47 a2, 2A2A; P3A TEKS Apply Doble-Angle and Half-Angle Formlas Before Yo evalated expressions sing sm and difference formlas Now Yo will se doble-angle and half-angle formlas Why? So yo can find the distance an object travels, as in Example 4 Key Vocablary sine, p 852 cosine, p 852 tangent, p 852 In this lesson, yo will se formlas for doble angles (angles of measre 2a) and half angles angles of measre a 2 KEY CONCEPT For Yor Notebook Doble-Angle and Half-Angle Formlas Doble-Angle Formlas sin 2a 5 2 sin a cos a cos 2a 5 2 2 sin 2 a tan 2a 5 2 tan a 2 tan 2 a cos 2a 5 2 cos 2 a 2 cos 2a 5 cos 2 a 2 sin 2 a Half-Angle Formlas sin a 56Î 2 cos a cos a 56Î cos a tan a 5 2 cos a sin a The signs of sin a and cos a depend on the tan a 5 sin a cos a qadrant in which a lies E XAMPLE Evalate trigonometric expressions Find the exact vale of (a) cos 658 and (b) tan p 2 a cos 658 5 cos (3308) b tan p 2 5 tan p 6 2 CHOOSE SIGNS Becase 658 is in Qadrant II and the vale of cosine is negative in Qadrant II, the following formla is sed: cos a 52Î cos a 52Î 2 cos p cos 3308 5 6 sin p 6 52Î Ï 3 2 Ï 3 5 2 2 52 Ï Ï 3 5 2 2 Ï 3 47 Apply Doble-Angle and Half-Angle Formlas 955
E XAMPLE 2 Evalate trigonometric expressions Given cos a 5 5 3 with 3p < a < 2p, find (a) sin 2a and (b) sin a Soltion MULTIPLY AN INEQUALITY In part (b), yo can mltiply throgh the ineqality 3p < a < 2π 2 by to get 3p < a < π 2 4 2 So, a is in Qadrant II a Using a Pythagorean identity gives sin a 52 2 3 sin 2a 5 2 sin a cos a 5 2 2 2 3 2 5 3 2 5220 69 b Becase a is in Qadrant II, sin a is positive 2 2 sin a 5 2 Î 2 2 cos a 5 2 Î 5 3 5 2 Î 4 3 5 2Ï 3 3 E XAMPLE 3 TAKS PRACTICE: Mltiple Choice Which expression is eqivalent to 2 cos 2 sin 2? A cos B tan C cot D sin Soltion 2 cos 2 5 2 ( 2 2 sin2 ) Use doble-angle formlas sin 2 2 sin cos 5 2 sin2 sin cos Simplify nmerator 5 sin cos Divide ot common factor 2 sin 5 tan Use tangent identity c The correct answer is B A B C D GUIDED PRACTICE for Examples, 2, and 3 Find the exact vale of the expression tan p 8 2 sin 5p 8 3 cos 58 4 Given sin a 5 Ï with 0 < a < p, find cos 2a and tan a 5 Given cos a 52 3 5 with π < a < 3p, find sin 2a and sin a Simplify the expression 6 cos 2 sin cos 7 tan 2x tan x 8 sin 2x tan x 956 Chapter 4 Trigonometric Graphs, Identities, and Eqations
PATH OF A PROJECTILE The path traveled by an object that is projected at an initial height of h 0 feet, an initial speed of v feet per second, and an initial angle θ is given by y 52 6 v 2 cos 2 x2 (tan )x h 0 where x is the horizontal distance (in feet) and y is the vertical distance (in feet) (This model neglects air resistance) E XAMPLE 4 Derive a trigonometric model SOCCER Write an eqation for the horizontal distance traveled by a soccer ball kicked from grond level (h 0 5 0) at speed v and angle Soltion USE ZERO PRODUCT PROPERTY One soltion of this eqation is x 5 0, which corresponds to the point where the ball leaves the grond This soltion is ignored in later steps, becase the problem reqires finding where the ball lands 2 6 v 2 cos 2 x2 (tan )x 0 5 0 Let h 0 5 0 2x 6 v 2 cos 2 x 2 tan 2 5 0 Factor 6 x 2 tan 5 0 v 2 cos 2 Zero prodct property 6 x 5 tan v 2 cos 2 Add tan to each side x 5 6 v 2 cos 2 tan Mltiply each side by 6 v 2 cos 2 x 5 6 v 2 cos sin Use cos tan 5 sin x 5 32 v 2 (2 cos sin ) Rewrite 6 as 32 p 2 x 5 32 v 2 sin 2 Use a doble-angle formla GUIDED PRACTICE for Example 4 9 WHAT IF? Sppose yo kick a soccer ball from grond level with an initial speed of 70 feet per second Can yo make the ball travel 200 feet? 0 REASONING Use the eqation x 5 v 2 sin 2 to explain why the projection 32 angle that maximizes the distance a soccer ball travels is 5 458 47 Apply Doble-Angle and Half-Angle Formlas 957
