Chapter 5 Resource Masters

Transcription

1 Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois

2 StudentWorks TM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice, and Enrichment masters. TeacherWorks TM All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorks CD-ROM. Copyright The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Advanced Mathematical Concepts. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH ISBN: X Advanced Mathematical Concepts Chapter Resource Masters XXX

3 Vocabulary Builder vii-x Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson -4 Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson -6 Study Guide Practice Enrichment Contents Lesson -7 Study Guide Practice Enrichment Lesson -8 Study Guide Practice Enrichment Chapter Assessment Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Extended Response Assessment Chapter Mid-Chapter Test Chapter Quizzes A & B Chapter Quizzes C & D Chapter SAT and ACT Practice Chapter Cumulative Review SAT and ACT Practice Answer Sheet, 0 Questions A SAT and ACT Practice Answer Sheet, 0 Questions A ANSWERS A-A7 Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

4 A Teacher s Guide to Using the Chapter Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter Resource Masters include the core materials needed for Chapter. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorks CD-ROM. Vocabulary Builder Pages vii-x include a student study tool that presents the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. When to Use Give these pages to students before beginning Lesson -. Remind them to add definitions and examples as they complete each lesson. Practice There is one master for each lesson. These problems more closely follow the structure of the Practice section of the Student Edition exercises. These exercises are of average difficulty. When to Use These provide additional practice options or may be used as homework for second day teaching of the lesson. Study Guide There is one Study Guide master for each lesson. When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for those students who have been absent. Enrichment There is one master for each lesson. These activities may extend the concepts in the lesson, offer a historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for use with all levels of students. When to Use These may be used as extra credit, short-term projects, or as activities for days when class periods are shortened. Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

5 Assessment Options The assessment section of the Chapter Resources Masters offers a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Intermediate Assessment A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of free-response questions. Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. Chapter Assessments Chapter Tests Forms A, B, and C Form tests contain multiple-choice questions. Form A is intended for use with honors-level students, Form B is intended for use with averagelevel students, and Form C is intended for use with basic-level students. These tests are similar in format to offer comparable testing situations. Forms A, B, and C Form tests are composed of free-response questions. Form A is intended for use with honors-level students, Form B is intended for use with average-level students, and Form C is intended for use with basic-level students. These tests are similar in format to offer comparable testing situations. All of the above tests include a challenging Bonus question. The Extended Response Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. Continuing Assessment The SAT and ACT Practice offers continuing review of concepts in various formats, which may appear on standardized tests that they may encounter. This practice includes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are provided on the master. The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of advanced mathematics. It can also be used as a test. The master includes free-response questions. Answers Page A is an answer sheet for the SAT and ACT Practice questions that appear in the Student Edition on page 4. Page A is an answer sheet for the SAT and ACT Practice master. These improve students familiarity with the answer formats they may encounter in test taking. The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer keys are provided for the assessment options in this booklet. Glencoe/McGraw-Hill v Advanced Mathematical Concepts

6 Chapter Leveled Worksheets Glencoe s leveled worksheets are helpful for meeting the needs of every student in a variety of ways. These worksheets, many of which are found in the FAST FILE Chapter Resource Masters, are shown in the chart below. Study Guide masters provide worked-out examples as well as practice problems. Each chapter s Vocabulary Builder master provides students the opportunity to write out key concepts and definitions in their own words. Practice masters provide average-level problems for students who are moving at a regular pace. Enrichment masters offer students the opportunity to extend their learning. Five Different Options to Meet the Needs of Every Student in a Variety of Ways primarily skills primarily concepts primarily applications BASIC AVERAGE ADVANCED Study Guide Vocabulary Builder Parent and Student Study Guide (online) 4 Practice Enrichment Glencoe/McGraw-Hill vi Advanced Mathematical Concepts

7 Chapter Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Vocabulary Term ambiguous case Found on Page Definition/Description/Example angle of depression angle of elevation apothem arccosine relation arcsine relation arctangent relation circular function cofunctions cosecant (continued on the next page) Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

8 Chapter Reading to Learn Mathematics Vocabulary Builder (continued) cosine Vocabulary Term Found on Page Definition/Description/Example cotangent coterminal angles degree Hero s Formula hypotenuse initial side inverse Law of Cosines Law of Sines leg (continued on the next page) Glencoe/McGraw-Hill viii Advanced Mathematical Concepts

9 Chapter Reading to Learn Mathematics Vocabulary Builder (continued) minute Vocabulary Term Found on Page Definition/Description/Example quadrant angle reference angle secant second side adjacent side opposite sine solve a triangle standard position tangent (continued on the next page) Glencoe/McGraw-Hill ix Advanced Mathematical Concepts

10 Chapter Reading to Learn Mathematics Vocabulary Builder (continued) terminal side Vocabulary Term Found on Page Definition/Description/Example trigonometric function trigonometric ratio unit circle vertex Glencoe/McGraw-Hill x Advanced Mathematical Concepts

11 - Study Guide Angles and Degree Measure Decimal degree measures can be expressed in degrees( ), minutes( ), and seconds( ). Example a. Change.0 to degrees, minutes, and seconds..0 (0.0 60) Multiply the decimal portion of. the degrees by 60 to find minutes. (0. 60) Multiply the decimal portion of the minutes by 60 to find seconds..0 can be written as. b. Write 4 as a decimal rounded to the nearest thousandth can be written as 4.9. An angle may be generated by the rotation of one ray multiple times about the origin. Example Give the angle measure represented by each rotation. a.. rotations clockwise Clockwise rotations have negative measures. The angle measure of. clockwise rotations is 88. b. 4. rotations counterclockwise Counterclockwise rotations have positive measures. The angle measure of 4. counterclockwise rotations is. If is a nonquadrantal angle in standard position, its reference angle is defined as the acute angle formed by the terminal side of the given angle and the x-axis. Reference Angle Rule For any angle, 0 < < 60, its reference angle is defined by a., when the terminal side is in Quadrant I, b. 80, when the terminal side is in Quadrant II, c. 80, when the terminal side is in Quadrant III, and d. 60, when the terminal side is in Quadrant IV. Example Find the measure of the reference angle for 0. Because 0 is between 80 and 70, the terminal side of the angle is in Quadrant III The reference angle is 40. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

12 - Practice Angles and Degree Measure Change each measure to degrees, minutes, and seconds Write each measure as a decimal degree to the nearest thousandth Give the angle measure represented by each rotation... rotations clockwise 6..6 rotations counterclockwise Identify all angles that are coterminal with each angle. Then find one positive angle and one negative angle that are coterminal with each angle If each angle is in standard position, determine a coterminal angle that is between 0 and 60, and state the quadrant in which the terminal side lies Find the measure of the reference angle for each angle Navigation For an upcoming trip, Jackie plans to sail from Santa Barbara Island, located at 8 N, 9 7 W, to Santa Catalina Island, located at.86 N, 8.40 W. Write the latitude and longitude for Santa Barbara Island as decimals to the nearest thousandth and the latitude and longitude for Santa Catalina Island as degrees, minutes, and seconds. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

13 - Reading Mathematics: If and Only If Statements If p and q are interchanged in a conditional statement so that p becomes the conclusion and q becomes the hypothesis, the new statement, q p, is called the converse of p q. If p q is true, q p may be either true or false. Example Enrichment Find the converse of each conditional. a. p q: All squares are rectangles. (true) q p: All rectangles are squares. (false) b. p q: If a function ƒ(x) is increasing on an interval I, then for every a and b contained in I, ƒ(a) ƒ(b) whenever a b. (true) q p: If for every a and b contained in an interval I, ƒ(a) ƒ(b) whenever a b then function ƒ(x) is increasing on I. (true) In Lesson -, you saw that the two statements in Example can be combined in a single statement using the words if and only if. A function ƒ(x) is increasing on an interval I if and only if for every a and b contained in I, ƒ(a) ƒ(b) whenever a b. The statement p if and only if q means that p implies q and q implies p. State the converse of each conditional. Then tell if the converse is true or false. If it is true, combine the statement and its converse into a single statement using the words if and only if.. All integers are rational numbers.. If for all x in the domain of a function ƒ(x), ƒ(x) ƒ(x), then the graph of ƒ(x) is symmetric with respect to the origin.. If ƒ(x) and ƒ (x) are inverse functions, then [ ƒ ƒ ](x) [ ƒ ƒ](x) x. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

14 - Study Guide Trigonometric Ratios in Right Triangles The ratios of the sides of right triangles can be used to define the trigonometric ratios known as the sine, cosine, and tangent. Example Find the values of the sine, cosine, and tangent for A. First find the length of BC. (AC) (BC) (AB) Pythagorean Theorem 0 (BC) 0 Substitute 0 for AC and 0 for AB. (BC) 00 BC 00 or 0 Take the square root of each side. Disregard the negative root. Then write each trigonometric ratio. sin A s ide opposite cos A si de adjacent side opposite tan A hypotenuse hypotenuse s ide adjacent sin A 0 0 or cos A 0 or 0 tan A 0 or 0 Trigonometric ratios are often simplified but never written as mixed numbers. Three other trigonometric ratios, called cosecant, secant, and cotangent, are reciprocals of sine, cosine, and tangent, respectively. Example Find the values of the six trigonometric ratios for R. First determine the length of the hypotenuse. (RT) (ST) (RS) Pythagorean Theorem (RS) RT, ST (RS) 4 RS 4 or 6 Disregard the negative root. sin R s ide opposite hypotenuse cos R si de adjacent hypotenuse side opposite tan R s ide adjacent sin R or 6 cos R or 6 tan R or hypotenuse csc R s ide opposite hypotenuse sec R si de adjacent cot R s ide adjacent side opposite csc R 6 or 6 sec R 6 or 6 cot R or Glencoe/McGraw-Hill 84 Advanced Mathematical Concepts

15 - Practice Trigonometric Ratios in Right Triangles Find the values of the sine, cosine, and tangent for each B.... If tan, find cot. 4. If sin,find csc. 8 Find the values of the six trigonometric ratios for each S Physics Suppose you are traveling in a car when a beam of light passes from the air to the windshield. The measure of the angle of incidence is, and the measure of the angle of refraction is sin i. Use Snell s Law, s n, to find the index of refraction n in r of the windshield to the nearest thousandth. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

16 - Enrichment Using Right Triangles to Find the Area of Another Triangle You can find the area of a right triangle by using the formula A bh. In the formula, one leg of the right triangle can be used as the base, and the other leg can be used as the height. The vertices of a triangle can be represented on the coordinate plane by three ordered pairs. In order to find the area of a general triangle, you can encase the triangle in a rectangle as shown in the diagram below. A rectangle is placed around the triangle so that the vertices of the triangle all touch the sides of the rectangle. Example Find the area of a triangle whose vertices are A(, ), B(4, 8), and C(8, ). Plot the points and draw the triangle. Encase the triangle in a rectangle whose sides are parallel to the axes, then find the coordinates of the vertices of the rectangle. Area ABC area ADEF area ADB area BEC area CFA, where ADB, BEC, and CFA are all right triangles. Area ABC (9) ()() (4)() ()(9) 7. square units Find the area of the triangle having vertices with each set of coordinates.. A(4, 6), B(, ), C(6, ). A(, 4), B(4, 7), C(6, ). A(4, ), B(6, 9), C(, 4) 4. A(, ), B(6, 8), C(, ) Glencoe/McGraw-Hill 86 Advanced Mathematical Concepts

