Types of Angles acute right obtuse straight Types of Triangles acute right obtuse hypotenuse legs


 Julius Hutchinson
 2 years ago
 Views:
Transcription
1 MTH 065 Class Notes Lecture 18 (4.5 and 4.6) Lesson 4.5: Triangles and the Pythagorean Theorem Types of Triangles Triangles can be classified either by their sides or by their angles. Types of Angles An acute angle is one whose measure is between 0º and 90 º. A right angle is one whose measure is exactly 90 º. An obtuse angle is one whose measure is between 90 º and 180 º. A straight angle is one whose measure is exactly 180 º. The sum of the three angles of a triangle must be 180 º. Types of Triangles An acute triangle is a triangle with three acute angles. A right triangle is a triangle with one right angle. An obtuse triangle is a triangle with one obtuse angle. Practice 1 Identify the following triangles as right, obtuse or acute. Use a protractor to measure the angles when necessary. is a right triangle because is a right angle. is an obtuse triangle because is an obtuse angle. is an acute triangle because all of the angles are less than 90. Pythagorean Theorem For the remainder of this section we are going to focus on right triangles. The longest side of a right triangle is called the hypotenuse. The hypotenuse is always opposite the right angle. The other two sides are called the legs of the right triangle.
2 Pythagorean Theorem If a triangle is a right triangle with legs of length a and b, and a hypotenuse of length c, then. B c a A b C Practice 2 Find the length of the unknown side in the given right triangle. If necessary, round to 3 decimal places. The hypotenuse is 5.5 cm. This creates the following: a = 4.5, b = c, c = 5.5 Converse of the Pythagorean Theorem If a triangle has sides of length, a, b, c and, then the triangle is a right triangle, and side c is the hypotenuse. Practice 3 Determine whether the given lengths of segments can form a right triangle. a) 15, 12, 9 a = 9, b = 12, c = 15 Yes, these lengths form a right triangle.
3 b) 4 a =, b =, c = 8 Yes, these lengths form a right triangle. c) 5, 5, 10 a =, b =, c = 10 No, these lengths do not form a right triangle. Distance Formula We are now ready to learn how to find the distance between any two points in the rectangular coordinate system. We will use the Pythagorean Theorem to help us find that distance. We can find the lengths of each leg of the right triangle by taking the difference of coordinate that is not shared by those endpoints. For example, with the above triangle, we can find the length of AB, by looking at the coordinates of A and B. A, (2, 5), and B, (10, 5) give us a difference of 8 since 10 2 = 8. We can use this same argument to obtain the length of BC = 6. We can now find the distance from A to C by using the Pythagorean Theorem. Note: We do not need to consider 10 as we are talking about length and cannot have a negative length. We can generalize this process in what is called the Distance Formula. Remember that we have our and
4 Distance Formula The distance between two points and, is Practice 4 Find the distance between (4,5) and (2,9). Give the answer both as an exact value and as an approximate value rounded to 3 decimal places. and Using Triangles to Solve Problems There are many applications that use triangles. We will look at some in this section and again in section 4.6. Practice 5 Richard is going to attach 3 guy wires to his CB antenna at a point 30 feet above the ground. According to the manufacturer of the antenna, the wires should be secured at ground level 17 feet from the base of the antenna. Richard wants to know how much wire he is going to need to secure his antenna. Round the answer to the nearest tenth of a foot. (Assume the antenna is located on level ground.) a =, b = With three guy wires needed for this project, this means Richard will need a total of 34.5(3) = ft. Lesson 4.6: Sine, Cosine, and Tangent Right triangle trigonometry is a branch of mathematics that deals with properties of triangles. In this section we will study one concept from trigonometry: the sine, cosine, and tangent ratios as they apply to a right triangle. The Ratios A ratio is a way of comparing two things. In this case, the lengths of the sides of a triangle. Let s start by identifying the parts of a right triangle. The best way to do this is through the perspective of one of the acute angles.
