Sulpcegu5e ppt 8_3
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This document discusses solving triangles using the Law of Cosines. It provides two examples to solve: a triangle with side lengths b = 5, c = 8 and angle A = 35 degrees; and a triangle with side lengths a = 6, b = 8, and c = 5. The document aims to teach the reader how to use the Law of Cosines to find missing side lengths or angles of a triangle when some combination of sides and angles are known.
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Sulpcegu5e ppt 8_2
This document discusses using the Law of Sines to solve triangles. It provides examples of solving triangles when given two angles and the side between them, as well as ambiguous cases where two sides and a non-included angle are given. The objectives are to use the Law of Sines to solve triangles when two parts are given as well as handle ambiguous SSA cases.
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We are missing one piece of information to completely specify the triangle. The Law of Sines requires knowing two angles and the side opposite one of those angles, or all three sides of the triangle.
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We are missing one piece of information to completely specify the triangle. The Law of Sines requires knowing two angles and the side opposite one of those angles, or all three sides. Here we only know two angles and two sides, so the triangle is not completely specified.
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The document discusses the sine and cosine laws for solving triangles. It provides examples of using these laws to calculate missing angles and sides of triangles when given certain information. However, one of the examples leads to two possible triangle solutions, showing that the information provided an ambiguous case with multiple valid options. The summary concludes that without more context, both triangle solutions are considered correct since the given information allows for more than one possibility.
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1.3 segments and their measures ppt 2
The document provides information about geometry concepts covered in a first semester class, including segments and their measures, postulates, axioms and theorems, the ruler postulate, distance between points, segment addition postulate, congruent segments, and the distance formula. Students are instructed to check their seating chart and complete assigned problems for homework.
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This document contains homework questions on evaluating logarithmic expressions, converting between logarithmic and exponential forms, and applying the Law of Sines to solve for missing sides and angles of triangles given various inputs. Students are asked to solve seven triangle problems by using the Law of Sines formula to find the unknown side or angle.
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This document discusses the Law of Cosines and how to use it to solve for unknown sides or angles of triangles. It provides examples of using the Law of Cosines to solve triangles where one side and its two angles are given, when three sides are given, and when one angle and two sides are given. It also covers Heron's formula for finding the area of a triangle given its three sides. Finally, it provides homework problems from the textbook involving using the Law of Cosines and finding areas of triangles.
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This document discusses using the Law of Sines to solve triangles. It provides examples of solving triangles when given two angles and the side between them, as well as ambiguous cases where two sides and a non-included angle are given. The objectives are to use the Law of Sines to solve triangles when two parts are given as well as handle ambiguous SSA cases.
Section_9-3B
We are missing one piece of information to completely specify the triangle. The Law of Sines requires knowing two angles and the side opposite one of those angles, or all three sides of the triangle.
Section_9-3B
We are missing one piece of information to completely specify the triangle. The Law of Sines requires knowing two angles and the side opposite one of those angles, or all three sides. Here we only know two angles and two sides, so the triangle is not completely specified.
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The document discusses the sine and cosine laws for solving triangles. It provides examples of using these laws to calculate missing angles and sides of triangles when given certain information. However, one of the examples leads to two possible triangle solutions, showing that the information provided an ambiguous case with multiple valid options. The summary concludes that without more context, both triangle solutions are considered correct since the given information allows for more than one possibility.
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This document contains homework questions on logarithms and the Law of Sines. For logarithms, it includes converting between log and exponential form, expanding and condensing logarithmic expressions. For the Law of Sines, it provides examples of using the law to solve for missing sides and angles of triangles given information about two angles and one side, or two sides and one angle. The document provides step-by-step work for students to practice applying the Law of Sines to ambiguous cases and multiple solutions. It assigns worksheet problems from the textbook for homework.
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The document provides information about geometry concepts covered in a first semester class, including segments and their measures, postulates, axioms and theorems, the ruler postulate, distance between points, segment addition postulate, congruent segments, and the distance formula. Students are instructed to check their seating chart and complete assigned problems for homework.
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This document contains homework questions on evaluating logarithmic expressions, converting between logarithmic and exponential forms, and applying the Law of Sines to solve for missing sides and angles of triangles given various inputs. Students are asked to solve seven triangle problems by using the Law of Sines formula to find the unknown side or angle.
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This document discusses the Law of Cosines and how to use it to solve for unknown sides or angles of triangles. It provides examples of using the Law of Cosines to solve triangles where one side and its two angles are given, when three sides are given, and when one angle and two sides are given. It also covers Heron's formula for finding the area of a triangle given its three sides. Finally, it provides homework problems from the textbook involving using the Law of Cosines and finding areas of triangles.
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The document discusses the Law of Cosines, which extends trigonometry to non-right triangles. It defines the Law of Cosines using three equations relating the sides and angles of a triangle. Examples are provided to demonstrate solving for missing sides and angles of triangles using the Law of Cosines given various known values.
