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.
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.
Who wants to be a millionare triangles reviewbendontje
This document contains a series of multiple choice questions testing concepts in geometry including:
- Finding the value of x
- Finding angle measures
- Using the Law of Sines to find side lengths and angle measures
- Using the Law of Cosines to find side lengths and angle measures
- Finding the area of triangles
For each question there are 4 possible answer choices labelled A-D. The document ends by congratulating the reader for completing the questions.
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.
The document provides solutions to 22 trigonometry problems involving topics like distance traveled along a circle, distance between initial and final locations along a circle, angles subtended by lines, heights of balloons using trigonometric ratios, angles subtended by flagpoles and pedestals, lengths of poles, velocities of boats factoring currents, distances between circle centers, angles of inclination of hills, distances between ships traveling at different velocities and directions, trigonometric identities, and solving for unknowns in trigonometric equations. The final problem uses the law of cosines to solve for the length of side a of a spherical triangle given two angles and the opposite side.
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.
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.
This document discusses the ambiguous case of the Law of Sines when given an acute angle (A) and sides a and c of a triangle, where a < c. It explains that:
1) When a < h, where h is the altitude of the triangle, no triangle can be formed.
2) When a = h, one right triangle is formed.
3) When a > h, two triangles are formed - one acute triangle and one obtuse triangle.
So in summary, given the measurements A, a and c where a < c, there will be 0 triangles formed if a < h, 1 triangle if a = h, and 2 triangles if a > h.
This document provides a proof of the angle property of a circle. It shows that the angle subtended by an arc at the center of a circle is twice the angle subtended at the circumference. This is proved using the fact that triangles formed by the radius and arc are isosceles triangles, making the base angles equal. It also discusses an alternative proof that can be used when the standard proof does not apply for a given case.
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.
Who wants to be a millionare triangles reviewbendontje
This document contains a series of multiple choice questions testing concepts in geometry including:
- Finding the value of x
- Finding angle measures
- Using the Law of Sines to find side lengths and angle measures
- Using the Law of Cosines to find side lengths and angle measures
- Finding the area of triangles
For each question there are 4 possible answer choices labelled A-D. The document ends by congratulating the reader for completing the questions.
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.
The document provides solutions to 22 trigonometry problems involving topics like distance traveled along a circle, distance between initial and final locations along a circle, angles subtended by lines, heights of balloons using trigonometric ratios, angles subtended by flagpoles and pedestals, lengths of poles, velocities of boats factoring currents, distances between circle centers, angles of inclination of hills, distances between ships traveling at different velocities and directions, trigonometric identities, and solving for unknowns in trigonometric equations. The final problem uses the law of cosines to solve for the length of side a of a spherical triangle given two angles and the opposite side.
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.
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.
This document discusses the ambiguous case of the Law of Sines when given an acute angle (A) and sides a and c of a triangle, where a < c. It explains that:
1) When a < h, where h is the altitude of the triangle, no triangle can be formed.
2) When a = h, one right triangle is formed.
3) When a > h, two triangles are formed - one acute triangle and one obtuse triangle.
So in summary, given the measurements A, a and c where a < c, there will be 0 triangles formed if a < h, 1 triangle if a = h, and 2 triangles if a > h.
This document provides a proof of the angle property of a circle. It shows that the angle subtended by an arc at the center of a circle is twice the angle subtended at the circumference. This is proved using the fact that triangles formed by the radius and arc are isosceles triangles, making the base angles equal. It also discusses an alternative proof that can be used when the standard proof does not apply for a given case.
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.
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.
The document discusses the Law of Sines, which can be used to find missing parts of any triangle. It provides two cases where the Law of Sines can be used: 1) when two angles and any side are known, and 2) when two sides and the angle between them are known. It then gives examples of using the Law of Sines to find specific values like side lengths and triangle areas.
1) No matter where point P is placed on a circle, the angle α formed between the radius OP and the chord AB is always 90 degrees.