E XAMPLE 5 Verify a trigonometric identity Verify the identity cos 3x 5 4 cos 3 x 2 3 cos x cos 3x 5 cos (2x x) Rewrite cos 3x as cos (2x x) 5 cos 2x cos x 2 sin 2x sin x Use a sm formla 5 (2 cos 2 x 2 ) cos x 2 (2 sinx cos x) sin x Use doble-angle formlas 5 2 cos 3 x 2 cos x 2 2 sin 2 x cos x Mltiply 5 2 cos 3 x 2 cos x 2 2( 2 cos 2 x) cos x Use a Pythagorean identity 5 2 cos 3 x 2 cos x 2 2 cos x 2 cos 3 x Distribtive property 5 4 cos 3 x 2 3 cos x Combine like terms E XAMPLE 6 Solve a trigonometric eqation Solve sin 2x 2 cos x 5 0 for 0 x < 2p Soltion sin 2x 2 cos x 5 0 Write original eqation 2 sin x cos x 2 cos x 5 0 Use a doble-angle formla 2cosx (sin x ) 5 0 Factor Set each factor eqal to 0 and solve for x 2cosx 5 0 sin x 5 0 cos x 5 0 sin x 52 x 5 p, 3p x 5 3p CHECK Graph the fnction y 5 sin 2x 2 cos x on a graphing calclator Then se the zero featre to find the x-vales on the interval 0 x < 2π for which y 5 0 The two x-vales are: x 5 p ø 57 and x 5 3p ø 47 Zero X=570796 Y=0 E XAMPLE 7 Find a general soltion Find the general soltion of 2 sin x 5 SOLVE EQUATIONS As seen in Example 7, some eqations that involve doble or half angles can be solved withot resorting to doble- or half-angle formlas 2 sin x 5 Write original eqation sin x 5 Divide each side by 2 x 5 p 6 2nπ or x 5 p 3 4nπ or 5p 6 2nπ General soltion for x 5p 3 4nπ General soltion for x 958 Chapter 4 Trigonometric Graphs, Identities, and Eqations
GUIDED PRACTICE for Examples 5, 6, and 7 Verify the identity sin 3x 5 3 sin x 2 4 sin 3 x 2 cos 0x 5 2 cos 2 5x Solve the eqation 3 tan 2x tan x 5 0 for 0 x < 2π 4 2 cos x 5 0 2 47 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p WS for Exs 7, 3, and 53 5 TAKS PRACTICE AND REASONING Exs, 27, 54, 55, 57, and 58 VOCABULARY Copy and complete: sin 2a 5 2 sin a cos a is called the? formla for sine 2 WRITING Explain how to determine the sign of the answer when evalating a half-angle formla for sine or cosine EXAMPLE on p 955 for Exs 3 EVALUATING EXPRESSIONS Find the exact vale of the expression 3 sin 058 4 tan 258 5 tan (2658) 6 cos (2758) 7 cos p 8 8 sin 5p 2 9 tan 25p 8 2 0 sin 2p 2 2 TAKS REASONING What is the exact vale of tan 58? A 2Ï 3 B 22 Ï 3 C Ï 3 D 2 Ï 3 EXAMPLE 2 on p 956 for Exs 2 20 HALF-ANGLE FORMULAS Find the exact vales of sin a, cos a, and tan a 2 cos a 5 4 5, 0 < a < p 3 cos a 5 3, 3p < a < 2π 4 sin a 5 2 3, p < a < π 5 sin a 52 3 5, π < a < 3p 6 ERROR ANALYSIS Describe and correct the error in finding the exact vale of sin a given that cos a 52 3 5 with p < a < π sin a 2 52Î 2 cos a 52Î 3 5 52Î 4 5 52 2Ï 5 5 DOUBLE-ANGLE FORMULAS Find the exact vales of sin 2a, cos 2a, and tan 2a 7 tan a 5 2, π < a < 3p 8 tan a 52Ï 3, p < a < π 9 sin a 52 2 3, π < a < 3p 20 cos a 5 2 5, 2 p < a < 0 47 Apply Doble-Angle and Half-Angle Formlas 959
EXAMPLE 3 on p 956 for Exs 2 29 SIMPLIFYING EXPRESSIONS Rewrite the expression withot doble angles or half angles, given that 0 < < p Then simplify the expression 2 cos 2 2 2 sin 2 22 sin 2 cos 23 ( 2 tan ) tan 2 24 cos 2 sin 2 cos 25 2tan csc 26 2 sin cos 27 TAKS REASONING Which expression is eqivalent to cot tan? A csc 2 B 2 csc 2 C sec 2 D 2 sec 2 ERROR ANALYSIS Describe and correct the error in simplifying the expression 28 cos 2x cos 2 x 5 cos2 x 2 sin 2 x cos 2 x 29 sin 2258 5 sin (458) 5 cos 2 x 5 sec 2 x 5 2 sin 458 cos 458 5 2 Ï 2 Ï 2 5 EXAMPLE 5 on p 958 for Exs 30 35 VERIFYING IDENTITIES Verify the identity 30 2 cos 2 5 cos 2 3 sin 3 5 sin (4 cos 2 2 ) 32 sin x 5 sin x cos x 33 2 sin 2 x tan x 5 2 sin x 2 sin 2x 2 3 3 3 2 cos 2 34 2 5 2 sin 2 csc 35 cos 4 5 cos 4 2 6 sin 2 cos 2 sin 4 sin EXAMPLE 6 on p 958 for Exs 36 4 SOLVING EQUATIONS Solve the eqation for 0 x < 2p 36 sin x 5 37 2 cos x 5 0 38 tan x 2 tan 2x 5 0 39 tan x 5 2 2 Ï sin x 40 cos 2x 522 cos 2 x 4 2 sin 2x sin x 5 3 cos x EXAMPLE 7 on p 958 for Exs 42 47 FINDING GENERAL SOLUTIONS Find the general soltion of the eqation 42 cos x 5 43 tan x 5 sin x 44 sin 2x 5 sin x 45 cos 2x cos x 5 0 46 cos x sin x 5 0 47 sin x cos x 5 0 48 REASONING Show that the three doble-angle formlas for cosine are eqivalent 49 CHALLENGE Use the diagram shown at the right to derive the formlas for sin, cos, and tan when is an acte angle 2 cos sin 960 5 WORKED-OUT SOLUTIONS 5 TAKS PRACTICE AND REASONING
