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Name:______________________________ NOTES 6 & 7: TRIGONOMETRIC FUNCTIONS OF ANGLES AND OF REAL NUMBERS Date:________________Period:_________ Mrs. Nguyen’s Initial:_________________ LESSON 6.4 – THE LAW OF SINES 1) SSS 2) SAS 3) ASA 4) AAS Never: SSA and AAA Triangles with no right angles. Drawings: Area of Oblique Triangles (SAS case) The area A of a triangle with sides of lengths a and b and with included angle 1 is: A ab sin 2 Example: Law of Sines (in the case of ASA, SAA, SSA) In triangle ABC we have: Example: Review: Shortcuts to prove triangles congruent Definition of Oblique Triangles Drawings: sin A sin B sin C a b c Practice Problems: Find the missing sides and angles in each problem. Round to 2 decimal places. 1. ABC , m A 54, m B 29, a 10 2. AHS , m A 25, m H 111, a 110 3. B 43 92 58 A C Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 1 Practice Problems: Find the area of each triangle. Round to 2 decimal places. 4. ABC , if b 10, c 6, and m A 65 6. The triangle has sides of length 10 cm, 3 cm, with included angle 120 . 5. ABC , if a 5, b 10, and m C 18 Practice Problems: Applying what you know. 7. A water tower 30 m tall is located at the top of a hill. From a distance of 120 m down the hill, it is observed that the angle formed between the top and the base of the tower is 8 . Find the angle of inclination of the hill. 8. A communications tower is located at the top of a steep hill. The angle of inclination of the hill is 58 . A guy wire is to be attached to the top of the tower & to the ground, 100 m downhill from the base of the tower. The angle of elevation from the bottom of the guy wire to the top of the tower is 70 . Find the length of the cable required for the guy wire. Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 2 Ambiguous Case of Law of Sines If you’re given SSA, then there can either be 0, 1, or 2 triangles formed. This is the ambiguous case. Drawings: Practice Problems: Solve for all possible triangles that satisfy the given conditions. Round all answers to 2 decimal places. 9. ERW , m R 35, e 5, r 4 10. DWC , m D 12, d 11, w 6 11. MLT , m M 15, m 10, l 15 12. A = 39, a = 10, b = 14 Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 3 LESSON 6.5 – THE LAW OF COSINES Law of Cosines (in the case of SAS or SSS) In triangle ABC we have: Example: a 2 b 2 c 2 2bc cos A b 2 a 2 c 2 2ac cos B c 2 a 2 b 2 2ab cos C Practice Problems: Solve each triangle. 1. In ABC , b 6, c 8, and m A 62 2. In ABC , a 6, c 8, and m B 109 3. In ABC , a 3, b 7, and c 5 4. In BAT , b 7, a 9, and t 12 Heron’s Formula: (SSS case) The area A of triangle ABC is given by: A s ( s a )( s b)( s c) where 1 (a b c) is the semi-perimeter of 2 the triangle; that is, s is half the perimeter. s Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 4 Example: Practice Problems: Find the area of the triangle whose sides have the given lengths. 5. MAP, if m 5, a 8, and p 12 7. CAT , c 29, a 45, and t 18 Heading and Bearing… 6. MEW , if m 5, e 7, and w 11 is a direction of navigation indicated by an acute angle measured from due north or due south. Practice Problems: Solve each triangle. 8. A pilot sets out from an airport and heads in the direction N15 W, flying at 250 mph. After one hour, he makes a course correction and heads in the direction of N45 W. Half an hour after that, he must make an emergency landing. (A) Find the distance between the airport & his final landing point. (B) Find the bearing from the airport to his final landing point. Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 5 9. Airport B is 300 mi from airport A at a bearing of N50 E. A pilot wishes to fly from A to B mistakenly flies due east at 200 mph for 30 minutes, when he notices his error. (A) How far is the pilot from his destination at the time he notices the error? (B) What bearing should he head his plane in order to arrive at airport B? 10. Two ships leave a harbor at the same time. One ship travels on a bearing of S12 W at 14 mph. The other ship travels on a bearing of N75 E at 10mph. How far apart will the ships be after three hours? 11. You are on a fishing boat that leaves its pier and heads east. After traveling for 25 miles, there is a report warning of rough seas directly south. The captain turns the boat & follows a bearing of S40 W for 13.5 miles. (A) At this time, how far are you from the boat’s pier? (B) What bearing could the boat have originally taken to arrive at this spot? Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 6 LESSON 7.1 – THE UNIT CIRCLE The Unit Circle The unit circle is the circle of radius 1 centered at the origin. The equation of the unit circle is: x2 y 2 1 Note: Every point on the unit circle can be linked to the values of cos and sin . If point P whose coordinates are (x, y) lies on the unit circle for a given angle , then we know that x cos and y sin Practice Problems: Find the missing coordinate of P, using the fact that P lies on the unit circle in the given quadrant. 1. P , 7 in QIV 25 2. 2 P , 5 in Q II Practice Problems: Find (a) the reference angle for each value of t, and (b) find the terminal point P(x, y) on the unit circle determined by the given value of t. 3. t 2 3 4. t 5 4 5. t 7 6 6. t 11 6 7. t 8. t 3 Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 7 2 9. t 3 2 t 10. THE UNIT CIRCLE y ( ( ( , , , ) ( ) , ) ) ( , ) 2 ( , ) ( 90 135 120 , ) 0 180 ( , 300 270 ( 3 ) ( ( , 2 , ) 315 240 ) , 0 330 225 , ( 30 210 ( ) 45 150 ( , 60 ( ) ( , ) Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 8 , ) , ) ) x Review: Definition of Reference Angle Let be an angle in standard position. Its reference angle is the acute angle ' formed by the terminal side of and the x-axis. Quadrant II Quadrant III ' rad ' rad ' 2 rad ' 180 deg ree ' 180 deg ree ' 360 deg ree Practice Problems: Find the reference angle for each of the given angles. 11. t 170 13. t 15. t 5 7 8 7 Quadrant IV 12. t 410 14. t 16. t 5.8 Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 9 11 9 LESSON 7.2 – TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS Definitions of Trigonometric Functions point on the unit circle corresponding to t. 1 sin t y csc t , y 0 y cos t x tan t Cofunctions Fundamental Trigonometric Identities x, y be the Let t be a real number and let y , x 0, x sec t 1 , x0 x cot t Remember: SOH CAH TOA sin opp hyp csc hyp opp cos adj hyp sec hyp adj tan opp adj cot adj opp x , y0 y sin 90 cos cos 90 sin sin cos 2 cos sin 2 tan 90 cot cot 90 tan tan cot 2 cot tan 2 sec 90 csc csc 90 sec sec csc 2 csc sec 2 Reciprocal Identities sin 1 csc csc 1 sin cos 1 sec sec 1 cos 1 cot sin tan cos cot tan Quotient Pythagorean sin 2 cos 2 1 1 tan 2 sec 2 1 cot 2 csc 2 Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 10 1 tan cos cot sin Practice Problems: Evaluate the six trig functions at each real number without using a calculator. Plot the ordered pair. 1. 2. 3. 5 t 6 t 3 2 3 t 2 sin csc cos sec tan cot sin csc cos sec tan cot sin csc cos sec tan cot Domain of the Trigonometric Functions sin, cos: All real numbers Definition of Periodic Function A function f is periodic if there exists a positive real number such that n for any integer n. 2 cot, csc: All real numbers other than n for any integer n. tan, sec: All real numbers other than f t c f t for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. Practice Problems: Evaluate the trigonometric function using its period as an aid. 4. cos5 5. sin Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 11 9 4 6. sin 3 7. 8 cos 3 The cosine and secant functions are even. Even and Odd Trigonometric Functions cos(t ) cos t sec(t ) sec t The sine, cosecant, tangent, and cotangent functions are odd. sin(t ) sin t tan(t ) tan t Remember: Even f (t ) f (t ) Odd f (t ) f (t ) csc(t ) csc t cot(t ) cot t Practice Problems: Use the value of the trig function to evaluate the indicated functions. 8. sin(t ) 9. cos t 3 8 4 5 sin t csct cos t cos t Practice Problems: Use a calculator to evaluate. Round to 4 decimal places. 10. sin 4 11. csc1.3 12. cos 2.5 13. cot1 Practice Problems: Use a calculator to evaluate. Round to 4 decimal places. 14. cos80 15. cot 66.5 16. Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 12 sec 7 Practice Problem 17: Let be an acute angle such that cos 0.6 . Find: sin csc cos sec tan cot Practice Problem 18: Given sin tan 13 and 2 13 sec , find 3 cos sec 90 cot tan 90 cos csc csc Practice Problem 19: Given tan 5 , find Practice Problems: Evaluate the value of in degrees 0 90 and radians 0 without using 2 a calculator. 20. csc 2 22. cot 1 21. tan 1 23. sin 3 2 Practice Problems: Use a calculator to evaluate the value of in degrees 0 90 and radians 0 . Round to the nearest degrees and 3 decimal places for radians. 2 24. sec 2.4578 25. sin 0.4565 26. cot 2.3545 27. sin 0.3746 Mrs. Nguyen – Honors Algebra II – Chapter 6 & 7 Notes – Page 13