2. 1 INTRODUCTION LAW OF SINES AND COSINES LAW OF SINES LAW OF COSINES PROBLEM 1b PROBLEM 1a PROBLEM 2a PROBLEM 2b PROBLEM 4b PROBLEM 4a PROBLEM 3b PROBLEM 3a SUMMARY L. SINES SUMMARY L. COSINES STANDARD 19 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3. 2 Standard 19: Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Los estudiantes usan funciones trigonometricas para resolver un triángulo rectángulo con un lado desconocido dados un ángulo y la longitud de un lado. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4. 3 55° 64°61° 110° 21° 49° RIGHT TRIANGLE ACUTE TRIANGLE OBTUSE TRIANGLE ANGLES CLASSIFYING TRIANGLES BY ANGLES PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5. 4 o u i C B A Tan C= i o Adjacent side Opposite side TANGENT Sin C= i u Hypotenuse Opposite side SINE Cos C= o u Hypotenuse COSINE Adjacent side STANDARD 18 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6. 5 ? ? STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7. 6 = C B A b a c a b Sin A Sin B LAW OF SINES STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8. 7 = C B A b a c a c Sin A Sin C LAW OF SINES STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9. 8 = C B A b a c c b Sin C Sin B LAW OF SINES STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10. 9 Sin A Sin B a b = C B A b a c LAW OF SINES Sin C c = STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11. 10 m A = 67°, a=14, b=9 C B A b=9 a=14 c 67° SOLVE: Sin A Sin 67° Sin B a 14 b 9 = = (9) Sin 67° Sin B 14 = (9).9205 Sin B 14 = (9) Sin B =.5917 m B = Sin( ).5917 m B = 36.3° m C = 180°-67°-36.3°= 76.7° STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12. 11 C B A b=9 a=14 c 67° 36.3° 76.7° Sin 67° Sin 76.7° 14 c = Sin 67° c = Sin 76.7° 14 Sin 67° Sin 76.7° 14 c = Sin 67° (.9731) (.9205) 14 c = c = 14.79 14.79= m A = 67°, a=14, b=9 SOLVE: Sin A Sin 67° Sin B 14 b 9 = = (9) Sin 67° Sin B 14 = (9).9205 Sin B 14 = (9) Sin B =.5918 m B = Sin( ).5918 m B = 36.3° m C = 180°-67°-36.3°= 76.7° a STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13. 12 m A = 58°, a=11, b=8 C B A b=8 a=11 c 58° SOLVE: Sin A Sin 58° Sin B a 11 b 8 = = (8) Sin 58° Sin B 11 = (8).8480 Sin B 11 = (8) Sin B =.6167 m B = Sin( ).6167 m B = 38° m C = 180°-58°-38°= 84° STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14. 13 Sin A Sin B a b = C B A b=8 a=11 c m A = 58°, a=11, b=8 58° SOLVE: Sin 58° Sin B 11 8 = Sin 58° Sin B 11 = (8).8480 Sin B 11 = (8) Sin B =.6167 m B = Sin( ).6167 m B = 38° m C = 180°-58°-38°= 84° Sin 58° Sin 84° 11 c = Sin 58° c = Sin 84° 11 Sin 58° Sin 84° 11 c = Sin 58° (.9945) (.8480) 11 c = c= 12.9 12.9= STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15. 14 79° c C B A b a=15 63° 38° Sin 79° c = Sin 38° 15 Sin 79° Sin 38° 15 c = Sin 79° (.6157) (.9816) c 15 = c = 9.40 9.40 = a=15, m B = 63°, m C = 38° SOLVE: Sin 79° Sin A Sin 38° Sin C c c 15 a m A = 180°-63°-38°= 79° = = STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16. 15 C B A b a=15 c 79° 63° 38° 9.4 = a=15, m B = 63°, m C = 38° SOLVE: = Sin 79° b Sin 63° 15 Sin 79° Sin 63° 15 b = Sin 79° (.8910) (.9816) 15 b = b = 13.61 Sin 79° Sin A Sin 63° Sin B b b = = 15 a =13.61 Sin 79° c = Sin 38° 15 Sin 79° Sin 38° 15 c = Sin 79° (.6157) (.9816) c 15 = c = 9.4 Sin 79° Sin A Sin 38° Sin C c c 15 a m A = 180°-63°-38°= 79° = = STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17. 16 75° c C B A b a=17 65° 40° Sin 75° c = Sin 40° 17 Sin 75° Sin 40° 17 c = Sin 75° (.6428) (.9659) c 17 = c = 11.31 11.31 = a=17, m B = 65°, m C = 40° SOLVE: Sin 75° Sin A Sin 40° Sin C c c 17 a m A = 180°-65°-40°= 75° = = STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18. 17 C B A b a=17 c 75° 65° 40° Sin 75° Sin 40° 17 c = Sin 75° c = Sin 40° 17 Sin 75° Sin 40° 17 c = Sin 75° (.6428) (.9659) 17 c = c = 11.31 11.31 = a=17, m B = 65°, m C = 40° SOLVE: Sin A Sin C c = a m A = 180°-65°-40°= 75° = Sin 75° b Sin 65° 17 Sin 75° Sin 65° 17 b = Sin 75° (.9063) (.9659) 17 b = b = 15.95 Sin 75° Sin A Sin 65° Sin B b b = = 17 a =15.95 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19. 18 LAW OF SINES 1.TWO KNOWN SIDES AND ONE KNOWN OPPOSITE ANGLE TO ONE OF THEM. (2 LADOS CONOCIDOS Y UN ÁNGULO OPUESTO A UNO DE ELLOS) 2. TWO KNOWN ANGLES AND ANY KNOWN SIDE. (2 ÁNGULOS CONOCIDOS Y UN LADO CUALQUIERA) STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20. 19 C B A b a c LAW OF COSINES Cos A a = 2 + - 2 b b 2 c c 2 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21. 20 C B A b a c LAW OF COSINES Cos B b = 2 a + - 2 a 2 c c 2 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22. 21 C B A b a c LAW OF COSINES Cos C c = 2 a + - 2 a 2 b b 2 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23. 22 b c + - 2 Cos A a = 2 b 2 c 2 a c + - 2 Cos B b = 2 a 2 c 2 a b + - 2 Cos C c = 2 a 2 b 2 C B A b a c LAW OF COSINES STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