2) AB is a diameter of the circle and forms two isosceles triangles, so the angles x and y are equal. Adding the angles of triangle APB gives x + x + y + y = 1800, so x + y = 900 and angle APB is always 90 degrees.
3) For any chord AB of a circle, the angle subtended (APB) at the circumference is equal to twice the angle subtended (AOB) at the centre.
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.
This module discusses solving oblique triangles using the law of sines. It begins by introducing acute and obtuse triangles and how to find the measure of the third angle given two angles. It then derives the law of sines and shows how it can be used to solve triangles where two angles and a side opposite one angle are given, two angles and the included side are given, or all three sides are given. Examples of solving various triangle scenarios are provided.
1. No matter where a point P is placed on a circle, the angle subtended at the center (α) is always 90 degrees.
2. If AB is a chord of a circle with center O, the angle subtended by the arcs APB is always equal to the angle subtended by the arcs AOB.
3. If AB and CD are chords of equal length in a circle, then the perpendicular distances of their midpoints from the center (d1 and d2) are also equal.
The document discusses trigonometry and using a scientific calculator to solve trigonometric problems involving right triangles. It defines the sides of a right triangle as the hypotenuse, opposite, and adjacent sides. It explains how to use the sine, cosine, and tangent functions on a calculator to find missing angles and sides of a triangle when given other information. Examples are provided for finding unknown angles from side lengths and unknown sides when given an angle measure. The document also addresses situations where the unknown value is on the bottom of a trigonometric ratio fraction.
This document provides instruction on using the Law of Cosines to solve triangles. It begins by explaining that the Law of Sines cannot be used to solve triangles when the side-side-side (SSS) or side-angle-side (SAS) configurations are given. The Law of Cosines formulas are then introduced. Examples are worked through demonstrating how to use the Law of Cosines to solve triangles in SSS and SAS formations. Key steps like verifying a triangle can be formed from given SSS measurements and properly applying order of operations are emphasized. Students are assigned homework problems applying these concepts.
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
This document provides examples and practice problems for identifying angle measures in parallelograms. It gives the side lengths of parallelogram ABCD and asks to find the three other angle measures. It also provides a practice problem to find missing angle measures for an additional parallelogram.
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.
3) Notation used to denote sides and angles of a triangle.
4) The sine rule and cosine rule for solving triangles given certain information about sides and angles.
5) Several other formulas and theorems for solving triangles, finding trig functions of half-angles, areas of triangles, and properties related to circumcircles, incircles, and more. It also provides example problems.
Precalculus 4 4 graphs pf sine and cosine v2excetes1
The document provides instructions on graphing sine and cosine functions by identifying key points, describing how amplitude, period, and phase shift affect the graph, and giving examples of constructing sinusoidal models from data. It explains how to find the amplitude, period, phase shift, and vertical shift from maximum and minimum values of a periodic function and choose between sine and cosine based on the behavior at a given time. Examples demonstrate modeling tide depth from high and low tide times and depths.
The document introduces the Segment Addition Postulate, which states that if point B is between points A and C, then the sum of the lengths of segments AB and BC equals the length of segment AC. It then provides an example of using this postulate to solve for the unknown value x in a geometry problem where one segment bisects another.
The document provides reasoning to show that a car was built after 1990 based on the presence of an mp3 player. It does this by:
1) Examining the conclusion that the car was built after 1990.
2) Assuming the opposite, that the car was built before 1990.
3) Developing a contradictory statement using logic - that if built before 1990 it would not have an mp3 player, but it does, so it must have been built after 1990.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The cosine law can be used to find missing side lengths or angles in any triangle where two sides and the angle between them are known. The sine law relates the sines of the angles of a triangle to the lengths of the sides opposite them.