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Trigonometry is used to determine the measures (sides and angles) of a right triangle. The document reviews three trigonometric functions: sine, cosine, and tangent. Sine relates an angle to the opposite side over the hypotenuse. Cosine relates an angle to the adjacent side over the hypotenuse. Tangent relates an angle to the opposite side over the adjacent side. An example problem is shown to find missing sides and angles of a right triangle when given one angle measure and the side opposite to it.
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1. The document discusses solving oblique triangles using the Law of Sines. It provides examples of solving triangles given: (1) two angles and a side (ASA case) and (2) two sides and a non-included angle (SSA case).
2. For the ASA case, it shows how to find the missing angle and sides using the given information. For the SSA case, it notes that SSA is not a unique case and there may be 0, 1, or 2 possible triangles depending on the side lengths.
3. It provides an example of solving a triangle with ASA given and finds the missing angle and sides. It also provides an example of an SSA case where
8.4 the law of cosines
The Law of Cosines is used to calculate the length of one side of a triangle when two sides and the included angle are known, or when all three sides are known. It states that the square of one side of a triangle equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
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The document summarizes two laws for solving triangles - the Law of Cosines and the Law of Sines. It also discusses an ambiguous case of the Law of Sines. The Law of Cosines can be used to find a missing side given two sides and the included angle, or to find a missing angle given all three sides. The Law of Sines can be used to find a missing side or angle given two angles and the side opposite one of them. The ambiguous case allows finding a missing angle given two sides and the angle opposite one of them.
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I. The property that the sum of the three angles of any triangle is 180 degrees is proven by drawing a parallel line through one vertex of a triangle. The angles opposite the parallel sides are equal, showing the three angles sum to 180 degrees.
II. This property can be used to calculate unknown angles of different types of triangles:
A) In a generic triangle, the sum of the three known angles is set equal to 180 degrees to solve for the unknown angle.
B) In a right triangle, the two acute angles sum to 90 degrees since one angle is 90 degrees by definition.
C) In an isosceles triangle, the two angles opposite the equal sides are set equal,
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This document provides instruction on using the Law of Sines to solve triangles. It begins with examples of using the Law of Sines to find missing side lengths or angle measures when two angles and a side, or two sides and an angle are known. It also covers cases where an ambiguous triangle could result from given side-side-angle information. The document demonstrates solving for the area of triangles using trigonometric functions. It concludes with practice problems applying the Law of Sines to find missing measurements and the number of possible triangles based on given side lengths and an angle measure.
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4) The sine rule and cosine rule for solving triangles given certain information about sides and angles.
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This document summarizes how to prove that the angle subtended by an arc at the centre of a circle is twice that subtended at the circumference, even when a line cannot be drawn through the centre cutting both angles. It does this by extending one arc line to the centre, then using properties of isosceles triangles and angles on a straight line to derive an equation that solves to the desired property.
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The document discusses the Law of Sines, which can be used to solve triangles that are not right triangles. It provides three formulas for finding the area of a triangle using two angles and one included side: Area = 1/2 * b * c * sin(A), 1/2 * a * c * sin(B), 1/2 * a * b * sin(C). It also gives the formula for the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). An example problem finds the area of a triangle given two side lengths and one angle measure.
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The law of sines, also known as the sine rule, relates the ratios of sides and opposite angles in any triangle. Given any two elements of a triangle (side or angle), the law of sines can be used to calculate the remaining unknown elements. The formula is a/sinA = b/sinB = c/sinC, where a, b, c are the sides and A, B, C are the opposite angles. The document provides examples of using the law of sines to solve for unknown sides and angles in various triangles. It also includes practice problems for students to work through applying the law of sines.
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The document discusses the Law of Cosines, which extends trigonometry to non-right triangles. It defines the Law of Cosines using three equations relating the sides and angles of a triangle. Examples are provided to demonstrate solving for missing sides and angles of triangles using the Law of Cosines given various known values.
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1. The document discusses solving oblique triangles using the Law of Sines. It provides examples of solving triangles given: (1) two angles and a side (ASA case) and (2) two sides and a non-included angle (SSA case).
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The Law of Cosines is used to calculate the length of one side of a triangle when two sides and the included angle are known, or when all three sides are known. It states that the square of one side of a triangle equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
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The document summarizes two laws for solving triangles - the Law of Cosines and the Law of Sines. It also discusses an ambiguous case of the Law of Sines. The Law of Cosines can be used to find a missing side given two sides and the included angle, or to find a missing angle given all three sides. The Law of Sines can be used to find a missing side or angle given two angles and the side opposite one of them. The ambiguous case allows finding a missing angle given two sides and the angle opposite one of them.
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I. The property that the sum of the three angles of any triangle is 180 degrees is proven by drawing a parallel line through one vertex of a triangle. The angles opposite the parallel sides are equal, showing the three angles sum to 180 degrees.
II. This property can be used to calculate unknown angles of different types of triangles:
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B) In a right triangle, the two acute angles sum to 90 degrees since one angle is 90 degrees by definition.
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This document provides instruction on using the Law of Sines to solve triangles. It begins with examples of using the Law of Sines to find missing side lengths or angle measures when two angles and a side, or two sides and an angle are known. It also covers cases where an ambiguous triangle could result from given side-side-angle information. The document demonstrates solving for the area of triangles using trigonometric functions. It concludes with practice problems applying the Law of Sines to find missing measurements and the number of possible triangles based on given side lengths and an angle measure.