PROBLEM SOLVING EXAMPLE 4 on p 957 for Exs 50 5 50 GOLF Use the eqation x 5 32 v 2 sin 2 from Example 4 on page 957 to find the horizontal distance a golf ball will travel if it is hit at an initial speed of 50 feet per second and at an initial angle of 408 5 SOCCER Sppose yo are attempting to kick a soccer ball from grond level Throgh what range of angles can yo kick the soccer ball with an initial speed of 80 feet per second to make it travel at least 50 feet? at classzonecom 52 MULTI-STEP PROBLEM At latitde L, the acceleration de to gravity g (in centimeters per second sqared) at sea level can be approximated by: g 5 978 57 sin 2 L 2 004 sin L cos L a Simplify the eqation above to show that g 5 978 57 sin 2 L 2 0007 sin 2L b Graph the fnction from part (a) c Use the graph to approximate the acceleration de to gravity when the latitde is 458, 308, and 08 N Eqator S 45 latitde 30 latitde 0 latitde 53 MACH NUMBER An airplane s Mach nmber M is the ratio of its speed to the speed of sond When an airplane travels faster than the speed of sond, the sond waves form a cone behind the airplane The Mach nmber is related to the apex angle of the cone by the eqation sin 5 Find the angle 2 M that corresponds to a Mach nmber of 25 54 TAKS REASONING AMercator projection is a map projection of the globe onto a plane that preserves angles On a globe with radis r, consider a point P that has latitde L and longitde T The coordinates (x, y) of the corresponding point P9 on the plane can be fond sing these eqations: P x 5 rt y 5 r lnftan p L P 9 2G y ΠΠ9 x a Use half-angle and sm formlas to show that the eqation for the y-coordinate can be written as y 5 r ln sin L cos L 2 b What is a reasonable domain for the eqation in part (a)? Explain 47 Apply Doble-Angle and Half-Angle Formlas 96
55 TAKS REASONING At a basketball game, a person has a chance to win million dollars by making a half cort shot The distance from half cort to the point below the 0-foot-high basketball rim is 475 feet a Write an eqation that models the path of the basketball if the person releases the ball 6 feet high with an initial speed of 40 feet per second b Simplify the eqation Use a calclator to find the angles at which the person can make the half cort shot c Assme the person releases the ball at one of the angles fond in part (b) What other assmption(s) mst yo make to say that the shot is made? 56 CHALLENGE A rectangle is inscribed in a semicircle with radis, as shown What vale of creates the rectangle with the largest area? y (cos, sin ) (cos, 0) x MIXED REVIEW FOR TAKS TAKS PRACTICE at classzonecom REVIEW Skills Review Handbook p 994; TAKS Workbook 57 TAKS PRACTICE MNO and PQR are spplementary angles Which of the following statements is tre? TAKS Obj 0 A MNO 5 PQR B MNO PQR C m MNO m PQR 5 908 D m MNO m PQR 5 808 REVIEW Lesson 32; TAKS Workbook 58 TAKS PRACTICE What is the approximate length of arc MN? TAKS Obj 8 F 305 ft G 330 ft P 4 ft 358 M H 396 ft J 550 ft N QUIZ for Lessons 46 47 Find the exact vale of the expression (pp 949, 955) sin p 2 2 sin (22258) 3 tan (23458) 4 cos p 8 Solve the eqation for 0 x < 2p 5 sin x p 2 2 2 sin x 2 p 2 2 5 0 (p 949) 6 cos 2x 5 3 sin x 2 (p 955) Find the exact vales of sin a, cos a, and tan 2a (p 955) 7 tan a 5 3 5, 0 < a < p 8 cos a 52 4 7, π < a < 3p 9 FOOTBALL Use the formla x 5 32 v 2 sin 2 to find the horizontal distance x (in feet) that a football travels if it is kicked from grond level with an initial speed of 25 feet per second at an angle of 308 (p 955) 962 Chapter 4EXTRA PRACTICE for Lesson 47, p 023 ONLINE QUIZ at classzonecom