Solving a right triangle when given an angle group 1 sam,john,molly,shmashjseal
This problem gives a right triangle with one angle (34.5 degrees) and one side (12.7 inches) known. It then shows how to calculate the remaining side (A = 7.19 inches) using sine, another side (B = 10.5 inches) using cosine, and the remaining angle (B = 55.5 degrees) using subtraction from 180 degrees. The key steps are using trigonometric functions of sine and cosine to solve for missing sides when an angle and one side are known in a right triangle.
This document discusses finding angle measures in parallelograms. It provides the side lengths of parallelogram ABCD, where DC = AD = CE = AC = BE = DB = 7, 14, 6, 12, 9, 18. It asks to find the three other angle measures in the parallelogram. It also asks to find the missing angle measures for additional parallelograms.
This document discusses using the law of sines to find unknown parts of a triangle. It explains that the law of sines can be used when two angles and a side are known to find the other sides. Two examples are provided: the first finds distances AC and BC of a triangle where two angles and one side are known, and the second determines possible measures of angle T when angle S, side s, and side t are known. The document ends by posing a practice problem to solve triangle RST when angle S, side r, and side s are given.
The Law of Sines is a principle of trigonometry stating that the length of the sides of any triangle are proportional to the sines of the opposite angles.
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.
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.
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.
The document discusses the Law of Sines, which can be used to find missing parts of any triangle. It provides two cases where the Law of Sines can be used: 1) when two angles and any side are known, and 2) when two sides and the angle between them are known. It then gives examples of using the Law of Sines to find specific values like side lengths and triangle areas.
1) No matter where point P is placed on a circle, the angle α formed between the radius OP and the chord AB is always 90 degrees.
2) AB is a diameter of the circle and forms two isosceles triangles, so the angles x and y are equal. Adding the angles of triangle APB gives x + x + y + y = 1800, so x + y = 900 and angle APB is always 90 degrees.
3) For any chord AB of a circle, the angle subtended (APB) at the circumference is equal to twice the angle subtended (AOB) at the centre.
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.
This module discusses solving oblique triangles using the law of sines. It begins by introducing acute and obtuse triangles and how to find the measure of the third angle given two angles. It then derives the law of sines and shows how it can be used to solve triangles where two angles and a side opposite one angle are given, two angles and the included side are given, or all three sides are given. Examples of solving various triangle scenarios are provided.
1. No matter where a point P is placed on a circle, the angle subtended at the center (α) is always 90 degrees.
2. If AB is a chord of a circle with center O, the angle subtended by the arcs APB is always equal to the angle subtended by the arcs AOB.
3. If AB and CD are chords of equal length in a circle, then the perpendicular distances of their midpoints from the center (d1 and d2) are also equal.
The document discusses trigonometry and using a scientific calculator to solve trigonometric problems involving right triangles. It defines the sides of a right triangle as the hypotenuse, opposite, and adjacent sides. It explains how to use the sine, cosine, and tangent functions on a calculator to find missing angles and sides of a triangle when given other information. Examples are provided for finding unknown angles from side lengths and unknown sides when given an angle measure. The document also addresses situations where the unknown value is on the bottom of a trigonometric ratio fraction.
This document provides instruction on using the Law of Cosines to solve triangles. It begins by explaining that the Law of Sines cannot be used to solve triangles when the side-side-side (SSS) or side-angle-side (SAS) configurations are given. The Law of Cosines formulas are then introduced. Examples are worked through demonstrating how to use the Law of Cosines to solve triangles in SSS and SAS formations. Key steps like verifying a triangle can be formed from given SSS measurements and properly applying order of operations are emphasized. Students are assigned homework problems applying these concepts.
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
This document provides examples and practice problems for identifying angle measures in parallelograms. It gives the side lengths of parallelogram ABCD and asks to find the three other angle measures. It also provides a practice problem to find missing angle measures for an additional parallelogram.
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.
3) Notation used to denote sides and angles of a triangle.
4) The sine rule and cosine rule for solving triangles given certain information about sides and angles.