17 - Study Guide Trigonometric Functions on the Unit Circle Example Use the unit circle to find cot (70 ). The terminal side of a 70 angle in standard position is the positive y-axis, which intersects the unit circle at (0, ). By definition, cot (70 ) y x or 0. Therefore, cot (70 ) 0. Trigonometric sin y r cos x r tan y x Functions of an Angle in csc y r sec x r cot x y Standard Position Example Find the values of the six trigonometric functions for angle in standard position if a point with coordinates (9, ) lies on its terminal side. We know that x 9 and y. We need to find r. r x y Pythagorean Theorem r (9) Substitute 9 for x and for y. r or Disregard the negative root. sin or 4 cos 9 or tan or 4 9 csc or 4 sec or 9 cot 9 or 4 Example Suppose is an angle in standard position whose terminal side lies in Quadrant I. If cos, find the values of the remaining five trigonometric functions of. r x y Pythagorean Theorem y Substitute for r and for x. 6 y 4 y Take the square root of each side. Since the terminal side of lies in Quadrant I, y must be positive. sin 4 tan 4 csc 4 sec cot 4 Glencoe/McGraw-Hill 87 Advanced Mathematical Concepts

18 - Practice Trigonometric Functions on the Unit Circle Use the unit circle to find each value.. csc 90. tan 70. sin (90) Use the unit circle to find the values of the six trigonometric functions for each angle Find the values of the six trigonometric functions for angle in standard position if a point with the given coordinates lies on its terminal side. 6. (, ) 7. (7, 0) 8. (, 4) Glencoe/McGraw-Hill 88 Advanced Mathematical Concepts

19 - Enrichment Areas of Polygons and Circles A regular polygon has sides of equal length and angles of equal measure. A regular polygon can be inscribed in or circumscribed about a circle. For n-sided regular polygons, the following area formulas can be used. Area of circle A C r nr Area of inscribed polygon A I sin Area of circumscribed polygon A C nr tan 60 n 80 n Use a calculator to complete the chart below for a unit circle (a circle of radius ). Number of Sides Area of Area of Circle Area of Area of Polygon Inscribed less Circumscribed less Polygon Area of Polygon Polygon Area of Circle What number do the areas of the circumscribed and inscribed polygons seem to be approaching as the number of sides of the polygon increases? Glencoe/McGraw-Hill 89 Advanced Mathematical Concepts

20 -4 Study Guide Applying Trigonometric Functions Trigonometric functions can be used to solve problems involving right triangles. Example If T 4 and u 0, find t to the nearest tenth. From the figure, we know the measures of an angle and the hypotenuse. We want to know the measure of the side opposite the given angle. The sine function relates the side opposite the angle and the hypotenuse. sin T t side opposite u sin hypotenuse sin 4 Substitute 4 for T and 0 for u. t0 0 sin 4 t Multiply each side by t Use a calculator. Therefore, t is about 4.. Example Geometry The apothem of a regular polygon is the measure of a line segment from the center of the polygon to the midpoint of one of its sides. The apothem of a regular hexagon is.6 centimeters. Find the radius of the circle circumscribed about the hexagon to the nearest tenth. First draw a diagram. Let a be the angle measure formed by a radius and its adjacent apothem. The measure of a is 60 or 0. Now we know the measures of an angle and the side adjacent to the angle. cos 0.6 side adjacent r cos hypotenuse r cos 0.6 Multiply each side by r. r. 6 co Divide each side by cos 0. s 0 r.004 Use a calculator. Therefore, the radius is about.0 centimeters. Glencoe/McGraw-Hill 90 Advanced Mathematical Concepts

21 -4 Practice Applying Trigonometric Functions Solve each problem. Round to the nearest tenth.. If A and c 6, find a.. If a 9 and B 49,find b.. If B 6 48 and c 6., find b. 4. If B 64 and b 9., find a.. If b 4 and A 6,find c. 6. Construction A 0-foot ladder leaning against the side of a house makes a 70 angle with the ground. a. How far up the side of the house does the ladder reach? b. What is the horizontal distance between the bottom of the ladder and the house? 7. Geometry A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters. a. Find the radius of the circumscribed circle. b. What is the length of a side of the hexagon? c. What is the perimeter of the hexagon? 8. Observation A person standing 00 feet from the bottom of a cliff notices a tower on top of the cliff. The angle of elevation to the top of the cliff is 0. The angle of elevation to the top of the tower is 8. How tall is the tower? Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

22 -4 Enrichment Making and Using a Hypsometer A hypsometer is a device that can be used to measure the height of an object. To construct your own hypsometer, you will need a rectangular piece of heavy cardboard that is at least 7 cm by 0 cm, a straw, transparent tape, a string about 0 cm long, and a small weight that can be attached to the string. Mark off -cm increments along one short side and one long side of the cardboard. Tape the straw to the other short side. Then attach the weight to one end of the string, and attach the other end of the string to one corner of the cardboard, as shown in the figure below. The diagram below shows how your hypsometer should look. To use the hypsometer, you will need to measure the distance from the base of the object whose height you are finding to where you stand when you use the hypsometer. Sight the top of the object through the straw. Note where the freehanging string crosses the bottom scale. Then use similar triangles to find the height of the object.. Draw a diagram to illustrate how you can use similar triangles and the hypsometer to find the height of a tall object. Use your hypsometer to find the height of each of the following.. your school s flagpole. a tree on your school s property 4. the highest point on the front wall of your school building. the goal posts on a football field 6. the hoop on a basketball court 7. the top of the highest window of your school building 8. the top of a school bus 9. the top of a set of bleachers at your school 0. the top of a utility pole near your school Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

