1. In right triangle ABC, a = 3, b = 5, and c is the length of hypotenuse. Evaluate sin A, cos A, and tan A. 2. Evaluate cos 60º.

Transcription

1 Lesson 13.3, Fo use with pages In ight tiangle ABC, a 3, b 5, and c is the length of hpotenuse. Evaluate sin A, cos A, and tan A. ANSWER sin A 3 34, cos A, tan A Evaluate cos 60º. ANSWER 1 2

2 Lesson 13.3, Fo use with pages Evaluate sec 45º. ANSWER 2 4. Evaluate cot 30º. ANSWER 3

3 Evaluating Distance using Tigonometic Functions Students building tomoow with Robotics!!!!!!

4 EXAMPLE 1 Evaluate tigonometic functions given a point Let ( 4, 3) be a point on the teminal side of an angle θ in standad position. Evaluate the si tigonometic functions of θ. SOLUTION Use the Pthagoean theoem to find the value of ( 4)

5 EXAMPLE 1 Evaluate tigonometic functions given a point Using 4, 3, and 5, ou can wite the following: sin θ 3 5 cos θ 4 5 tan θ 3 csc θ sec θ 5 cot θ 4 4 3

6 EXAMPLE 2 Use the unit cicle Use the unit cicle to evaluate the si tigonometic functions of θ 270. SOLUTION Daw the unit cicle, then daw the angle θ 270 in standad position. The teminal side of θ intesects the unit cicle at (0, 1), so use 0 and 1 to evaluate the tigonometic functions.

7 EXAMPLE 2 Use the unit cicle sin θ csc θ cos θ 0 1 sec θ 0 undefined 1 0 tan θ 1 0 undefined cot θ 0 1 0

8 GUIDED PRACTICE fo Eamples 1 and 2 Evaluate the si tigonometic functions of. θ 1. SOLUTION Use the Pthagoean Theoem to find the value of ( 3)

9 GUIDED PRACTICE fo Eamples 1 and 2 Using 3, 3, and 3 2, ou can wite the following: sin θ cos θ tan θ 3 1 csc θ sec θ cot θ 1 3 3

10 GUIDED PRACTICE fo Eamples 1 and 2 2. SOLUTION Use the Pthagoean theoem to find the value of. ( 8) 2 + (15)

11 GUIDED PRACTICE fo Eamples 1 and 2 Using 8, 15, and 17, ou can wite the following: sin θ cos θ 8 17 tan θ 15 csc θ sec θ 17 cot θ

12 GUIDED PRACTICE fo Eamples 1 and 2 3. SOLUTION Use the Pthagoean theoem to find the value of ( 5) 2 + ( 12)

13 GUIDED PRACTICE fo Eamples 1 and 2 Using 5, 12, and 13, ou can wite the following: sin θ cos θ 5 13 tan θ 12 5 csc θ sec θ 13 5 cot θ 5 12

14 GUIDED PRACTICE fo Eamples 1 and 2 4. Use the unit cicle to evaluate the si tigonometic functions of θ 180. SOLUTION Daw the unit cicle, then daw the angle θ 180 in standad position. The teminal side of θ intesects the unit cicle at ( 1, 0), so use 1 and 0 to evaluate the tigonometic functions.

15 GUIDED PRACTICE fo Eamples 1 and 2 sin θ cos θ tan θ 0 1 csc θ 1 0 undefined sec θ cot θ 0 undefined

16 EXAMPLE 3 Find efeence angles Find the efeence angle θ' fo (a) θ and (b) θ π 3 SOLUTION a. The teminal side of θ lies in Quadant IV. So, θ' 2π 5π. 3 π 3 b. Note that θ is coteminal with 230, whose teminal side lies in Quadant III. So, θ'

17 EXAMPLE 4 Use efeence angles to evaluate functions Evaluate (a) tan ( 240 ) and (b) csc 17π. 6 SOLUTION a. The angle 240 is coteminal with 120. The efeence angle is θ' The tangent function is negative in Quadant II, so ou can wite: tan ( 240 ) tan 60 3

18 EXAMPLE 4 Use efeence angles to evaluate functions b. 17π The angle is conteminal 6 with 5π. 6 5π The efeence angle is θ' π 6 The cosecant function is positive in Quadant II, so ou can wite: π 6 17π 5π csc csc 2 6 6

19 GUIDED PRACTICE fo Eamples 3 and 4 Sketch the angle. Then find its efeence angle The teminal side of θ lies in Quadant III, so θ'

20 GUIDED PRACTICE fo Eamples 3 and 4 Sketch the angle. Then find its efeence angle is coteminal with 100, whose teminal side of θ lies in Quadant III, so θ'

21 GUIDED PRACTICE fo Eamples 3 and 4 Sketch the angle. Then find its efeence angle. 7π π 11π The angle 9 is conteminal with 9. The teminal side lies in Quadant III, so θ' 11π π 2π 9 9

22 GUIDED PRACTICE fo Eamples 3 and 4 Sketch the angle. Then find its efeence angle π 4 The teminal side lies in Quadant III, so θ' 2π 15π π 4 4

23 GUIDED PRACTICE fo Eamples 3 and 4 9. Evaluate cos ( 210 ) without using a calculato. 210 is coteminal with 150. The teminal side lies in Quadant II, which means it will have a negative value. So, cos ( 210 ) 3 2

24 EXAMPLE 5 Calculate hoizontal distance taveled Robotics The fogbot is a obot designed fo eploing ough teain on othe planets. It can jump at a 45 angle and with an initial speed of 16 feet pe second. On Eath, the hoizontal distance d (in feet) taveled b a pojectile launched at an angle θ and with an initial speed v (in feet pe second) is given b: d v 2 32 sin 2θ How fa can the fogbot jump on Eath?

25 EXAMPLE 5 Calculate hoizontal distance taveled SOLUTION d v 2 32 sin 2θ Wite model fo hoizontal distance. d sin (2 45 ) Substitute 16 fo v and 45 fo θ. 8 Simplif. The fogbot can jump a hoizontal distance of 8 feet on Eath.

26 EXAMPLE 6 Model with a tigonometic function Rock climbing A ock climbe is using a ock climbing teadmill that is 10.5 feet long. The climbe begins b ling hoizontall on the teadmill, which is then otated about its midpoint b 110 so that the ock climbe is climbing towads the top. If the midpoint of the teadmill is 6 feet above the gound, how high above the gound is the top of the teadmill?

27 EXAMPLE 6 Model with a tigonometic function SOLUTION sin θ sin Use definition of sine Substitute 110 fo θ and 5.25 fo Solve fo. The top of the teadmill is about feet above the gound.

28 GUIDED PRACTICE fo Eamples 5 and 6 TRACK AND FIELD 10. Estimate the hoizontal distance taveled b a tack and field long jumpe who jumps at an angle of 20 and with an initial speed of 27 feet pe second. SOLUTION d d v sin 2θ sin (2 20 ) Wite model fo hoizontal distance. Substitute 27 fo v and 20 fo θ. Simplif. The long jumpe can jump feet.

29 GUIDED PRACTICE fo Eamples 5 and WHAT IF? In Eample 6, how high is the top of the ock climbing teadmill if it is otated 100 about its midpoint? SOLUTION sin θ sin Use definition of sine Substitute 100 fo θ and 5.25 fo Solve fo. The top of the teadmill is about feet above the gound.