5) Several other formulas and theorems for solving triangles, finding trig functions of half-angles, areas of triangles, and properties related to circumcircles, incircles, and more. It also provides example problems.
Precalculus 4 4 graphs pf sine and cosine v2excetes1
The document provides instructions on graphing sine and cosine functions by identifying key points, describing how amplitude, period, and phase shift affect the graph, and giving examples of constructing sinusoidal models from data. It explains how to find the amplitude, period, phase shift, and vertical shift from maximum and minimum values of a periodic function and choose between sine and cosine based on the behavior at a given time. Examples demonstrate modeling tide depth from high and low tide times and depths.
The document introduces the Segment Addition Postulate, which states that if point B is between points A and C, then the sum of the lengths of segments AB and BC equals the length of segment AC. It then provides an example of using this postulate to solve for the unknown value x in a geometry problem where one segment bisects another.
The document provides reasoning to show that a car was built after 1990 based on the presence of an mp3 player. It does this by:
1) Examining the conclusion that the car was built after 1990.
2) Assuming the opposite, that the car was built before 1990.
3) Developing a contradictory statement using logic - that if built before 1990 it would not have an mp3 player, but it does, so it must have been built after 1990.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The cosine law can be used to find missing side lengths or angles in any triangle where two sides and the angle between them are known. The sine law relates the sines of the angles of a triangle to the lengths of the sides opposite them.
Solving a right triangle when given an angle group 1 sam,john,molly,shmashjseal
This problem gives a right triangle with one angle (34.5 degrees) and one side (12.7 inches) known. It then shows how to calculate the remaining side (A = 7.19 inches) using sine, another side (B = 10.5 inches) using cosine, and the remaining angle (B = 55.5 degrees) using subtraction from 180 degrees. The key steps are using trigonometric functions of sine and cosine to solve for missing sides when an angle and one side are known in a right triangle.
This document discusses finding angle measures in parallelograms. It provides the side lengths of parallelogram ABCD, where DC = AD = CE = AC = BE = DB = 7, 14, 6, 12, 9, 18. It asks to find the three other angle measures in the parallelogram. It also asks to find the missing angle measures for additional parallelograms.
This document discusses using the law of sines to find unknown parts of a triangle. It explains that the law of sines can be used when two angles and a side are known to find the other sides. Two examples are provided: the first finds distances AC and BC of a triangle where two angles and one side are known, and the second determines possible measures of angle T when angle S, side s, and side t are known. The document ends by posing a practice problem to solve triangle RST when angle S, side r, and side s are given.
The Law of Sines is a principle of trigonometry stating that the length of the sides of any triangle are proportional to the sines of the opposite angles.
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.
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.
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
The document derives the Law of Sines and Law of Cosines, which relate the angles and sides of triangles. It discusses using these laws to solve oblique triangles given certain information like two angles and a side, two sides and the angle opposite one of the sides, two sides and the angle between them, or all three sides. It also covers finding the area of a triangle given two sides and the included angle, or using Heron's Formula with all three sides. Application problems demonstrate using these concepts to solve real-world geometry problems.
This document provides an introduction and overview of Module 7: Triangle Trigonometry which covers using trigonometric ratios to solve problems involving right triangles and oblique triangles. The module is divided into 5 lessons that cover the six trigonometric ratios, ratios of special angles, angles of elevation and depression, application word problems involving right triangles, and the laws of sines and cosines for solving problems with oblique triangles. A pre-assessment with 19 multiple choice questions is also provided to gauge students' prior knowledge on triangle trigonometry concepts before beginning the lessons.
Using Kinect with Flash to create amazing experiencesHuy Nguyen
This document discusses using the Kinect motion sensor with Flash to create interactive experiences. It provides links to OpenKinect and ActionScript 3 libraries for accessing Kinect data from Flash, as well as examples of demos created with these libraries and ideas for other types of experiences that can be built. The document aims to introduce developers to programming options for using Kinect with Flash.