23 - Study Guide Solving Right Triangles When we know a trigonometric value of an angle but not the value of the angle, we need to use the inverse of the trigonometric function. Trigonometric Function y sin x y cos x y tan x Inverse Trigonometric Relation x sin y or x arcsin y x cos y or x arccos y x tan y or x arctan y Example Solve tan x. If tan x, then x is an angle whose tangent is. x arctan From a table of values, you can determine that x equals 60, 40, or any angle coterminal with these angles. Example If c and b, find B. In this problem, we know the side opposite the angle and the hypotenuse. The sine function relates the side opposite the angle and the hypotenuse. sin B b c sin s ide opposite hypotenuse sin B Substitute for b and for c. B sin Definition of inverse B.07 or about.. Example Solve the triangle where b 0 and c, given the triangle above. a b c a 0 a 8 a B 90 B Therefore, a 8.7, A., and B 4.8. cos A b c cos A 0 A cos - 0 A.0094 Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

24 - Practice Solving Right Triangles Solve each equation if 0x 60.. cos x. tan x. sin x Evaluate each expression. Assume that all angles are in Quadrant I. 4. tan tan. tan cos 6. cos arcsin Solve each problem. Round to the nearest tenth. 7. If q 0 and s, find S. 8. If r and s 4, find R. 9. If q 0 and r, find S. Solve each triangle described, given the triangle at the right. Round to the nearest tenth, if necessary. 0. a 9, B 49. A 6, c 4. a, b 7. Recreation The swimming pool at Perris Hill Plunge is 0 feet long and feet wide. The bottom of the pool is slanted so that the water depth is feet at the shallow end and feet at the deep end. What is the angle of elevation at the bottom of the pool? Glencoe/McGraw-Hill 94 Advanced Mathematical Concepts

25 - Enrichment Disproving Angle Trisection Most geometry texts state that it is impossible to trisect an arbitrary angle using only a compass and straightedge. This fact has been known since ancient times, but since it is usually stated without proof, some geometry students do not believe it. If the students set out to find a method for trisecting angles, they will probably try the following method. It is based on the familiar construction which allows a segment to be divided into any desired number of congruent segments. You can use inverse trigonometric functions to show that application of the method to the trisection of angles is not valid. Given: A Claim: A can be trisected using the following method. Method: Choose point C on one ray of A. Through C construct a perpendicular to the other ray, intersecting it at B. Construct M and N, the points that divide CB into three congruent segments. Draw AM and AN, which trisect CAB into the congruent angles,, and. The proposed method has been used to construct the figure below. CM MN NB. AB. Follow the instructions to show that the three angles,, and, are not congruent. Find angle measures to the nearest tenth of a degree.. Express m as the value of an inverse function.. Find the measure of.. Write mmab as the value of an inverse function. 4. Find the measure of MAB.. Find the measure of. 6. Find mcab and use it to find m. 7. Explain why the proposed method for trisecting an angle fails. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

26 -6 The Law of Sines Study Guide Given the measures of two angles and one side of a triangle, we can use the Law of Sines to find one unique solution for the triangle. Law of Sines a b c sin A sin B sin C Example Solve ABC if A 0, B 00, and a. First find the measure of C. C 80 (0 00 ) or 0 Use the Law of Sines to find b and c. a sin b c A sin B sin a C sin A sin b c 0 sin 00 sin 0 sin 0 sin 00 b c sin 0 sin 0 sin b c.989 Therefore, C 0, b 9., and c.0. The area of any triangle can be expressed in terms of two sides of a triangle and the measure of the included angle. Area (K) of a Triangle K bc sin A K ac sin B K ab sin C Example Find the area of ABC if a 6.8, b 9., and C 7. K ab sin C K (6.8)(9.) sin 7 K The area of ABC is about 6. square units. Glencoe/McGraw-Hill 96 Advanced Mathematical Concepts

27 -6 Practice The Law of Sines Solve each triangle. Round to the nearest tenth.. A 8, B 6, c. A, B 9, b 4. A 0, C 0, a A 0, B 4, a 0 Find the area of each triangle. Round to the nearest tenth.. c 4, A 7, B C 8, a, B 9 7. A 0, b, c 4 8. b 4, C 0, B 9. b, c 0, A 0. a 68, c 0, B 4.. Street Lighting A lamppost tilts toward the sun at a angle from the vertical and casts a -foot shadow. The angle from the tip of the shadow to the top of the lamppost is 4.Find the length of the lamppost. Glencoe/McGraw-Hill 97 Advanced Mathematical Concepts

28 -6 Enrichment Triangle Challenge A surveyor took the following measurements from two irregularlyshaped pieces of land. Some of the lengths and angle measurements are missing. Find all missing lengths and angle measurements. Round lengths to the nearest tenth and angle measurements to the nearest minute.. e G a H. f c e b J Glencoe/McGraw-Hill 98 Advanced Mathematical Concepts

