INSTRUCTIONAL PLAN Day 2 Subject: Trignmetry Tpic: Other Trignmetric Ratis, Relatinships between Trignmetric Ratis, and Inverses Target Learners: Cllege Students Objectives: At the end f the lessn, students will be able t define the three ther trignmetric ratis and derive all the five ther ratis given just ne rati; and explain the meaning f an inverse and be able t use the inverses f sine, csine and tangent in slving right triangles. Instructinal Activity Descriptin f the Activity 1. Mtivatin Use the Pwerpint presentatin (Time fr a Walk.pptx) and the fllwing as a hk: Frm a parking lt yu want t walk t a huse n the cean. The huse is lcated 1500 feet dwn a paved path that parallels the beach, which is 500 feet wide. Alng the path yu can walk 300 feet per minute, but in the sand n the beach yu can nly walk 100 feet per minute. What is the fastest time yu can walk frm the parking lt t the huse? Fr what angle θ wuld this fastest time be? (c) What wuld be the distance walked n the paved path? On the beach? (Surce: Sullivan and Sullivan, 2009) The presentatin will shw an illustratin f the prblem with prper labels. Say, In rder t answer the first tw questins, we need t knw hw t find ut the values f ther trignmetric ratis, given just ne f them, and we als need t understand the cncept f inverses. Technlgical requirements: Sftware MS Pwerpint Hardware cmputer, prjectr 2. Objective At the end f the lessn, students will be able t: define ctangent, secant and csecant. find the values f ther trignmetric ratis, given just ne f them, by understanding the relatinship between all six trignmetric ratis. explain the cncept f inverses and be able t use the inverses f sine, csine and tangent in slving right triangles. 3. Prerequisite Students will need prir knwledge abut angles, triangles, rati and prprtin, expnents and radicals, slutins t algebraic equatins, prperties f right triangles, Pythagrean Therem, and the three primary trignmetric ratis. 4. Infrmatin and examples 1. Define the three ther trignmetric ratis. 2. Allw students t make the cnnectin between sine and csecant, csine and secant, and tangent and
ctangent. 3. Shw students the ratinal relatinship f sine and csine t tangent and ctangent, and let them derive the ther ratinal relatinships between the trignmetric ratis. 4. Sample prblem: Given, find the values f the ther five trignmetric ratis. Slutin: Draw a right triangle and label ne f the angles as θ Since, θ the side ppsite the 4 angle wuld be 3 units and the side adjacent t the angle wuld be 4 units. Use Pythagrean Therem t get the length f the hyptenuse (5 units). Since we nw knw the lengths f each side, we can use the definitins t find the values f the ther trignmetric ratis: 5. Call 2 t 3 pairs f students t read their hmewrk regarding inverses. Ask the class fr reactins regarding the pairs definitin and examples f inverses. Sample answer: An inverse is a reverse actin f an peratin. In essence, it undes what the riginal peratin did. Fr instance, t get frm his hme t his schl, a student needs t walk 500 meters east and 200 meters nrth. T get frm his schl t his hme, the student must then travel the same distance but in reverse rder and n the ppsite directins, that is, 200 meters suth and then 500 meters west. This actin brings him back t where he started, which is his hme, thus unding what has been dne n his walk t schl. (Surce: McKeague & Turner, 2008) 6. Define the inverse f sine, csine and tangent as the angle that wuld prduce the ratis sine, csine r tangent, respectively. 3
7. Sample prblem: A right triangle has legs 2.73 and 3.41 units, respectively. Find the hyptenuse and the acute angles f the right triangle. Slutin: Frm the illustratin, we see that therefre, By Pythagrean Therem, Ask students if they culd find ther ways t slve the prblem. (Surce: McKeague & Turner, 2008) 8. Build the students vcabulary with the fllwing terms: Inverse an peratin that undes what the riginal peratin did. Sine Inverse (sin -1 ) the inverse peratin f sine. It is an angle that wuld prduce the sine rati. Csine Inverse (cs -1 ) the inverse peratin f csine. It is an angle that wuld prduce the csine rati. Tangent Inverse (tan -1 ) the inverse peratin f tangent. It is an angle that wuld prduce the tangent rati. Technlgical requirement: Hardware scientific calculatr 5. Practice and feedback 1. Allw the students t answer the fllwing prblems: Find the six trignmetric ratis f the angle θ in each figure. Find the measure f the angle θ as well. (c) (d) Answers: (c) ; ; ; ; ; ; ; ; ; ; ; ;
; ; ; (d) ; ; ; ; ; 2. A resistr and an inductr cnnected in a series netwrk impede the flw f an alternating current. This impedance Z is determined by the reactance X f the inductr and the resistance R f the resistr. The three quantities, all measured in hms, can be represented by the sides f a right triangle as illustrated. Frm the Pythagrean Therem, Z 2 = X 2 + R 2. The angle φ is called the phase angle. Suppse a series netwrk has an inductive reactance f X = 400 hms and a resistance f R = 600 hms. Find the impedance Z. ( hms) Find the phase angle φ. (33.7 ) 3. Ask the students t answer the questin psted at the beginning f the lessn. Suggest that they use a graphing calculatr fr and : Frm a parking lt yu want t walk t a huse n the cean. The huse is lcated 1500 feet dwn a paved path that parallels the beach, which is 500 feet wide. Alng the path yu can walk 300 feet per minute, but in the sand n the beach yu can nly walk 100 feet per minute. What is the fastest time yu can walk frm the parking lt t the huse? (9.7 min) Fr what angle θ wuld this fastest time be? (70.5 ) (c) What wuld be the distance walked n the paved path? (1323 ft) On the beach? (530 ft) (Surce: Sullivan & Sullivan, 2009) Technlgical requirements: Hardware scientific calculatr, graphing calculatr 6. Additinal examples 1. Express h in terms f d, and the ctangents f α, and β in the fllwing figures: ( ) ( ) Surce: Barnett, R. et al (2011)
2. If a circle f radius 4 centimeters has a chrd f length 3 centimeters, find the central angle that is ppsite this chrd (t the nearest degree). (44 ) Surce: Barnett, R. et al (2011) 3. Find the measure f angle A t the nearest degree: (49 ) (34 ) Surce: McKeague & Turner (2008) Technlgical requirement: Hardware scientific calculatr 7. Additinal practice and feedback Shrt Pp Quiz: 1. Penalty Kick. A penalty kick is taken frm a crner f the penalty area at psitin A (see figure belw). The galkeeper stands 6 feet frm the galpst nearest the shter and can thus blck a sht anywhere between the middle f the gal and the nearest galpst (segment CD). T scre, the shter must kick the ball within the angle CAE. Find the measure f this angle t the nearest tenth f a degree. Answer: 4.6 Surce: McKeague & Turner (2008) 2. Calculating Pl Shts. A pl player lcated at X wants t sht the white ball ff the tp cushin and hit the red ball dead center. He knws frm physics that the white ball will cme ff a cushin at the same angle as it hits a cushin. Where n the tp cushin shuld he hit the white ball? Answer: 4.125 ft frm the upper-left crner Surce: Sullivan & Sullivan (2009) Technlgical requirement: Hardware scientific calculatr
8. Summary 1. Ctangent, secant and csecant are the three ther trignmetric ratis. Ctangent is the reciprcal f tangent; secant is the reciprcal f csine; and csecant is the reciprcal f sine. 2. Tangent can als be expressed as the rati f sine t csine, r f secant t csecant. Ctangent is likewise expressed as the rati f csine t sine, r f csecant t secant 3. Inverses und the peratin dne. Thus, the inverse f a trignmetric rati is the angle at which the rati between the specific sides was defined. 4. Knwledge f all the trignmetric ratis, their relatin t each ther, and the inverses f the trignmetric ratis can help slve prblems invlving right triangles and even thse prblems wherein right triangles are nt apparent, such as sprts, electricity, etc. 9. Hmewrk 1. Separate the class int grups f 4 r 5 students. They will wrk n the Design A Game prject, which will be submitted electrnically at 10PM n the night befre Day 5 and t be presented n Day 5 fr this unit. 2. Jurnal/blg entry in their CMS pages abut what they have learned frm the sessin and their insights n hw this new knwledge can be applied in their lives. Evaluatin/Assessment: Class participatin, t be assessed using the Classwrk/Participatin Rubric (generated frm irubric https://www.irubric.cm/), and shrt pp quiz wrth 5 pints each item. References: Aufmann, Richard N., Barker, Vernn C. and Natin, Richard D. (2011). Cllege Algebra and Trignmetry, 11 th ed. Barnett, Raymnd A. et al (2008). Cllege Algebra with Trignmetry, 9 th ed. Larsn, Rn (2012). Algebra and Trignmetry: Real Mathematics, Real Peple, 6 th ed. McKeague, Charles P. and Turner, Mark D. (2008). Trignmetry, 7th ed. Sullivan, Michael and Sullivan, Michael III (2009). Algebra & Trignmetry, 6 th ed. Prepared by: Cyrus B. Alvarez