March 4 and 5, 2009
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The document contains examples and explanations of using trigonometry to solve for unknown side lengths of triangles. It includes the definitions of the trigonometric ratios sine, cosine and tangent. There are also practice problems provided for students to work through applying trigonometric concepts. Homework assignments are listed at the end for additional practice.
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How to test
To solve a quadratic equation by completing the square, rewrite the left side in the form of x^2 + bx, add (b/2)^2 to both sides to create a perfect square on the left, then take the square root of both sides. To solve by the quadratic formula, identify the coefficients a, b, and c and plug them into the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the solutions.
Sketching a sinusoidal graph
The document provides instructions for sketching a sinusoidal graph in 3 steps: 1) Identify the key characteristics of the sinusoidal function, including median line, amplitude, period, and start point. 2) Draw the median line and mark the start, end, maximum, and minimum points. 3) Join the points with a smooth curve. It then gives an example sinusoidal function to sketch.
Lsn 11-5: Law of Sines
Here are the steps to solve these Law of Sines problems:
1. Given: a = 13, b = 20, A = 75°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(75°)/13 = sin(B)/20
sin(75°)*20/13 = sin(B)
B = sin-1(0.8) = 67°
2. Given: a = 25, B = 38°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(38°)/25 = sin(m°)/20
sin(38°)*20/25 = sin
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This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Boolean, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, describing how to eliminate values of that type into a dependent type.
Eliminators into dependent types
This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Booleans, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, allowing values of that type to be analyzed in a dependent context.
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Cones and frustum slides
The document defines and provides information about cones, conical frustums, and how to calculate their volume and total surface area. It explains that a cone tapers from a circular base to a point, and can be a right cone or oblique cone. Formulas are given for calculating the volume and total surface area of cones and conical frustums. Examples are worked through applying the formulas to calculate volume and surface area of different cone and frustum shapes.
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How to test
To solve a quadratic equation by completing the square, rewrite the left side in the form of x^2 + bx, add (b/2)^2 to both sides to create a perfect square on the left, then take the square root of both sides. To solve by the quadratic formula, identify the coefficients a, b, and c and plug them into the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the solutions.
Sketching a sinusoidal graph
The document provides instructions for sketching a sinusoidal graph in 3 steps: 1) Identify the key characteristics of the sinusoidal function, including median line, amplitude, period, and start point. 2) Draw the median line and mark the start, end, maximum, and minimum points. 3) Join the points with a smooth curve. It then gives an example sinusoidal function to sketch.
Lsn 11-5: Law of Sines
Here are the steps to solve these Law of Sines problems:
1. Given: a = 13, b = 20, A = 75°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(75°)/13 = sin(B)/20
sin(75°)*20/13 = sin(B)
B = sin-1(0.8) = 67°
2. Given: a = 25, B = 38°
Use the Law of Sines: sin(A)/a = sin(B)/b
sin(38°)/25 = sin(m°)/20
sin(38°)*20/25 = sin
19 - Scala. Eliminators into dependent types (induction)
This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
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- Dependent pairs and dependent functions
- Boolean, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, describing how to eliminate values of that type into a dependent type.
Eliminators into dependent types
This document describes eliminators (also called induction principles) for dependent types in dependent type theory, including:
- Empty, unit, sum, product, and function types
- Dependent pairs and dependent functions
- Booleans, natural numbers, lists, and vectors
- Identity types
For each type, it provides the eliminator signature and definition, allowing values of that type to be analyzed in a dependent context.
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The document defines and provides information about cones, conical frustums, and how to calculate their volume and total surface area. It explains that a cone tapers from a circular base to a point, and can be a right cone or oblique cone. Formulas are given for calculating the volume and total surface area of cones and conical frustums. Examples are worked through applying the formulas to calculate volume and surface area of different cone and frustum shapes.
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The document defines and provides information about cones, conical frustums, and how to calculate their volume and total surface area. It explains that a cone tapers from a circular base to a point, and can be a right cone or oblique cone. Formulas are given for calculating the volume and total surface area of cones and conical frustums. Examples are worked through applying the formulas to calculate volume and surface area of different cone and frustum shapes.
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The document contains instructions for students to complete worksheet problems involving adding and subtracting fractions. It lists several fraction addition and subtraction problems for students to work through independently, including problems finding the sum of fractions with common denominators and fractions with algebraic expressions in the denominators. It also instructs students to find the vertex of a parabolic function and explain what the graph of that function will look like.