29 -7 Study Guide The Ambiguous Case for the Law of Sines If we know the measures of two sides and a nonincluded angle of a triangle, three situations are possible: no triangle exists, exactly one triangle exists, or two triangles exist. A triangle with two solutions is called the ambiguous case. Example Find all solutions for the triangle if a 0, b 0, and A 40. If no solutions exist, write none. Since 40 90, consider Case. b sin A 0 sin 40 b sin A Since , there are two solutions for the triangle. Use the Law of Sines to find B. 0 sin 0 40 si n B sin B 0 s in 40 0 B sin 0 s in 40 0 B a sin b A sin B So, B Since we know there are two solutions, there must be another possible measurement for B. In the second case, B must be less than 80 and have the same sine value. Since we know that if 90, sin sin (80 ), or 0.4 is another possible measure for B. Now solve the triangle for each possible measure of B. Case : A 90 for a, b, and A a b sin A no solution a b sin A one solution a b one solution b sin A a b two solutions Case : A 90 a b no solution a b one solution Solution I C 80 ( ) or 6.4 a sin c A sin C 0 sin c 40 sin 6.4 c 0 sin 6.4 sin 40 c One solution is B 74.6, C 6.4, and c 8.. Solution II C 80 ( ) or 4.6 a sin c A sin C 0 sin c 40 sin 4.6 c 0 sin 4.6 sin 40 c Another solution is B 0.4, C 4.6, and c 7.7. Glencoe/McGraw-Hill 99 Advanced Mathematical Concepts

30 -7 Practice The Ambiguous Case for the Law of Sines Determine the number of possible solutions for each triangle.. A 4, a, b. a, b, A 8. A 8, a 4., b 4. A 0, a 4, c 4 Find all solutions for each triangle. If no solutions exist, write none. Round to the nearest tenth.. b 0, a, A 6. a, A, b 0 7. a, c 0, A 8. a, b, A 9. A 4, a, b 0. b, c, C 0. Property Maintenance The McDougalls plan to fence a triangular parcel of their land. One side of the property is 7 feet in length. It forms a 8 angle with another side of the property, which has not yet been measured. The remaining side of the property is 9 feet in length. Approximate to the nearest tenth the length of fence needed to enclose this parcel of the McDougalls lot. Glencoe/McGraw-Hill 00 Advanced Mathematical Concepts

31 -7 Enrichment Spherical Triangles Spherical trigonometry is an extension of plane trigonometry. Figures are drawn on the surface of a sphere. Arcs of great circles correspond to line segments in the plane. The arcs of three great circles intersecting on a sphere form a spherical triangle. Angles have the same measure as the tangent lines drawn to each great circle at the vertex. Since the sides are arcs, they too can be measured in degrees. The sum of the sides of a spherical triangle is less than 60. The sum of the angles is greater than 80 and less than 40. The Law of Sines for spherical triangles is as follows. sin a sin A sin b sin B sin c sin C There is also a Law of Cosines for spherical triangles. cos a cos b cos c sin b sin c cos A cos b cos a cos c sin a sin c cos B cos c cos a cos b sin a sin b cos C Example Solve the spherical triangle given a 7, b 0, and c 6. Use the Law of Cosines ( 0.88)(0.4848) (0.969)(0.8746) cos A cos A 0.4 A (0.090)(0.4848) (0.9)(0.8746) cos B cos B 0.49 B (0.090)( 0.88) (0.9)(0.969) cos C cos C C Check by using the Law of Sines. sin 7 sin 9 sin 0 sin 9 sin 6 sin. Solve each spherical triangle.. a 6, b, c 94. a 0, b, c 97. a 76, b 0, C b 94, c, A 48 Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

32 -8 The Law of Cosines Study Guide When we know the measures of two sides of a triangle and the included angle, we can use the Law of Cosines to find the measure of the third side. Often times we will use both the Law of Cosines and the Law of Sines to solve a triangle. Law of Cosines a b c bc cos A b a c ac cos B c a b ab cos C Example Solve ABC if B 40, a, and c 6. b a c ac cos B Law of Cosines b 6 ()(6) cos 40 b b So, b 8.. b sin c B sin Law of Sines C 8. sin 6 40 sin C sin C 6si n C sin 6si n C So, C 7.7. A 80 ( ). The solution of this triangle is b 8., A., and C 7.7. Example Find the area of ABC if a, b 8, and c 0. First, find the semiperimeter of ABC. s (a b c) s ( 8 0) s. Now, apply Hero s Formula k s( s a)( s b)( s c) k.(. )(. 8)(. 0) k 9.47 k The area of the triangle is about 9.8 square units. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

33 -8 Practice The Law of Cosines Solve each triangle. Round to the nearest tenth.. a 0, b, c 8. a 0, c 8, B 00. c 49, b 40, A 4. a, b 7, c 0 Find the area of each triangle. Round to the nearest tenth.. a, b, c 6. a, b, c 6 7. a 4, b 9, c 8 8. a 8, b 7, c 9. The sides of a triangle measure.4 centimeters, 8.7 centimeters, and 6. centimeters. Find the measure of the angle with the least measure. 0. Orienteering During an orienteering hike, two hikers start at point A and head in a direction 0 west of south to point B. They hike 6 miles from point A to point B.From point B, they hike to point C and then from point C back to point A, which is 8 miles directly north of point C. How many miles did they hike from point B to point C? Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