5 This is defined from the perspective of B. From that perspective, AC is considered the opposite side or leg, AB is considered the hypotenuse as it is opposite from the 90 angle, and BC is considered adjacent side or leg because it is next to B. With these sides established we are ready to write down the trigonometric ratios of angle B. The sine of B: The cosine of B: The tangent of B: Trigonometric Ratios Note: The opposite leg and adjacent leg change depending on which acute angle we are talking about. For example, if we were to look from the perspective of A, then the opposite leg and adjacent leg would change. Practice 1 For the triangle in Figure 4 write: (a) the sine, cosine, and tangent ratios of A (b) the sine, cosine, and tangent ratios of B. (a)
6 (b) Finding the Sine, Cosine, and Tangent of an Angle on a Calculator If we don't know the lengths of the sides in the right triangle, but we know the measure of the angle, we can use a calculator to find the trigonometric ratio for that angle. We will be working with angles measured in degrees. We need to make sure our calculator is in the degree mode. For a TI 83/84, to do this press the MODE key. The 3 rd line has two options: Radian and Degree. If your calculator has Radian selected, you need to select Degree. Practice 2 Using a calculator, evaluate each trigonometric function. Round answers to four decimal places where appropriate. (a) (b) (c) Finding a Missing Length in a Triangle In a right triangle we know the measure of one of the angles is 90º. If we know the length of one of the sides and we know the measure of one of the angles (other than the right angle), we can find the lengths of the other sides. Practice 3 A. A right triangle is drawn below. Find the length of side a. Round the answer to two decimal places.
7 B. A triangle is drawn below. Find the length of side b. Round the answer to one decimal place. Finding the Height or a Distance Many kinds of problems can be solved using trigonometry, from finding the height of a tree to measuring the distance across a canyon. Practice 4 (a) A 24foot ladder leaning against a vertical wall forms an angle of 65º with the ground. To the nearest tenth of a foot, how far is the base of the ladder from the wall? 24 ft. x 65
8 (b) Steven rides his bicycle home from school every day. He normally rides 0.6 miles along Arnold Way, then turns the corner (90º) onto Willamette Avenue and rides a short distance to his home. If he could ride in a straight line from his school to his home that line would form a 23º angle with his route along Arnold Way. How far from the corner of Arnold Way and Willamette Avenue does Steven live? Round your answer to one decimal place. School Arnold Way mil. x Willamette Avenue Home
Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook
Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.
More information11 Trigonometric Functions of Acute Angles
Arkansas Tech University MATH 10: Trigonometry Dr. Marcel B. Finan 11 Trigonometric Functions of Acute Angles In this section you will learn (1) how to find the trigonometric functions using right triangles,
More informationName Period Right Triangles and Trigonometry Section 9.1 Similar right Triangles
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationYou can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure
Solving a Right Triangle A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve
More informationRight Triangle Trigonometry Test Review
Class: Date: Right Triangle Trigonometry Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the length of the missing side. Leave your answer
More informationas a fraction and as a decimal to the nearest hundredth.
Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. sin A The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. So, 2. tan C The tangent of an
More informationUNIT 8 RIGHT TRIANGLES NAME PER. I can define, identify and illustrate the following terms
UNIT 8 RIGHT TRIANGLES NAME PER I can define, identify and illustrate the following terms leg of a right triangle short leg long leg radical square root hypotenuse Pythagorean theorem Special Right Triangles
More informationRight Triangles and SOHCAHTOA: Finding the Measure of an Angle Given any Two Sides (ONLY for ACUTE TRIANGLES Why?)
Name Period Date Right Triangles and SOHCAHTOA: Finding the Measure of an Angle Given any Two Sides (ONLY for ACUTE TRIANGLES Why?) Preliminary Information: SOH CAH TOA is an acronym to represent the following
More informationSolve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
Solve each right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree. 42. The sum of the measures of the angles of a triangle is 180. Therefore, The sine of an angle
More informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8  Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationUnit 2: Right Triangle Trigonometry RIGHT TRIANGLE RELATIONSHIPS
Unit 2: Right Triangle Trigonometry This unit investigates the properties of right triangles. The trigonometric ratios sine, cosine, and tangent along with the Pythagorean Theorem are used to solve right
More informationIntermediate Algebra with Trigonometry. J. Avery 4/99 (last revised 11/03)
Intermediate lgebra with Trigonometry J. very 4/99 (last revised 11/0) TOPIC PGE TRIGONOMETRIC FUNCTIONS OF CUTE NGLES.................. SPECIL TRINGLES............................................ 6 FINDING
More informationRight Triangles LongTerm Memory Review Review 1
Review 1 1. Is the statement true or false? If it is false, rewrite it to make it true. A right triangle has two acute angles. 2 2. The Pythagorean Theorem for the triangle shown would be a b c. Fill in
More information1 Math 116 Supplemental Textbook (Pythagorean Theorem)
1 Math 116 Supplemental Textbook (Pythagorean Theorem) 1.1 Pythagorean Theorem 1.1.1 Right Triangles Before we begin to study the Pythagorean Theorem, let s discuss some facts about right triangles. The
More informationLaw of Cosines. If the included angle is a right angle then the Law of Cosines is the same as the Pythagorean Theorem.
Law of Cosines In the previous section, we learned how the Law of Sines could be used to solve oblique triangles in three different situations () where a side and two angles (SAA) were known, () where
More informationB a. This means that if the measures of two of the sides of a right triangle are known, the measure of the third side can be found.
The Pythagorean Theorem One special kind of triangle is a right triangle a triangle with one interior angle of 90º. B A Note: In a polygon (like a triangle), the vertices (and corresponding angles) are
More information41 Right Triangle Trigonometry
Find the exact values of the six trigonometric functions of θ. 1. The length of the side opposite θ is 8 is 18., the length of the side adjacent to θ is 14, and the length of the hypotenuse 3. The length
More informationG E O M E T R Y CHAPTER 9 RIGHT TRIANGLES AND TRIGONOMETRY. Notes & Study Guide
G E O M E T R Y CHAPTER 9 RIGHT TRIANGLES AND TRIGONOMETRY Notes & Study Guide 2 TABLE OF CONTENTS SIMILAR RIGHT TRIANGLES... 3 THE PYTHAGOREAN THEOREM... 4 SPECIAL RIGHT TRIANGLES... 5 TRIGONOMETRIC RATIOS...
More informationθ. The angle is denoted in two ways: angle θ
1.1 Angles, Degrees and Special Triangles (1 of 24) 1.1 Angles, Degrees and Special Triangles Definitions An angle is formed by two rays with the same end point. The common endpoint is called the vertex
More information7.1 Apply the Pythagorean Theorem
7.1 Apply the Pythagorean Theorem Obj.: Find side lengths in right triangles. Key Vocabulary Pythagorean triple  A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationGeometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles
Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.
More informationThe Six Trigonometric Functions
CHAPTER 1 The Six Trigonometric Functions Copyright Cengage Learning. All rights reserved. SECTION 1.1 Angles, Degrees, and Special Triangles Copyright Cengage Learning. All rights reserved. Learning Objectives
More informationPreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to:
PreAlgebra Interactive Chalkboard Copyright by The McGrawHill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGrawHill 8787 Orion Place Columbus, Ohio 43240 Click the mouse button
More information15 Trigonometry Pythagoras' Theorem. Example 1. Solution. Example 2
15 Trigonometry 15.1 Pythagoras' Theorem MEP Y9 Practice Book B Pythagoras' Theorem describes the important relationship between the lengths of the sides of a rightangled triangle. Pythagoras' Theorem
More information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More informationUsing Trigonometry to Find Missing Sides of Right Triangles
Using Trigonometry to Find Missing Sides of Right Triangles A. Using a Calculator to Compute Trigonometric Ratios 1. Introduction: Find the following trigonometric ratios by using the definitions of sin(),
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationSolution Guide for Chapter 6: The Geometry of Right Triangles
Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab
More informationTrigonometry. Week 1 Right Triangle Trigonometry
Trigonometry Introduction Trigonometry is the study of triangle measurement, but it has expanded far beyond that. It is not an independent subject of mathematics. In fact, it depends on your knowledge
More informationPythagorean Theorem & Trigonometric Ratios
Algebra 20122013 Pythagorean Theorem & Trigonometric Ratios Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the length of a side a right triangle using the Pythagorean Theorem Pgs: 14 HW:
More informationMA Lesson 19 Summer 2016 Angles and Trigonometric Functions
DEFINITIONS: An angle is defined as the set of points determined by two rays, or halflines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common
More informationLesson 19. Triangle Properties. Objectives
Student Name: Date: Contact Person Name: Phone Number: Lesson 19 Triangle Properties Objectives Understand the definition of a triangle Distinguish between different types of triangles Use the Pythagorean
More informationCreated by Ethan Fahy
Created by Ethan Fahy To proceed to the next slide click the button. Next NCTM: Use trigonometric relationships to determine lengths and angle measures. NCTM: Use geometric ideas to solve problems in,
More informationRight Triangles Test Review
Class: Date: Right Triangles Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the length of the missing side. The triangle is not drawn
More informationTrigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:
First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin
More informationTopics Covered on Geometry Placement Exam
Topics Covered on Geometry Placement Exam  Use segments and congruence  Use midpoint and distance formulas  Measure and classify angles  Describe angle pair relationships  Use parallel lines and transversals
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationSOLVING RIGHT TRIANGLES
PYTHAGOREAN THEOREM SOLVING RIGHT TRIANGLES An triangle tat as a rigt angle is called a RIGHT c TRIANGLE. Te two sides tat form te rigt angle, a and b, a are called LEGS, and te side opposite (tat is,
More informationAny two right triangles, with one other angle congruent, are similar by AA Similarity. This means that their side lengths are.
Lesson 1 Trigonometric Functions 1. I CAN state the trig ratios of a right triangle 2. I CAN explain why any right triangle yields the same trig values 3. I CAN explain the relationship of sine and cosine
More informationThe Pythagorean Packet Everything Pythagorean Theorem
Name Date The Pythagorean Packet Everything Pythagorean Theorem Directions: Fill in each blank for the right triangle by using the words in the Vocab Bo. A Right Triangle These sides are called the of
More informationCHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY
CHAPTER 8: ACUTE TRIANGLE TRIGONOMETRY Specific Expectations Addressed in the Chapter Explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that
More informationa c Pythagorean Theorem: a 2 + b 2 = c 2
Section 2.1: The Pythagorean Theorem The Pythagorean Theorem is a formula that gives a relationship between the sides of a right triangle The Pythagorean Theorem only applies to RIGHT triangles. A RIGHT
More informationMATH 10 COMMON TRIGONOMETRY CHAPTER 2. is always opposite side b.
MATH 10 OMMON TRIGONOMETRY HAPTER 2 (11 Days) Day 1 Introduction to the Tangent Ratio Review: How to set up your triangles: Angles are always upper case ( A,, etc.) and sides are always lower case (a,b,c).
More informationCLIL MultiKey lesson plan
LESSON PLAN Subject: Mathematics Topic: Triangle Age of students: 16 Language level: B1, B2 Time: 4560 min Contents aims: After completing the lesson, the student will be able to: Classify and compare
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationLaw of Cosines TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TINspire Navigator System
Math Objectives Students will be able to state the Law of Cosines Students will be able to apply the Law of Cosines to find missing sides and angles in a triangle Students will understand why the Law of
More informationRight Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle
Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle Preliminar Information: is an acronm to represent the following three trigonometric ratios or formulas: opposite
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More information41 Right Triangle Trigonometry
Find the measure of angle θ. Round to the nearest degree, if necessary. 31. Because the lengths of the sides opposite θ and the hypotenuse are given, the sine function can be used to find θ. 35. Because
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More informationChapter 8. Right Triangles
Chapter 8 Right Triangles Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the
More information4.1 Converse of the Pyth TH and Special Right Triangles
Name Per 4.1 Converse of the Pyth TH and Special Right Triangles CONVERSE OF THE PYTHGOREN THEOREM Can be used to check if a figure is a right triangle. If triangle., then BC is a Eample 1: Tell whether
More informationSimilar Right Triangles
9.1 Similar Right Triangles Goals p Solve problems involving similar right triangles formed b the altitude drawn to the hpotenuse of a right triangle. p Use a geometric mean to solve problems. THEOREM
More informationSquare Roots and the Pythagorean Theorem
4.8 Square Roots and the Pythagorean Theorem 4.8 OBJECTIVES 1. Find the square root of a perfect square 2. Use the Pythagorean theorem to find the length of a missing side of a right triangle 3. Approximate
More informationLine AB (no Endpoints) Ray with Endpoint A. Line Segments with Endpoints A and B. Angle is formed by TWO Rays with a common Endpoint.
Section 8 1 Lines and Angles Point is a specific location in space.. Line is a straight path (infinite number of points). Line Segment is part of a line between TWO points. Ray is part of the line that
More information41 Right Triangle Trigonometry
Find the exact values of the six trigonometric functions of θ. 3. The length of the side opposite θ is 9, the length of the side adjacent to θ is 4, and the length of the hypotenuse is. 7. The length of
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More information9 Right Triangle Trigonometry
www.ck12.org CHAPTER 9 Right Triangle Trigonometry Chapter Outline 9.1 THE PYTHAGOREAN THEOREM 9.2 CONVERSE OF THE PYTHAGOREAN THEOREM 9.3 USING SIMILAR RIGHT TRIANGLES 9.4 SPECIAL RIGHT TRIANGLES 9.5
More informationApplications of Trigonometry
chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced
More information3 Pythagoras' Theorem
3 Pythagoras' Theorem 3.1 Pythagoras' Theorem Pythagoras' Theorem relates the length of the hypotenuse of a rightangled triangle to the lengths of the other two sides. Hypotenuse The hypotenuse is always
More information2.1 The Tangent Ratio
2.1 The Tangent Ratio In this Unit, we will study Right Angled Triangles. Right angled triangles are triangles which contain a right angle which measures 90 (the little box in the corner means that angle
More informationExploring Geometric Mean
Exploring Geometric Mean Lesson Summary: The students will explore the Geometric Mean through the use of Cabrii II software or TI 92 Calculators and inquiry based activities. Keywords: Geometric Mean,
More informationFunctions  Inverse Trigonometry
10.9 Functions  Inverse Trigonometry We used a special function, one of the trig functions, to take an angle of a triangle and find the side length. Here we will do the opposite, take the side lengths
More informationInverse Trigonometric Functions
SECTION 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the exact value of an inverse trigonometric function. Use a calculator to approximate
More informationInstructions for SA Completion
Instructions for SA Completion 1 Take notes on these Pythagorean Theorem Course Materials then do and check the associated practice questions for an explanation on how to do the Pythagorean Theorem Substantive
More informationMidChapter Quiz: Lessons 41 through 44
Find the exact values of the six trigonometric functions of θ. Find the value of x. Round to the nearest tenth if necessary. 1. The length of the side opposite is 24, the length of the side adjacent to
More informationBasic Trigonometry, Significant Figures, and Rounding  A Quick Review
Basic Trigonometry, Significant Figures, and Rounding  A Quick Review (Free of Charge and Not for Credit) by Professor Patrick L. Glon, P.E. Basic Trigonometry, Significant Figures, and Rounding A Quick
More informationSolutions of Right Triangles with Practical Applications
Trigonometry Module T09 Solutions of Right Triangles with Practical pplications Copyright This publication The Northern lberta Institute of Technology 2002. ll Rights Reserved. LST REVISED December, 2008
More informationBasic Electrical Theory
Basic Electrical Theory Mathematics Review PJM State & Member Training Dept. Objectives By the end of this presentation the Learner should be able to: Use the basics of trigonometry to calculate the different
More informationExpress each ratio as a fraction and as a decimal to the nearest hundredth.
Express each ratio as a fraction and as a decimal to the nearest hundredth. 1.sin A SOLUTION: The sine of an angle is defined as the ratio of the opposite side to the hypotenuse. So, ANSWER: 2.tan C SOLUTION:
More informationPythagorean Theorem, Distance and Midpoints Chapter Questions. 3. What types of lines do we need to use the distance and midpoint formulas for?
Pythagorean Theorem, Distance and Midpoints Chapter Questions 1. How is the formula for the Pythagorean Theorem derived? 2. What type of triangle uses the Pythagorean Theorem? 3. What types of lines do
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationSkills Practice Simplifying Radical Expressions. Lesson Simplify c 2 d x 4 y m 5 n
111 Simplify. Skills Practice Simplifying Radical Expressions 1. 28 2. 40 3. 72 4. 99 5. 2 10 6. 5 60 7. 3 5 5 8. 6 4 24 Lesson 111 9. 2 3 3 15 10. 16b 4 11. 81c 2 d 4 12. 40x 4 y 6 13. 75m 5 n 2 5 14.
More informationy = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y
More informationRight Triangle Trigonometry
Right Triangle Trigonometry MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: evaluate trigonometric functions of acute angles, use
More informationTrigonometry. An easy way to remember trigonometric properties is:
Trigonometry It is possible to solve many force and velocity problems by drawing vector diagrams. However, the degree of accuracy is dependent upon the exactness of the person doing the drawing and measuring.
More informationLaw of Sines. Definition of the Law of Sines:
Law of Sines So far we have been using the trigonometric functions to solve right triangles. However, what happens when the triangle does not have a right angle? When solving oblique triangles we cannot
More informationContent Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade
Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources
More informationCHAPTER 10 GEOMETRY: ANGLES, TRIANGLES, AND DISTANCE (3 WEEKS)...
Table of Contents CHAPTER 10 GEOMETRY: ANGLES, TRIANGLES, AND DISTANCE (3 WEEKS)... 10.0 ANCHOR PROBLEM: REASONING WITH ANGLES OF A TRIANGLE AND RECTANGLES... 6 10.1 ANGLES AND TRIANGLES... 7 10.1a Class
More information114 Areas of Regular Polygons and Composite Figures
1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,
More informationAnswer Key. Lesson 7.1. Study Guide
Answer Key Lesson 7.1 Study Guide 1. leg; 30 2. hypotenuse; 3 Ï } 13 3. hypotenuse; 52 4. leg; 20 Ï } 6 5. leg; 5 Ï } 3 6. hypotenuse; 39 7. 1452 yd 2 8. 540 mi 2 9. 5, 12, 13; 130 cm 10. 7, 24, 25; 96
More informationGeometry review There are 2 restaurants in River City located at map points (2, 5) and (2, 9).
Geometry review 2 Name: ate: 1. There are 2 restaurants in River City located at map points (2, 5) and (2, 9). 2. Aleta was completing a puzzle picture by connecting ordered pairs of points. Her next point
More information55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.
Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit
More informationN33. Trigonometry Preview Assignment. Part 1: Right Triangles. x = FMP1O NAME:
x= 4. Trigonometry Preview ssignment FMP1O NME: B) Label all the sides of each right triangle (Hypotenuse, djacent, Opposite). 1. 2. N33 ) Find the measure of each unknown angle (variable) in the triangle.
More informationGeometry Second Semester Final Exam Review
Name: Class: Date: ID: A Geometry Second Semester Final Exam Review 1. Mr. Jones has taken a survey of college students and found that 1 out of 6 students are liberal arts majors. If a college has 7000
More informationMATH 150 TOPIC 9 RIGHT TRIANGLE TRIGONOMETRY. 9a. Right Triangle Definitions of the Trigonometric Functions
Math 50 T9aRight Triangle Trigonometry Review Page MTH 50 TOPIC 9 RIGHT TRINGLE TRIGONOMETRY 9a. Right Triangle Definitions of the Trigonometric Functions 9a. Practice Problems 9b. 5 5 90 and 0 60 90
More informationChapter 9 Party Planner
Name: Date:. Given A 53, B 78, and a 6., use the Law of Sines to solve the triangle for the value of b. Round answer to two decimal places. C b a A c b sin a B sin A b 6. sin 78 sin 53 6.sin 78 b sin 53
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationThe integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself.
Exponents An integer is multiplied by itself one or more times. The integer is the base number and the exponent (or power). The exponent tells how many times the base number is multiplied by itself. Example:
More informationGeometry Mathematics Curriculum Guide Unit 6 Trig & Spec. Right Triangles 2016 2017
Unit 6: Trigonometry and Special Right Time Frame: 14 Days Primary Focus This topic extends the idea of triangle similarity to indirect measurements. Students develop properties of special right triangles,
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationBasic Lesson: Pythagorean Theorem
Basic Lesson: Pythagorean Theorem Basic skill One leg of a triangle is 10 cm and other leg is of 24 cm. Find out the hypotenuse? Here we have AB = 10 and BC = 24 Using the Pythagorean Theorem AC 2 = AB
More informationApplications of the Pythagorean Theorem
9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem
More informationFLC Ch 1 & 3.1. A ray AB, denoted, is the union of and all points on such that is between and. The endpoint of the ray AB is A.
Math 335 Trigonometry Sec 1.1: Angles Definitions A line is an infinite set of points where between any two points, there is another point on the line that lies between them. Line AB, A line segment is
More information(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units
1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units
More informationGeometry: Classifying, Identifying, and Constructing Triangles
Geometry: Classifying, Identifying, and Constructing Triangles Lesson Objectives Teacher's Notes Lesson Notes 1) Identify acute, right, and obtuse triangles. 2) Identify scalene, isosceles, equilateral
More informationSimilarity, Right Triangles, and Trigonometry
Instruction Goal: To provide opportunities for students to develop concepts and skills related to trigonometric ratios for right triangles and angles of elevation and depression Common Core Standards Congruence
More information111 Areas of Parallelograms and Triangles. Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary.
Find the perimeter and area of each parallelogram or triangle. Round to the nearest tenth if necessary. 2. 1. Use the Pythagorean Theorem to find the height h, of the parallelogram. Each pair of opposite
More information