24. 23 Cos C c = 2 a + - 2 a 2 b b 2 a=29, b=42, m C = 65° SOLVE: C B A b=42 a=29 c 65° (29) (42) + - 2 Cos (65°) c = 2 29 2 42 2 c = 2 (841) + (1764) - (2436) (.423) c = 2 (2605) - (1030.43) c = 2 1574.57 c = 39.7 39.7= c = 39.7 c = -39.7 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

25. 24 Cos C c = 2 a + - 2 a 2 b b 2 C B A b=42 a=29 c 65° 39.7= Sin C Sin 65° c 39.7 a 29 = = (29) Sin 65° Sin A 39.7 = (29).9063 Sin A 39.7 = (29) Sin A =.662 m A = Sin( ).662 m A = 41.5° m B = 180°-65°- 41.5°= 73.5° Sin A 73.5° 41.5° (29) (42) + - 2 Cos (65°) c = 2 29 2 42 2 c = 2 (841) + (1764) - (2436) (.423) c = 2 (2605) - (1030.43) c = 2 1574.57 c = 39.7 c = 39.7 c = -39.7 a=29, b=42, m C = 65° SOLVE: STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26. 25 Cos C c = 2 a + - 2 a 2 b b 2 a=30, b=40, m C = 70° SOLVE: C B A b=40 a=30 c 70° (30) (40) + - 2 Cos (70°) c = 2 30 2 40 2 c = 2 (900) + (1600) - (2400) (.342) c = 2 (2500) - (820.85) c = 2 1679.15 c = 41 41= c = 41 c = -41 STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

27. 26 Cos C c = 2 a + - 2 a 2 b b 2 a=30, b=40, m C = 70° SOLVE: (30) (40) + - 2 Cos (70°) c = 2 30 2 40 2 c = 2 (900) + (1600) - (2400) (.342) c = 2 (2500) - (820.85) c = 2 1679.15 c = 41 C B A b=40 a=30 c 70° 41= c = 41 -41 Sin C Sin 70° c 41 a 30 = = (30) Sin 70° Sin A 41 = (30).94 Sin A 41 = (30) Sin A =.688 m A = Sin( ).688 m A = 43.5° m B = 180°-70°- 43.5°= 66.5° Sin A 66.5° 43.5° STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

28. 27 a=23, b=18, c = 35 SOLVE: + - 2 35 = 2 (23) 23 2 (18) 18 2 C B A b=18 a=23 c 35= + - 2 c = 2 a a 2 b b 2 Cos C C 1225 = 853 - Cos C (828) -853 372 = Cos C -828 Cos C= -.4493 m C = Cos( ) -.4493 m C = 116.7° C 1225 = (529) + (324) - (828) Cos STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

29. 28 Cos C c = 2 a + - 2 a 2 b b 2 C B A b=18 a=23 c 35= 116.7° = = (18) Sin 116.7° Sin B 35 = (18).8934 Sin B 35 = (18) Sin B =.4595 m B = Sin( ).4595 m B = 27.4° m A = 180°-116.7°-27.4°= 35.9° Sin C Sin 116.7° Sin B b 18 35 27.4° 35.9° a=23, b=18, c = 35 SOLVE: 35 = 2 (23) 23 2 (18) 18 2 Cos C + - 2 1225 = 853 - Cos C (828) -853 372 = Cos C -828 Cos C= -.4493 m C = Cos( ) -.4493 m C = 116.7° C 1225 = (529) + (324) - (828) Cos STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

30. 29 a=21, b=16, c = 30 SOLVE: + - 2 30 = 2 (21) 21 2 (16) 16 2 C B A b=16 a=21 c 30= + - 2 c = 2 a a 2 b b 2 Cos C C 900 = 697 - Cos C (672) -697 203 = Cos C -672 Cos C= -.3020 m C = Cos( ) -.3020 m C = 107.6° C 900 = (441) + (256) - (672) Cos STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

31. 30 Cos C c = 2 a + - 2 a 2 b b 2 a=21, b=16, c = 30 SOLVE: C B A b=16 a=21 (21) (16) + - 2 Cos C 30 = 2 21 2 16 2 900 = (441) + (256) - (672) c 30= Cos C 900 = 697 - Cos C (672) -697 203 = Cos C -672 Cos C= -.3020 m C = Cos( ) -.3020 m C = 107.6° = = (16) Sin 107.6° Sin B 30 = (16).9532 Sin B 30 = (16) Sin B =.5084 m B = Sin( ).5084 m B = 30.6° m A = 180°-107.6°-30.6°= 41.8° Sin C Sin 107.6° Sin B b 16 30 30.6° 41.8° STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

32. 31 LAW OF COSINES 1.TWO KNOWN SIDES AND THE ANGLE IN BETWEEN. (DOS LADOS CONOCIDOS Y EL ÁNGULO ENTRE ELLOS) 2. THREE KNOWN SIDES. Always take the largest side with the first angle to find. (TRES LADOS CONOCIDOS. Siempre tomar el lado más largo con el primer ángulo a encontrar). STANDARD 19 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved