6.4.1 sum and difference formulas
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This document outlines sum and difference formulas for cosine, sine, and tangent. It provides examples of using the formulas to find exact values of trigonometric functions of sum and difference of angles. It also gives two examples of using the formulas to find exact values of trigonometric functions. In the examples, it applies the sum and difference formulas to calculate sin65° and cos40°.
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This document explains the order of operations PEMDAS (Parentheses, Exponents, Multiply, Divide, Add, Subtract) and works through 3 example problems step-by-step using this order:
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1) Find equivalent fractions with a common denominator
2) Add the numerators and use the sum as the new numerator
3) Keep the common denominator as the denominator
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Worked examples demonstrate applying the steps to add several pairs of fractions.
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This document outlines sum and difference formulas for cosine, sine, and tangent. It provides examples of using the formulas to find exact values of trigonometric functions of sum and difference of angles. It also gives two examples of using the formulas to find exact values of trigonometric functions. In the examples, the formulas are used to find sin65° and cos40°.
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3) Rationalizing denominators by multiplying the denominator and numerator by the radical in the denominator to remove radicals from the denominator.
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3) Students are instructed to complete several practice problems subtracting two-digit numbers without regrouping and to check their work.
Insertion sort
Insertion sort is a sorting algorithm that builds the final sorted array one element at a time by taking each element and inserting it into the proper position in the sorted portion of the array. It assumes the first element is sorted and compares subsequent elements to prior elements, swapping elements as needed to place them in ascending order. The algorithm iterates through the array from index 1 to length, comparing each element to its predecessor and swapping as required until the entire array is sorted.
Prime factorization
There are several methods for factorizing numbers into their prime number components. These include using divisibility rules to check for factors of 2, 3, 5, and other primes. Two common methods are downward division, which divides the number successively by primes starting from the largest factor, and a division tree which splits the number into prime factors. Being able to quickly factor numbers is important for tasks like simplifying fractions and radicals.
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Add & subtract mixed numbers
The document discusses mixed numbers and how to add and convert fractions. It explains that a mixed number has a whole number part and fractional part. It provides examples of finding common denominators and equivalent fractions to add fractions with different denominators. It demonstrates how to convert improper fractions to mixed numbers.
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Lesson 1 introduces ratios and proportions. A ratio compares two or more quantities and can be written in fraction or colon form. Ratios can be simplified by dividing both quantities by their greatest common factor. Two ratios are equivalent if they can be expressed as the same simple ratio. A proportion is a statement of equality between two ratios. It involves setting the product of the means equal to the product of the extremes, where the extremes are the first and last terms and the means are the second and third terms. To solve a proportion, the ratios are set up as fractions, cross-multiplied, and then solved.
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The document discusses tests of divisibility to determine if a number is perfectly divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12 without remainder. It provides the specific tests, which involve examining digits in certain places of the number or adding all digits together. The tests are then demonstrated with examples of numbers that are and are not divisible by each number.
Adding Fractions With Unlike Denominators
To add or subtract fractions with unlike denominators:
1. Find the least common multiple (LCM) of the denominators.
2. Write the fractions with this LCM as the new denominator by multiplying the numerators and denominators.
3. Add or subtract the new numerators and put over the common denominator.
4. Simplify the final fraction if possible by dividing the numerator and denominator by common factors.
Number Theory
The document discusses key concepts in number theory including natural numbers, divisibility, factors and multiples, prime and composite numbers, and divisibility tests. It defines natural numbers as the set of counting numbers {1,2,3,4...} and notes that number theory deals with studying the properties of these numbers, with a focus on divisibility. Divisibility is defined as one number being divisible by another if dividing the first by the second leaves no remainder. Factors are numbers that divide another number, while multiples are numbers that are divisible by another number. Prime numbers have only 1 and themselves as factors, while composite numbers have more than two factors. Divisibility tests provide ways to determine if one number is divisible by
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This document provides instructions and examples for subtracting mixed numbers. It explains the steps as: 1) finding the least common denominator, 2) rewriting the mixed numbers with the common denominator, 3) subtracting the fractions, 4) subtracting the whole numbers, and 5) simplifying the final fraction if possible. Two examples are shown working through each step. The document concludes by asking the reader to try subtracting two additional mixed numbers on their own.
Rules of divisibility
The document provides examples of divisibility tests for numbers 2, 3, 5, and 10. It then asks which of these numbers can divide 28, 130, 120, and 45. For 28, it is divisible by 2. For 130, it is divisible by 2 but not 3, 5, or 10. For 120, it is divisible by both 2 and 3. And for 45, it is only divisible by 3.
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This document provides steps for adding fractions with unlike denominators:
1) Find equivalent fractions with a common denominator
2) Add the numerators and use the sum as the new numerator
3) Keep the common denominator as the denominator
4) Simplify the resulting fraction if possible by reducing to lowest terms
Worked examples demonstrate applying the steps to add several pairs of fractions.
Divisibility tests
This document provides an overview of divisibility tests for numbers 2 through 11. It explains the tests to check divisibility by each number without performing long division. For each number, it gives the rule to use (e.g. for 2 check the last digit is even), provides an example, then has practice problems checking divisibility for 5 numbers. The tests allow determining divisibility in a faster way than long division.
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Setting the Scene: The Day of the Kidnapping
The incident of Dwayne Johnson's kidnapping began on an ordinary day. Johnson was filming his latest high-octane action film set to break box office records. The location was a remote yet scenic area. chosen for its rugged terrain and breathtaking vistas. perfect for the film's climactic scenes.
But, beneath the veneer of normalcy, a sinister plot was unfolding. Unbeknownst to Johnson and his team, a group of criminals had planned his abduction. hoping to leverage his celebrity status for a hefty ransom. The stage was set for an event that would soon dominate worldwide headlines and social media feeds.
The Abduction: Unfolding the Dwayne Johnson Kidnapping
The Moment of Capture
On the day of the kidnapping, everything seemed to be proceeding as usual on set. Johnson and his co-stars and crew were engrossed in shooting a particularly demanding scene. As the day wore on, the production team took a short break. providing the kidnappers with the perfect opportunity to strike.
The abduction was executed with military precision. A group of masked men, armed and organized, infiltrated the set. They created chaos, taking advantage of the confusion to isolate Johnson. Johnson was outnumbered and caught off guard despite his formidable strength and fighting skills. The kidnappers overpowered him, bundled him into a waiting vehicle. and sped away, leaving everyone on set in a state of shock and disbelief.
The Immediate Aftermath
The immediate aftermath of the Dwayne Johnson kidnappin
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Orpah Winfrey Dwayne Johnson: Titans of Influence and Inspiration
Introduction
In the realm of entertainment, few names resonate as Orpah Winfrey Dwayne Johnson. Both figures have carved unique paths in the industry. achieving unparalleled success and becoming iconic symbols of perseverance, resilience, and inspiration. This article delves into the lives, careers. and enduring legacies of Orpah Winfrey Dwayne Johnson. exploring how their journeys intersect and what we can learn from their remarkable stories.
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Early Life and Backgrounds
Orpah Winfrey: From Humble Beginnings to Media Mogul
Orpah Winfrey, often known as Oprah due to a misspelling on her birth certificate. was born on January 29, 1954, in Kosciusko, Mississippi. Raised in poverty by her grandmother, Winfrey's early life was marked by hardship and adversity. Despite these challenges. she demonstrated a keen intellect and an early talent for public speaking.
Winfrey's journey to success began with a scholarship to Tennessee State University. where she studied communication. Her first job in media was as a co-anchor for the local evening news in Nashville. This role paved the way for her eventual transition to talk show hosting. where she found her true calling.
Dwayne Johnson: From Wrestling Royalty to Hollywood Superstar
Dwayne Johnson, also known by his ring name "The Rock," was born on May 2, 1972, in Hayward, California. He comes from a family of professional wrestlers, with both his father, Rocky Johnson. and his grandfather, Peter Maivia, being notable figures in the wrestling world. Johnson's early life was spent moving between New Zealand and the United States. experiencing a variety of cultural influences.
Before entering the world of professional wrestling. Johnson had aspirations of becoming a professional football player. He played college football at the University of Miami. where he was part of a national championship team. But, injuries curtailed his football career, leading him to follow in his family's footsteps and enter the wrestling ring.
Career Milestones
Orpah Winfrey: The Queen of All Media
Winfrey's career breakthrough came in 1986 when she launched "The Oprah Winfrey Show." The show became a cultural phenomenon. drawing millions of viewers daily and earning many awards. Winfrey's empathetic and candid interviewing style resonated with audiences. helping her tackle diverse and often challenging topics.
Beyond her talk show, Winfrey expanded her empire to include the creation of Harpo Productions. a multimedia production company. She also launched "O, The Oprah Magazine" and OWN: Oprah Winfrey Network, further solidifying her status as a media mogul.
Dwayne Johnson: From The Ring to The Big Screen
Dwayne Johnson's wrestling career took off in the late 1990s. when he became one of the most charismatic and popular figures in WWE. His larger-than-life persona and catchphrases endeared him to fans. making him a household name. But, Johnson had ambitions beyond the wrestling ring.
In the early 20
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- 1. 6.4 Sum and Difference Formulas Using Sum/Difference to Find Exact Values (Angles Measured in Degrees)
- 2. Sum and Difference Formulas for Cosine cos cos cos sin sin cos cos cos sin sin
- 3. Use Sum Formula to find exact value cos105 cos 60 45 cos60 cos45 sin60 sin45 1 2 3 2 2 2 2 2 2 6 4 4 1 4 2 6
- 4. Use Difference Formula to find exact valuecos105 cos 135 30 cos135 cos30 sin135 sin30 2 3 2 1 2 2 2 2 6 2 4 4 1 4 2 6
- 5. Sum and Difference Formulas for Sine sin sin cos cos sin sin sin cos cos sin
- 6. Use Sum Formula to find exact value sin75 sin 30 45 sin30 cos45 cos30 sin45 1 2 3 2 2 2 2 2 2 6 4 4 1 4 2 6
- 7. Use Difference Formula to find exact valuesin75 sin 120 45 sin120 cos45 cos120 sin45 3 2 1 2 2 2 2 2 6 2 4 4 1 4 6 2
- 8. Sum and Difference Formulas for Tangent tan tan tan 1 tan tan tan tan tan 1 tan tan
- 9. Use Sum Formula to find exact value tan195 tan 135 60 tan135 tan60 1 tan135 tan60 1 3 1 1 3 1 3 1 3 1 3 1 3 4 2 3 1 3 4 2 3 2 2 3
- 10. Use Difference Formula to find exact valuetan195 tan 240 45 tan 240 tan 45 1 tan 240 tan 45 3 1 1 3 1 3 1 1 3 1 3 1 3 4 2 3 1 3 4 2 3 2 2 3
- 11. Find the Exact Value sin65 cos25 cos65 sin25 sin cos cos sin sin sin 65 25 sin 90 1
- 12. Find the Exact Value cos40 cos20 sin40 sin20 cos cos sin sin cos cos 40 20 cos 60 1 2
- 13. ASSIGNMENT p. 481 # 2 - 8, 13 - 16, 21 - 26