed and for groups to begin the next homework assignment while working together. A quote by Henry Ford is included about dividing difficult tasks into smaller jobs.
The document discusses summation notation and properties of sums. It provides examples of writing sums using sigma notation, such as expressing the sum 2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 as the summation of 3k - 1 from k = 1 to 9. It also covers properties of sums, such as the property that the sum of a sum of a terms and b terms is equal to the sum of a terms plus the sum of b terms. The document provides guidance on calculating sums using sigma notation on a calculator.
The document provides an explanation of the binomial theorem formula for finding a specific term in the expansion of a binomial expression. It gives the formula as:
⎛ n ⎞ n−r r
⎜ r ⎟ x y
⎝ ⎠
Where n is the total number of terms, r is 1 less than the term number being found, x and y are the terms being added or subtracted. It provides an example of finding the 5th term of (a + b)6. It also provides an example of finding the 5th term of (3x - 5y)
This document contains two problems about hyperbolas:
[1] It gives the vertices and foci of a hyperbola and asks to find the standard form equation. The vertices are (±2, 0) and the foci are (±3, 0). The standard form equation is calculated to be x^2/4 - y^2/5 = 1.
[2] It gives the vertices and asymptotes of another hyperbola and asks to find the equation and foci. The vertices are (0, ±4) and the asymptotes are y = ±4x. The standard form equation is calculated to be y^2/16 - x^2 = 1, and the
The document defines and explains hyperbolas through the following key points:
1. A hyperbola is the set of points where the absolute difference between the distance to two fixed points (foci) is a constant.
2. Key parts of a hyperbola include vertices, foci, transverse axis, and conjugate axis.
3. The standard equation of a hyperbola is (x2/a2) - (y2/b2) = 1
4. Examples are worked through to graph specific hyperbolas using their equations.
The document discusses homework assignments and working in groups. It reminds students to ensure all homework questions have been addressed and directs groups to start working together on homework number 5. It also includes a quote about the importance of direction over current position.
🔥🔥🔥🔥🔥🔥🔥🔥🔥
إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
1. 7.3 Double Angle, Half Angle, and
Product Sum Formulas
On your help sheet ...
Psalm 119:28
My soul is weary with sorrow; strengthen me according to
your word.
2. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
3. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
4. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
−5
−12 13
5. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5
−12 13
6. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
7. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
144
= 1− 2 ⋅
169
8. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
144
= 1− 2 ⋅
169
169 288
= −
169 169
9. 12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of cos 2θ
2
cos 2θ = 1− 2sin θ
−5 2
⎛ 12 ⎞
= 1− 2 ⎜ − ⎟
−12 13 ⎝ 13 ⎠
144
= 1− 2 ⋅
169
169 288
= −
169 169
119
=−
169
10. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of sin 2θ
−5
−12 13
11. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of sin 2θ
−5 ⎛ 12 ⎞ ⎛ 5 ⎞
sin 2θ = 2 ⎜ − ⎟ ⎜ − ⎟
⎝ 13 ⎠ ⎝ 13 ⎠
−12 13
120
=
169
12. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of tan 2θ
−5
−12 13
13. Do this with your group:
12 3π
Given sin θ = − and π < θ < ,
13 2
determine the exact value of tan 2θ
⎛ −12 ⎞
−5 2 ⎜ ⎟
⎝ −5 ⎠
tan 2θ = 2
−12 13 ⎛ −12 ⎞
1− ⎜ ⎟
⎝ −5 ⎠
24 24
120
= 5 = 5 =−
144 119 119
1− −
25 25
15. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
16. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
2
2 cos x
4
=
2 cos x
17. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
2
2 cos x
4
=
2 cos x
1
2
=
cos x
18. Prove the identity:
cos 2x + 1 2
4
= 1+ tan x
2 cos x
2
2 cos x − 1+ 1
4
=
2 cos x
2
2 cos x
4
=
2 cos x
1
2
=
cos x
2
1+ tan x =
19. The formulas for Lowering Powers are used to
derive the Half Angle formulas.
I’ve put these Lowering Powers formulas on your
help sheet so you can see them and have them, but
we won’t be doing any problems which require them.
20. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
21. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
22. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
2
23. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
− 3
2
24. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
25. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
26. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
3
=±
4
27. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
3
=±
4
3
=±
2
28. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2
1
1+
=± 2
2
Which?
3 + or - ??
=±
4
3
=±
2
29. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1
1+
=± 2
2
Which?
3 + or - ??
=±
4
3
=±
2
30. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2
Which?
3 + or - ??
=±
4
3
=±
2
31. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2 3π α
Which? and < <π
3 + or - ?? 4 2
=±
4
3
=±
2
32. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2 3π α
Which? and < <π
3 + or - ?? 4 2
=±
4 this is in QII
3 α
=± so cos is negative
2 2
33. α
If sec α = 2 and tan α < 0 , find the exact value of cos
2
1
α 1+ cos α − 3
cos = ± 2
2 2 α is in QIV
1 3π
1+ ∴ < α < 2π
2 2
=±
2 3π α
Which? and < <π
3 + or - ?? 4 2
=±
4 this is in QII
3 α
=± so cos is negative
2 2
3
=−
2
34. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of sin
2
1
− 3
2
35. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of sin
2
1
1 α 1−
sin = 2
− 3 2 2
2
1
=
4
1
=
2
36. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of tan
2
1
− 3
2
37. Do this with your group:
α
If sec α = 2 and tan α < 0 , find the exact value of tan
2
3
1 α −
tan = 2
− 3 2 1+ 1
2
2
3
−
= 2
3
2
3
=−
3
38. HW #5
Optimism is the faith that leads to achievement.
Nothing can be done without hope and confidence.
Helen Keller
Editor's Notes
\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n
1. Hand out Trig Identity Help Sheet for the start of this slide.\n2. Review why Even-Odd identities are true.\n3. Explain why Cofunction identities are true.\n\n