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The document discusses trigonometric formulas and identities for solving triangles. It provides:
1) Definitions of trigonometric functions in terms of acute angles of a right triangle.
2) Formulas relating trig functions of complementary angles.
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4) The sine rule and cosine rule for solving triangles given certain information about sides and angles.
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This document provides instructions for using trigonometric functions (sine, cosine, tangent) to solve for unknown sides of a right triangle. It gives an example triangle with a hypotenuse of 9 meters and an angle of 35 degrees. It uses sine to solve for side CB (5.16 m), cosine to solve for side AC (7.37 m), and tangent to solve for the final angle (55 degrees).
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The document contains 25 solved trigonometric problems involving the law of sines, law of cosines, and graphs of sine and cosine functions. It provides step-by-step solutions to problems finding missing sides and angles of triangles using trigonometric identities and relationships. Several examples calculate lengths, angles, and areas using information about one or two sides or angles of a triangle.
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A) proving angle properties of circles 2
This document summarizes how to prove that the angle subtended by an arc at the centre of a circle is twice that subtended at the circumference, even when a line cannot be drawn through the centre cutting both angles. It does this by extending one arc line to the centre, then using properties of isosceles triangles and angles on a straight line to derive an equation that solves to the desired property.
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The document discusses the Law of Sines, which can be used to solve triangles that are not right triangles. It provides three formulas for finding the area of a triangle using two angles and one included side: Area = 1/2 * b * c * sin(A), 1/2 * a * c * sin(B), 1/2 * a * b * sin(C). It also gives the formula for the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). An example problem finds the area of a triangle given two side lengths and one angle measure.
Presentation2
The law of sines, also known as the sine rule, relates the ratios of sides and opposite angles in any triangle. Given any two elements of a triangle (side or angle), the law of sines can be used to calculate the remaining unknown elements. The formula is a/sinA = b/sinB = c/sinC, where a, b, c are the sides and A, B, C are the opposite angles. The document provides examples of using the law of sines to solve for unknown sides and angles in various triangles. It also includes practice problems for students to work through applying the law of sines.
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This document provides information on solving problems involving right triangles using trigonometry, including the Law of Sines and Law of Cosines. It includes examples of using trigonometry to solve problems involving angles of elevation/depression, finding areas and volumes, and determining distances. Tables of trigonometric function values are presented along with explanations of evaluating functions in different quadrants using reference angles. Proofs and examples are given for applying the Law of Sines and Law of Cosines to find missing sides and angles of triangles.
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This document discusses partial fraction decomposition and outlines 4 objectives: 1) decompose rational functions into partial fractions, 2) find the coefficients of the partial fractions, 3) use partial fraction decomposition to evaluate integrals, and 4) use partial fraction decomposition to solve application problems.
Sulpcegu5e ppt 11_4
This document outlines objectives for learning matrix algebra and solving a word problem. It discusses the zero matrix, matrices, and calculates total revenue from a clothing store's sales of 200 blouses for $40 each, 100 scarves for $20 each, and 10 suits for $300 each last month.
Sulpcegu5e ppt 11_3
This document discusses systems of linear equations and determinants. It covers 5 objectives: 1) introducing determinants, 2) using Cramer's Rule to solve systems of equations, 3) properties of determinants such as that multiplying all elements of any row or column by the same non-zero scalar does not change the determinant's value. It also covers additional properties of determinants.
Sulpcegu5e ppt 11_2
This document discusses systems of linear equations and matrices. It outlines 4 objectives: 1) represent systems of linear equations using matrices, 2) solve systems of linear equations using row reduction, 3) use matrices to represent and solve applications involving systems of linear equations, and 4) determine if a given system of linear equations has one solution, infinitely many solutions, or no solution.
Sulpcegu5e ppt 11_1
The document discusses a movie theater that sold tickets for $9 each with a $2 senior discount. It provides that the theater took in $4760 in revenue and sold 600 tickets total that evening. It asks to write two equations, one relating the number of $9 tickets (x) and $7 discounted tickets (y) based on revenue, and another relating x and y based on the total number of tickets sold.
Sulpcegu5e ppt 10_7
This document discusses plane curves and parametric equations, with 5 objectives. It provides two examples of motion around ellipses, one clockwise over 2 seconds starting at (4,0), the other counterclockwise over 1 second starting at (0,3).
Sulpcegu5e ppt 10_6
Polar equations can be used to represent ellipses and hyperbolas. The eccentricity (e) of a conic is defined as the distance from the focus to the center (c) divided by the distance from the center to the vertex (a). The polar equation of a conic with its focus at the pole is r = p/(1-e cos θ), where r is the radial distance, p is the perpendicular distance to the directrix, and e is the eccentricity.
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- 1. Section 8.3 The Law of Cosines
- 5. OBJECTIVE 1
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- 7. OBJECTIVE 2
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- 9. OBJECTIVE 3