This document contains notes from a geometry lesson on the law of sines and law of cosines. It includes reflection questions about what makes these laws powerful and which parts of a triangle they apply to. The notes state that the law of sines deals with two sides and two angles of a triangle, while the law of cosines deals with one angle and three sides. It also explains that in the law of cosines, the angle needs to be located across from the side that stands alone on one side of the equation. The document provides further notes and examples to explain these concepts.
This song discusses the artist feeling that he has fallen out of favor with the music industry and society. In the first verse, he talks about having success globally but feeling that he no longer fits with the current generation. He describes facing a "sophomore slump" and struggles to create new music that will be popular. The rest of the song elaborates on his wordplay and lyrical skills, with the artist portraying himself as a "wizard" of sounds and lyrics. He hopes to reconnect with fans and rebuild his career through his talented use of language and poetry.
The document discusses the Law of Cosines formula for solving for unknown sides and angles of triangles, including right triangles and non-right triangles. It introduces the Law of Cosines formula, showing the variables representing the sides and angles of a triangle, and walks through applying it and simplifying the formula. It instructs pausing to discuss and then continuing to problem 8 of a worksheet on the Law of Cosines.
Presentation on two interactive class activities for students learning a second language - a Taste Test and a Murder Mystery (originally done in Japanese)
Presentation for a workshop, geared toward elementary grades, focusing on web-based resources, games, and activities for most interactive whiteboards. The focus is on student-centered instruction.
Unpacking an Interactive 4-Part Math LessonKyle Pearce
This document describes a 4-part interactive math lesson plan consisting of definitions/procedures, examples, tasks, and consolidation. It provides an example lesson on sharing chocolates between 4 people. Students are asked to predict and calculate how many chocolates each person would receive. They then solve related problems to determine the number of each color chocolate. The lesson is designed according to a vision of math education that is interconnected, visual/concrete, and contextualized.
SBTP - Activity sheet for proving law of sines and cosines DavNor DivDods Dodong
This document provides an activity sheet to teach students how to solve for unknown parts of an oblique triangle. It introduces an oblique triangle ABC with sides of length a, b, c and altitude h. Students are asked to derive equations relating the sides and angles using properties of trigonometry and similar triangles. These include deriving one of the Law of Cosines formulas by relating b^2 to h^2 and x^2, then substituting x in terms of b and angle A. Students are also asked to derive the other two Law of Cosines formulas and the Law of Sines by drawing additional altitudes and writing proportional relationships between sides and opposite angles.
The document discusses the Law of Sines for solving oblique triangles. It explains that to find missing parts of an oblique triangle, you need two known parts (side or angle) and the Law of Sines relationship that the ratio of any side to its opposite angle is equal to the ratios of the other two sides and their opposite angles. Examples are provided to demonstrate using the Law of Sines to find missing sides and angles of triangles.
This document appears to be instructions for a past simple speaking game. Players take turns rolling dice and moving along a board, landing on squares that prompt them to speak for 30 seconds about a past event or experience from their life without stopping or making grammatical mistakes. The first player to reach the finish square wins. The board includes prompts like "the last time you played a game or sport" and "where you went on holiday last year".
The document discusses the Law of Sines, which is a rule used to find unknown angles and sides of triangles when some combination of angles and sides are known. The Law of Sines states that the ratio of any side to its opposite angle is equal to the ratio of any other side to its opposite angle. An example problem demonstrates using the Law of Sines to solve a triangle when two angles and one side are given. Additional resources are provided to learn more about solving triangles with the Law of Sines.
The sine rule can be used to solve for unknown sides or angles of any triangle when enough information is given, such as two sides and the angle between them, or one side and two angles. It relates the ratios of sides to opposite angles: a/sinA = b/sinB = c/sinC. To use it, corresponding sides and angles are matched up and the ratios set equal. Then cross-multiplication and inverse trigonometric functions are used to solve for the unknown. Worked examples show how to calculate missing sides or angles.
