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This document discusses multiples of exact angles in degrees and radians. It provides examples of multiples of angles and their values in both degrees and radians. The document also contains a summary table comparing degrees and radians and asks the reader to state the value of positive rotations between 0 and 2 radians.
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This document discusses multiples of angles in degrees and radians. It provides examples of multiples of angles in both units of measurement. The document concludes with a reminder that for questions, answers should be given in radians between 0 and 2π and specifies whether the rotation is positive or negative.
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This document discusses converting between degrees and radians for angles and locating angles in the x-y plane based on positive rotation. It provides examples of multiples of common angles like 30, 45, 60, 90 degrees and their radian equivalents, and asks the reader to express answers for questions in a given radian range with positive rotation.
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This document discusses key concepts in trigonometry including:
1. Angles are measured in degrees with positive numbers for counterclockwise rotations and negative numbers for clockwise rotations.
2. Coterminal angles have the same terminal side but can be arrived at by different amounts of rotation.
3. The values of the trig functions sine, cosine, and tangent can be determined using the ratios of sides of a right triangle with an angle of measure θ drawn in the unit circle.
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Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
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There are two systems for measuring angles - one uses degrees as the unit and has no symbol, while the other uses radians denoted by the symbol °. Radians are defined as the ratio of the arc length subtended by a central angle to the radius of the circle, while degrees are used to divide a full circle into 360 equal parts. The document provides examples of converting between degrees and radians, with 1 radian equaling approximately 57.3 degrees.
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Day 3 examples u4f13
This document discusses multiples of angles in degrees and radians. It provides examples of multiples of angles in both units of measurement. The document concludes with a reminder that for questions, answers should be given in radians between 0 and 2π and specifies whether the rotation is positive or negative.
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This document discusses converting between degrees and radians for angles and locating angles in the x-y plane based on positive rotation. It provides examples of multiples of common angles like 30, 45, 60, 90 degrees and their radian equivalents, and asks the reader to express answers for questions in a given radian range with positive rotation.
Notes 9-2
This document discusses key concepts in trigonometry including:
1. Angles are measured in degrees with positive numbers for counterclockwise rotations and negative numbers for clockwise rotations.
2. Coterminal angles have the same terminal side but can be arrived at by different amounts of rotation.
3. The values of the trig functions sine, cosine, and tangent can be determined using the ratios of sides of a right triangle with an angle of measure θ drawn in the unit circle.
Graphs of trigonometric functions
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Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
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There are two systems for measuring angles - one uses degrees as the unit and has no symbol, while the other uses radians denoted by the symbol °. Radians are defined as the ratio of the arc length subtended by a central angle to the radius of the circle, while degrees are used to divide a full circle into 360 equal parts. The document provides examples of converting between degrees and radians, with 1 radian equaling approximately 57.3 degrees.
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This document discusses strategies for verifying identities in trigonometry. It notes that verifying identities involves showing that the left-hand side and right-hand side are equal. Common strategies include working with the more complicated side of the identity, writing all terms in terms of basic trig functions like sine, cosine, and tangent, and replacing combinations with simpler expressions or common denominators to show the two sides are equal. The document provides some examples of identities to verify and assigns problems numbers 6 through 12.
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This document provides the exact coordinates for points on the unit circle corresponding to angles in radians. It gives the coordinates for 0, π/2, π, and 3π/2 radians as (1,0), (0,1), (-1,0), and (0,-1) respectively, based on the definitions of sine and cosine. It then generalizes the coordinates to any multiple of π/2 by using the fact that tan(nπ/2) = ±1. Examples are worked out for finding specific trig values over intervals using the coordinate rules.
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Permutations involve arranging objects or letters in a definite order. The number of permutations of n objects taken r at a time is written as nPr. To calculate nPr, use the formula nPr = n!/(n-r)!. Permutations are used to calculate possibilities like license plate numbers, phone numbers, and locker combinations. When objects are identical, the number of permutations is divided by the factorial of the number of identical objects.
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This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.
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The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
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This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
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Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
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Permutations involve arranging objects or letters in a definite order. The number of permutations of n objects taken r at a time is written as nPr. To calculate nPr, use the formula nPr = n!/(n-r)!. Permutations are used to calculate possibilities like license plate numbers, phone numbers, and locker combinations. When objects are identical, the number of permutations is divided by the factorial of the number of identical objects.
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The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
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This document provides an introduction and instructions for practicing Sheershasana, or the headstand yoga pose. It explains that Sheershasana means "topsy-turvy pose" and should be learned and practiced gradually and carefully to receive optimal health benefits. The document then outlines 12 steps to perform Sheershasana safely and properly, and provides warnings to ensure it is not attempted by certain individuals or in unsafe conditions. Potential health benefits are also listed.
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The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
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The document discusses the meaning and significance of the linga pooja ritual in Hinduism. It states that the linga represents the supreme being Shiva and symbolizes the phenomena of formation, maintenance, and destruction in the universe. While some see it as a vestige of primitive phallus worship, the document explains it actually signifies the procreation that creates life and is a ritual meant to help people remember and reconnect with their true nature. Specific days and places are believed to facilitate this realization of one's self, which is why there are designated times like Mahashivratri and sites like the Jyotirlinga for linga worship.
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The document discusses planning, recruitment and hiring of salespeople. It emphasizes the importance of determining personnel needs through planning, analyzing sales jobs, and developing job descriptions and specifications to identify qualified candidates. The goal is to hire above-average performers through an effective recruitment strategy that considers legal obligations and sources to find both internal and external applicants.
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- 1. Exact Angle - Multiples and x-y Plane In Degrees - Multiples - Multiples - Multiples - Multiples
- 2. In Radians - multiples - multiples - multiples - multiples
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