34 -8 Enrichment The Law of Cosines and the Pythagorean Theorem The law of cosines bears strong similarities to the Pythagorean Theorem. According to the Law of Cosines, if two sides of a triangle have lengths a and b and if the angle between them has a measure of x, then the length, y, of the third side of the triangle can be found by using the equation y a b ab cos (x ). Answer the following questions to clarify the relationship between the Law of Cosines and the Pythagorean Theorem.. If the value of x becomes less and less, what number is cos (x ) close to?. If the value of x is very close to zero but then increases, what happens to cos (x ) as x approaches 90?. If x equals 90, what is the value of cos (x )? What does the equation of y a b ab cos (x ) simplify to if x equals 90? 4. What happens to the value of cos (x ) as x increases beyond 90 and approaches 80?. Consider some particular values of a and b,say 7 for a and 9 for b. Use a graphing calculator to graph the equation you get by solving y 7 9 (7)(9) cos (x ) for y. a. In view of the geometry of the situation, what range of values should you use for X on a graphing calculator? b. Display the graph and use the TRACE function. What do the maximum and minimum values appear to be for the function? c. How do the answers for part b relate to the lengths 7 and 9? Are the maximum and minimum values from part b ever actually attained in the geometric situation? Glencoe/McGraw-Hill 04 Advanced Mathematical Concepts

35 Chapter Chapter Test, Form A Write the letter for the correct answer in the blank at the right of each problem.. Change 8.4 to degrees, minutes, and seconds.. A. 8 8 B. 8 9 C. 8 9 D Write 4 8 as a decimal to the nearest thousandth of a degree.. A B C D Give the angle measure represented by. rotations clockwise.. A. 70 B. 90 C. 90 D Identify all coterminal angles between 60 and for the angle 40. A. 60 and 00 B. 0 and 0 C. 0 and 0 D. 60 and 00. Find the measure of the reference angle for A. 6 B. 6 C. 4 D Find the value of the tangent for A. 6. A. B. C. D. 7. Find the value of the secant for R. 7. A. 70 B. 4 4 C. D Which of the following is equal to csc θ? 8. A. sin B. θ co C. s θ ta D. n θ se c θ 9. If cot θ 0.8, find tan θ. 9. A B. 0.8 C..76 D Find cos (70 ). 0. A. undefined B. C. D. 0. Find the exact value of sec 00.. A. B. C. D.. Find the value of csc θ for angle θ in standard position if. the point at (, ) lies on its terminal side. A. 9 B. 9 C. 9 9 D Suppose θ is an angle in standard position whose terminal side. lies in Quadrant II. If sin θ,find the value of sec θ. A. B. C. D. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

36 Chapter Chapter Test, Form A (continued) For Exercises 4 and, refer to the figure. The angle of elevation from the end of the shadow to the top of the building is 6 and the distance is 0 feet. 4. Find the height of the building to the nearest foot. 4. A. 00 ft B. 96 ft C. 4 ft D. ft. Find the length of the shadow to the nearest foot.. A. 00 ft B. 96 ft C. 4 ft D. ft 6. If 0 x 60, solve the equation sec x. 6. A. 0 and 0 B. 0 and 0 C. 0 and 40 D. 40 and Assuming an angle in Quadrant I, evaluate csc cot A. B. C. 4 D Given the triangle at the right, find B to the 8. nearest tenth of a degree if b 0 and c 4. A B.. C. 4. D. 4.6 For Exercises 9 and 0, round answers to the nearest tenth. 9. In ABC, A 7, B 78, and c 9. Find a. 9. A. 8.6 B. 9. C..8 D If A 4., B.6, and a 4., find the area of ABC. 0. A. 8.8 units B units C. 7.4 units D units. Determine the number of possible solutions if A 6, a 4,. and b 6. A. none B. one C. two D. three. Determine the greatest possible value for B if A 0, a,. and b 8. A.. B.. C. 6.9 D For Exercises -, round answers to the nearest tenth.. In ABC, A 47, b, and c 8. Find a.. A. 6. B. 8.7 C. 8.8 D In ABC, a 7.8, b 4., and c.9. Find B. 4. A.. B C D. 6.. If a, b 4, and c 0, find the area of ABC.. A. units B..0 units C. 0. units D. 4.8 units Bonus The terminal side of an angle θ in standard position Bonus: coincides with the line 4x y 0 in Quadrant II. Find sec θ to the nearest thousandth. A. 0.4 B. 4. C. 0.4 D. 4. Glencoe/McGraw-Hill 06 Advanced Mathematical Concepts

37 Chapter Chapter Test, Form B Write the letter for the correct answer in the blank at the right of each problem.. Change 0. to degrees, minutes, and seconds.. A B. 0 8 C. 0 8 D Write as a decimal to the nearest thousandth of a degree.. A B C D Give the angle measure represented by.7 rotations counterclockwise.. A. 60 B. 90 C. 90 D Identify the coterminal angle between 60 and 60 for the angle A. 40 B. 60 C. 40 D. 00. Find the measure of the reference angle for 9.. A. B. 6 C. D Find the value of the cosine for A. 6. A. B. C. D. 7. Find the value of the cosecant for R. 7. A. 6 B. 6 C. D Which of the following is equal to sec? 8. A. sin B. co C. s ta D. n se c 9. If tan 0., find cot. 9. A. 0. B. 4 C. 0. D Find cot (80 ). 0. A. undefined B. C. D. 0. Find the exact value of tan 40.. A. B. C. D.. Find the value of sec for angle in standard position if the. point at (, 4) lies on its terminal side. A. B. C. D.. Suppose is an angle in standard position whose terminal side. lies in Quadrant III. If sin, find the value of cot. A. B. C. D. Glencoe/McGraw-Hill 07 Advanced Mathematical Concepts