This document provides information on solving triangular problems using the sine and cosine rules. It explains that the sine rule can be used to calculate sides and angles in triangles where there is no right angle when two angles and a side or two sides and a non-included angle are known. Examples are provided to demonstrate using the sine rule to find missing sides and angles. The cosine rule is introduced as another method that can be used when three sides are known or two sides and an included angle. Examples demonstrate using the cosine rule to find a missing side or angle.
The document summarizes the sine rule and cosine rule for solving triangles.
The sine rule enables the calculation of sides and angles in triangles where there is no right angle. It can be used when two angles and a side are known, or two sides and the included angle are known.
The cosine rule is used when two sides and the included angle are given, or all three sides are given. It expresses the length of one side as a function of the other two sides and the included angle.
Worked examples are provided to demonstrate how to apply the sine rule and cosine rule to find missing sides or angles in triangles. Drill problems are also included for additional practice.
1. Draw two right triangles sharing the hypotenuse of 45m
2. For the first triangle, use tan to find the height of building A as 77.94m
3. For the second triangle, use tan to find the height of building B
This document discusses the Law of Sines and Law of Cosines for solving oblique triangles. It provides examples of using these laws to:
- Solve triangles given two angles and one side (AAS) or two sides and an angle opposite (SSA) using the Law of Sines.
- Determine if a triangle is valid or ambiguous given two sides and an angle (SSA) using the Law of Sines.
- Find the area of oblique triangles using the relationship between a side and its opposite angle from the Law of Sines or by Heron's formula when all three sides are given.
This document discusses using the Law of Sines and Law of Cosines to solve oblique triangles. It covers the four cases for solving triangles: two angles and a side (AAS/ASA), two sides and an angle opposite (SSA), three sides (SSS), and two sides and their included angle (SAS). The Law of Sines can be used for AAS/ASA and SSA cases, while the Law of Cosines is needed for SSS and SAS cases. It also discusses finding the area of triangles using the Law of Sines and Heron's formula for SSS cases.
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.
The document provides a review of trigonometric ratios such as sine, cosine, and tangent that are used to solve problems involving right triangles, includes examples of applying these ratios to solve practice problems involving missing side lengths and angle measures, and discusses special right triangles and the mnemonic device SOH CAH TOA to remember the trigonometric functions.
The document discusses two trigonometric identities, the Law of Sines and the Law of Cosines, that can be used to solve for unknown sides and angles in any triangle when sufficient information is given. The Law of Sines relates the ratios of the sines of the angles to the lengths of the sides opposite those angles. The Law of Cosines relates all three sides of a triangle and the angle opposite one of the sides. Examples are provided to demonstrate solving problems using each law.
The document provides a review of trigonometric ratios such as sine, cosine, and tangent that are used to solve problems involving right triangles, includes examples of applying these ratios to solve practice problems involving missing side lengths and angle measures, and reviews special right triangles and the SOH CAH TOA method for remembering the trigonometric functions.
1) The document discusses solving triangles using the Law of Sines. It provides examples of solving triangles given different combinations of angle and side measurements, known as the AAS, ASA, SSA, and SAS cases.
2) The SSA case is sometimes called the "ambiguous case" because it can result in zero, one, or two possible triangles depending on the angle and side measurements.
3) The document also discusses finding the area of triangles using trigonometric functions, providing examples of calculating area given different side lengths and included angles.
This document discusses the Law of Sines and Law of Cosines, which can be used to solve for missing sides and angles of oblique triangles (triangles without right angles). The Law of Sines relates the ratios of sides to opposite angles, while the Law of Cosines relates sides and angles. Several examples show how to apply these laws to find missing measurements in triangles given certain known values. The area of oblique triangles can also be found using these formulas.