Alg May 13, 2009
This document contains math homework assignments and practice problems involving algebra concepts like solving equations, adding and subtracting fractions, working with rational expressions, and factoring polynomials. Students are assigned worksheet problems 4, 7, 10, and 14-16 from section 12-6 for homework. There will be a quiz tomorrow on multiplying, dividing, adding, and subtracting. The document then provides 4 sample word problems for students to practice these skills on their own.
Alg May 4, 2009
The document provides instructions on factoring the expression 2x^2 - 39x - 21 and reviewing simplifying rational expressions. It also provides tips for the PSAT exam, including that there are 38 math questions over two sections, 28 are standard multiple choice and 10 are grid-in questions. It expects easy questions early in sets and advises eliminating wrong answers and only changing answers if you have a reason. The homework assigned is to complete odd problems 13-29 and problem 26 from worksheet 12-2.
Alg May 6, 2009
The document is a homework assignment that includes:
1) A quiz tomorrow on solving radical equations and simplifying rational expressions.
2) Several math problems involving multiplying, dividing, and simplifying fractions.
3) A note to work problems 1, 7, and 16 from worksheet 12-3 on your own.
Geo May 20, 2009
The document discusses surface area and volume formulas for different 3D shapes. It defines lateral area as the area of the side rectangles combined, and base area as the area of the base polygon or circles. Surface area is defined as the total area of all surfaces, including both lateral and base areas. Formulas are provided for calculating the surface area of prisms, cylinders, and other shapes using the base area, lateral area, perimeter, and height. Examples are given showing how to use the formulas to find missing values.
Geo May 19, 2009
The document describes different 3D shapes: a prism with sides A, B, C, and D measuring 10 feet and bases measuring 4 feet; pyramids with triangular sides meeting at a point and one polygon base; cylinders with two congruent circular bases; cones with one circular base; and spheres with equal distance from the middle to the outside. It assigns homework on surface area and notes test corrections are due on Friday.
Alg May 21, 2009
The document provides a date of May 21, 2009. No other information is given in the single line document, which simply states the date with no other context provided. In just one line, the document communicates only the date of May 21, 2009 without any additional details.
Alg May 20, 2009
The document contains a math homework assignment and schedule. It assigns factoring practice problems involving GCF, grouping, binomials, and trinomials. It also notes an upcoming quiz on solving rational equations and upcoming tests on radicals and rationals. Review of factoring techniques like GCF, difference of squares, and grouping are also listed.
Alg May 19, 2009
This document provides instructions for solving rational equations in 3 steps: 1) factor denominators, 2) multiply by the lowest common denominator to clear fractions, 3) solve the resulting equation using known methods. It also includes the formula for distance equals rate times time.
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March 4 and 5, 2009
- 1. March 4, 2009 C B 60o D A8E In the figure above, EBCD is a square and AE=8. What is the area of EBCD?
- 2. 60o 10 60o 2 short side = Ratios: long side short side = hypotenuse long side hypotenuse =
- 3. Trigonometry sine: ratio between the opposite side and the hypotenuse cosine: ratio between the adjacent side and the hypotenuse tangent: ratio between the opposite side and the adjacent side SOH CAH TOA
- 4. March 5, 2009 B xo yo In the figure to the right, AD =1 and xo o DC=√3. What is the z A C D value of z? Note: Figure not drawn to scale (a) 15 (b) 20 (c) 25 (d) 30 (e) 35
- 5. opposite side A For For adjacent side For hypotenuse this can't happen! C B SOH CAH TOA
- 6. A opposite side adjacent side hypotenuse B C For For For SOH CAH TOA
- 7. opposite side C A adjacent side B hypotenuse For For For SOH CAH TOA
- 8. SOH CAH TOA F G tan H = tan G = sin H = H sin G = cos H = cos G =
- 9. CW: pg. 469 #1, 35, 13 pg. 477 #1, 39
- 10. A Using Trigonometry How do you find the value of x ? 35o x tan A = 12 y 12 x . o . 12 tan 35 = 12 12 55o o 12 tan 35 = x x C B 8.4 ≈x
- 11. A Using Trigonometry How would you find the value of y ? 35o y 12 55o x C B
- 12. A Using Trigonometry Would x be the same using ? 35o 12 tan C = x y 12 12 x o x tan 55 = x 55o o x tan 55 = 12 x C B tan 55o tan 55o x ≈ 8.4
- 13. What would you use? Set it up! 20o x x 8 70o 7 x x 57o 45o 12 5
- 16. HW: pg. 470 #1828 (even), 31, 32 pg. 478 #1821, 30, 33, 34
- 17. March 6, 2009 What is the perimeter of the figure below? 6 (a) 24 (b) 25 (c) 28 (d) 30 6 6 (e) 36 30o 30o