38 Chapter Chapter Test, Form B (continued) For Exercises 4 and, refer to the figure. The angle of elevation from the end of the shadow to the top of the building is 6 and the distance is 0 feet. 4. Find the height of the building to the nearest foot. 4. A. 99 ft B. 67 ft C. 78 ft D. 8 ft. Find the length of the shadow to the nearest foot.. A. 99 ft B. 67 ft C. 78 ft D. 8 ft 6. If 0 x 60, solve the equation tan x. 6. A. and B. 4 and C. 4 and D. and 7. Assuming an angle in Quadrant I, evaluate tan cos A. 4 B. C. 4 D Given the triangle at the right, find A to the nearest 8. tenth of a degree if b 0 and c 4. A B.. C. 4. D. 4.6 For Exercises 9 and 0, round answers to the nearest tenth. 9. In ABC, A 4, B 07 9, and c 9. Find b. 9. A. 0.0 B. 4. C.. D If A.6, B 49.8, and a.8, find the area of ABC. 0. A. 7.9 units B. 8. units C. 6.4 units D units. Determine the number of possible solutions if A 6, a 7, and b 6.. A. none B. one C. two D. three For Exercises, round answers to the nearest tenth.. Determine the least possible value for B if A 0, a,. and b 8. A.. B.. C. 6.9 D In ABC, B, a 4, and c 9. Find b.. A. 8. B..0 C.. D In ABC, a 7.8, b 4., and c.9. Find A. 4. A.. B C D. 6.. If a, b 6, and c 40, find the area of ABC.. A. 49 units B..0 units C units D. 4. units Bonus The terminal side of an angle in standard position Bonus: coincides with the line x y 0 in Quadrant III. Find cos to the nearest ten-thousandth. A B C D Glencoe/McGraw-Hill 08 Advanced Mathematical Concepts

39 Chapter Chapter Test, Form C Write the letter for the correct answer in the blank at the right of each problem.. Change 6. to degrees, minutes, and seconds.. A B C D Write 44 as a decimal to the nearest thousandth of a degree.. A..7 B..7 C..7 D..74. Give the angle measure represented by 0. rotation clockwise.. A. 80 B. 90 C. 90 D Identify the coterminal angle between 0 and 60 for the angle A. 0 B. 60 C. 0 D. 40. Find the measure of the reference angle for.. A. B. C. D. 6. Find the value of the sine for A. 6. A. 9 B. 9 C. D. 7. Find the value of the cotangent for R. 7. A. 4 B. 4 C. 4 D Which of the following is equal to cot? 8. A. sin B. co C. s se D. c tan 9. If cos 0., find sec. 9. A. 0. B. 0. C. D. 0. Find tan A. undefined B. C. D. 0. Find the exact value of cos.. A. B. C. D.. Find the value of csc for angle in standard position if the point at. (, ) lies on its terminal side. A. 0 B. 0 0 C. 0 D.. Suppose is an angle in standard position whose terminal side lies.,find the value of tan. A. B. C. D. Glencoe/McGraw-Hill 09 Advanced Mathematical Concepts

40 Chapter Chapter Test, Form C (continued) For Exercises 4 and, refer to the figure. The angle of elevation from the end of the shadow to the top of the building is 70 and the distance is 80 feet. 4. Find the height of the building to the nearest foot. 4. A. 6 ft B. 66 ft C. 69 ft D. 49 ft. Find the length of the shadow to the nearest foot.. A. 6 ft B. 66 ft C. 69 ft D. 49 ft 6. If 0 x 60, solve the equation sin x. 6. A. 0 and 40 B. 40 and 00 C. 0 and 0 D. 0 and 0 7. Assuming an angle in Quadrant I, evaluate cos tan A. B. C. 4 D Given the triangle at the right, find B to the 8. nearest tenth of a degree if b 8 and c. A..7 B. 4.8 C. 48. D. 6. For Exercises 9 and 0, round answers to the nearest tenth. 9. In ABC, A 0, B, and c 9.8. Find a. 9. A. 8.0 B..8 C D If A., b., and c., find the area of ABC. 0. A. 9. units B. 0.8 units C units D..6 units. Determine the number of possible solutions if A 48, a, and b 6.. A. none B. one C. two D. three. Determine the least possible value for B if A 0,. a 7, and b. A..6 B.. C. 47. D For Exercises, round answers to the nearest tenth.. In ABC, A, b 9, and c 4. Find a.. A. 6. B. 8.7 C. 8.8 D In ABC, a.4, b 8., and c 0.. Find B. 4. A.. B..7 C. 8.9 D... If a, b 0, and c, find the area of ABC.. A..7 units B. 9.4 units C.. units D. 4.8 units Bonus The terminal side of an angle in standard position Bonus: coincides with the line y x in Quadrant I. Find sin to the nearest thousandth. A. 0. B C D Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

41 Chapter Chapter Test, Form A. Change.69 to degrees, minutes, and seconds... Write 6 as a decimal to the nearest thousandth of a degree... State the angle measure represented by.4 rotations clockwise.. 4. Identify all coterminal angles between 60 and 60 for the 4. angle 40.. Find the measure of the reference angle for 6.. For Exercises 6 8, refer to the figure. 6. Find the value of the sine for A Find the value of the cotangent for A Find the value of the secant for A. 8. Exercises If csc, find sin Find sin (70). 0.. Find the exact value of cot 0... Find the exact value of sec for angle in standard position if. the point at (, ) lies on its terminal side.. Suppose is an angle in standard position whose terminal side. lies in Quadrant IV. If cos,find the value of csc. Glencoe/McGraw-Hill Advanced Mathematical Concepts