- The Sine Rule and Cosine Rule can be used to find unknown sides and angles in triangles that are not right-angled.
- The Sine Rule states that the ratio of the sine of an angle to its opposite side is equal to the ratio of any other angle-side pair. It is generally easier to use than the Cosine Rule.
- The Cosine Rule relates all three sides of a triangle to one of its interior angles. It can be used to find a single unknown when three other parts of the triangle are known.
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.
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.
The document discusses the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. It provides examples of using the Law of Sines to solve for missing sides and angles of triangles. It also discusses ambiguous cases, area formulas for triangles, and provides sample problems for students to work.
This module introduces triangle trigonometry and solving for unknown sides and angles of right triangles. It covers determining the appropriate trigonometric function to use given known parts of a right triangle, such as the hypotenuse and one leg. Examples are provided to demonstrate applying trigonometric functions like sine, cosine, and tangent to find missing lengths and angles. The module also addresses solving right triangle problems involving angles of elevation and depression that are commonly seen in fields like surveying.
The document discusses the law of sines, which provides a way to solve triangles when the law of cosines does not apply. It derives the law of sines, which states that the ratio of the sine of any angle of a triangle to its opposite side is equal to the ratios of the sines of the other angles to their opposite sides. Examples are provided to demonstrate using the law of sines to solve for missing angles and sides of triangles when one angle and the side opposite it are known. Special cases are discussed where there may be one, two, or no possible triangle solutions depending on the given information.
The document provides information about using the sine rule to solve problems involving trigonometry in triangles. It explains that the sine rule can be used to find unknown sides or angles of any triangle as long as enough information is given. Examples are provided to demonstrate using the sine rule to calculate missing side lengths or angles by setting up ratios of sines of known quantities to the sines of the unknown quantity and rearranging the equation. Starter questions and learning objectives are also included to reinforce understanding of using the sine rule.
This module introduces triangle trigonometry and solving for unknown sides and angles of right triangles. It provides 5 lessons:
1. Determining the appropriate trigonometric equation based on the given information in a right triangle.
2. Solving right triangles when given the hypotenuse and one leg.
3. Solving right triangles when given the hypotenuse and one acute angle.
4. Solving right triangles when given one leg and one acute angle.
5. Solving right triangles when given the measures of both legs. It includes examples and practice problems for each case.
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 alleviate symptoms of mental illness and boost overall mental well-being.
The document discusses the results of a study on the effects of exercise on memory and thinking abilities in older adults. The study found that regular exercise can help reduce the decline in thinking abilities that often occurs with age. Older adults who exercised regularly performed better on cognitive tests and brain scans showed they had greater activity in important areas for memory and learning compared to less active peers.
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.
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.
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.
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.
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 stimulates the production of endorphins in the brain which elevate mood and reduce stress levels.
1. Section 9.3
The Law of Sines
Objective:
Use the Law of Sines to find unknown parts
of triangles.
A
B
C
2. Law of Sines
C
Use our new formula to
find the area of this
triangle three different
ways.
B A
Divide all three sides by 1wabc to get:
Law of Sines
3. Law of Sines
C
A = 40° Solve the
C = 60° triangle.
a = 10
B A
Answers
4. Law of Sines
C
A = 35° Solve the
C = 70° triangle.
b = 12
B A
Answers
5. Law of Sines
In the first example, we knew AAS about our
triangle.
In the second example, we knew ASA about
our triangle.
What other combinations of sides and angles
completely specify a triangle?
6. Law of Sines
C
B = 30° Solve the
a = 10 triangle.
b=4
B A
Answers
7. Law of Sines
C
B = 30° Solve the
a = 10 triangle.
b=4
B A
What is the problem here?
8. Law of Sines
C
B = 30° Solve the
a = 10 triangle.
b=4
B A
What is the problem here?
9. Law of Sines
C
B = 30° Solve the
a = 10 triangle.
b=4
B A
What is the problem here?