42 Chapter Chapter Test, Form A (continued) For Exercises 4 and, refer to the figure. The angle of elevation from the far side of the pool to the top of the waterfall is 7, and the distance is 8 feet. 4. Find the height of the waterfall 4. to the nearest foot.. Find the width across the pool. to the nearest foot. 6. If 0 x 60, solve cot x Assuming an angle in Quadrant I, evaluate sec tan Given triangle at the right, 8. find B to the nearest tenth of a degree if a 8 and b 0. For Exercises 9 and 0, round answers to the nearest tenth. 9. In ABC, A 47, B 8, and c. Find a If A 7., B 7.9, and a., find the area of ABC. 0.. Determine the number of possible solutions if A 47,. a 4, and b.. Determine the least possible value for c if A 0,. a, and b 8. For Exercises -, round answers to the nearest tenth.. In ABC, A 8, b 8, and c 6. Find a.. 4. In ABC, a 9, b, and c. Find B. 4.. If a, b 4, and c 0, find the area of ABC.. Bonus The terminal side of an angle in standard Bonus: position coincides with the line x y 0 in Quadrant II. Find csc to the nearest thousandth. Glencoe/McGraw-Hill Advanced Mathematical Concepts

43 Chapter Chapter Test, Form B. Change 4.6 to degrees, minutes, and seconds... Write 48 as a decimal to the nearest thousandth of. a degree.. State the angle measure represented by. rotations. clockwise. 4. Identify all coterminal angles between 60 and 60 for 4. the angle 60.. Find the measure of the reference angle for 0.. For Exercises 6 8, refer to the figure. 6. Find the value of the cosine for A Find the value of the cosecant for A Find the value of the cotangent for A. 8. Exercises If sec 4, find cos Find tan (80 ). 0.. Find the exact value of sec Find the exact value of sec for angle in standard. position if the point at (4, ) lies on its terminal side.. Suppose is an angle in standard position whose terminal. side lies in Quadrant IV. If cos,find the value of cot. Glencoe/McGraw-Hill Advanced Mathematical Concepts

44 Chapter Chapter Test, Form B (continued) For Exercises 4 and, refer to the figure. The angle of elevation from the far side of the pool to the top of the waterfall is 4, and the distance is 0 feet. 4. Find the height of the waterfall 4. to the nearest foot.. Find the width across the pool. to the nearest foot. 6. If 0 x 60, solve the equation csc x Assuming an angle in Quadrant I, evaluate cos cot Given the triangle at the right, 8. find B to the nearest tenth of a degree if a and c. For Exercises 9 and 0, round answers to the nearest tenth. 9. In ABC, A 4, B 68, and c. Find a If A 7., B 67.4, and a.8, find the area of ABC. 0.. Determine the number of possible solutions if A 0,. a, and b 4. For Exercises, round answers to the nearest tenth.. Determine the greatest possible value for c if A 0,. a, and b 8.. In ABC, A 9, b, and c 4. Find a.. 4. In ABC, a 4, b, and c 8. Find B. 4.. If a, b, and c 8, find the area of ABC.. Bonus The terminal side of an angle in standard Bonus: position coincides with the line x y 0 in Quadrant III. Find sin to the nearest ten-thousandth. Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts

45 Chapter Chapter Test, Form C. Change.6 to degrees, minutes, and seconds... Write 7 0 as a decimal to the nearest thousandth of a. degree.. State the angle measure represented by. rotations. counterclockwise. 4. Identify a coterminal angle between 0 and 60 for the 4. angle.. Find the measure of the reference angle for.. For Excercises 6 8, refer to the figure. 6. Find the value of the sine for A Find the value of the cotangent for A Find the value of the secant for A. 8. Exercises If tan, find cot Find tan Find the exact value of cos 0... Find the exact value of sin for angle in standard position. if the point at (, 4) lies on its terminal side.. Suppose is an angle in standard position whose. terminal side lies in Quadrant II. If sin,find the value of sec. Glencoe/McGraw-Hill Advanced Mathematical Concepts

46 Chapter Chapter Test, Form C (continued) For Exercises 4 and, refer to the figure. The angle of elevation from the far side of the pool to the top of the waterfall is 68 and the distance is 00 feet. 4. Find the height of the waterfall 4. to the nearest foot.. Find the width across the pool. to the nearest foot. 6. If 0 x 60, solve sin x Assuming an angle in Quadrant I, evaluate cos tan Given the triangle at the right, 8. find B to the nearest tenth of a degree if b and c 8. For Exercises 9 and 0, round answers to the nearest tenth. 9. In ABC, A 47, B 8, and b. Find a If C 7., a 7.9, and b., find the area of ABC. 0.. Determine the number of possible solutions if A 47,. a, and b 4.. Determine the greatest possible value for c if A,. a 8, and b. For Exercises, round answers to the nearest tenth.. In ABC, A 67, b 0, and c. Find a.. 4. In ABC, a 8, b 6, and c. Find C. 4.. If a 8, b, and c 0, find the area of ABC.. Bonus The terminal side of an angle in standard Bonus: position coincides with the line y x in Quadrant I. Find tan to the nearest thousandth. Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts

47 Chapter Chapter Open-Ended Assessment Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.. The point at (, ) lies on the terminal side of an angle in standard position. a. Give the degree measure of three angles that fit the description. b. Tell how to find the cosine of such angles. Give the cosine of these angles. c. Name angles in the first, second, and fourth quadrants that have the same reference angle as those above. d. Write the coordinates of a point in Quadrant II. Find the values of the six trigonometric functions of an angle in standard position with this point on its terminal side.. A children s play area is being built next to a circular fountain in the park. A fence will be erected around the play area for safety. A diagram of the area is shown below. a. How long will the fence need to be in order to enclose the area? b. The park commission is planning to enlarge the play area. Do you think it should be enlarged to the east or to the west? Why? Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts