INTRODUCCIÓN A LA MATEMÁTICA - TEVI 2017 - UNRN TRIGONOMETRIA - C.MAGGI -P.ALARCON
Ejercicios resueltos de Trigonometría. Resolución de triángulos oblicuángulos. Teorema del seno y coseno. Problema n° 1) Calcular el otro lado del triángulo ABC: Lado
Lado
Angulo
C ² = A ² + B ² - 2.A.B.cos c a - A = 11 cm B = 6 cm c = 42° b- A= 7m C= 8m b = 52° 20´ A ² = B ² + C ² - 2.B.C.cos a c - B = 10 cm C = 15 cm a = 123° 40´ B ² = A ² + C ² - 2.A.C.cos b
Solución aC ² = A ² + B ² - 2.A.B.cos c C ² = (11 cm) ² + (6 cm) ² - 2.(11 cm).(6 cm).cos 42° C ² = 121 cm ² + 36 cm ² - 132 cm ².0,7431 C ² = 58,9049 cm ² C = 7,675 cm bB ² = A ² + C ² - 2.A.C.cos b B ² = (7 m) ² + (8 m) ² - 2.(7 m).(8 m).cos 52° 20´ B ² = 49 m ² + 64 m ² - 112 m ².0,6111 B ² = 44,5605 m ² B = 6,675 m cA ² = B ² + C ² - 2.B.C.cos a A ² = (10 cm) ² + (15 cm) ² - 2.(10 cm).(15 cm).cos 123° 40´ A ² = 100 cm ² + 225 cm ² - 300 cm ².(-0,5544) A ² = 491,3081 cm ² A = 22,165 cm
INTRODUCCIÓN A LA MATEMÁTICA - TEVI 2017 - UNRN TRIGONOMETRIA - C.MAGGI -P.ALARCON Problema n° 2) Resolver los siguientes triángulos. a-
A = 325 m a = 30° 45´ 20" c = 87° 30´
c-
B = 601 m C = 1000 m c = 95° 02´ 08"
b-
A = 40 cm B = 38 cm C = 27 cm
d-
A = 12,33 cm C = 24,05 cm b = 76° 45´ 30"
Solución aa + b + c = 180° b = 180° - a - c b = 180° - 30° 45´ 20" - 87° 30´ b = 61° 44´ 40" A/sen a = C/sen c C = A.sen c/sen a C = (325 m.sen 87° 30´)/(sen 30° 45´ 20") C = (325 m.0,9990)/0,5114 C = 634,935 m A/sen a = B/sen b B = A.sen b/sen a B = (325 m.sen 61° 44´ 40")/(sen 30° 45´ 20") B = (325 m.0,8808)/0,5114 B = 559,812 m bA ² = B ² + C ² - 2.B.C.cos a B ² + C ² - A ² = 2.B.C.cos a cos a = (B ² + C ² - A ²)/2.B.C cos a = [(38 cm) ² + (27 cm) ² - (40 cm) ²]/2.(38 cm).(27 cm) cos a = (1444 cm ² + 729 cm ² - 1600 cm ²)/(2052 cm ²) cos a = (573 cm ²)/(2052 cm ²) cos a = 0,2792 a = arccos 0,2792 a = 73° 47´ 7" B ² = A ² + C ² - 2.A.C.cos b A ² + C ² - B ² = 2.A.C.cos b cos b = (A ² + C ² - B ²)/2.A.C cos b = [(40 cm) ² + (27 cm) ² - (38 cm) ²]/2.(40 cm).(27 cm) cos b = (1600 cm ² + 729 cm ² - 1444 cm ²)/(2160 cm ²)
INTRODUCCIÓN A LA MATEMÁTICA - TEVI 2017 - UNRN TRIGONOMETRIA - C.MAGGI -P.ALARCON cos b = (885 cm ²)/(2160 cm ²) cos b = 0,4097 b = arccos 0,4097 b = 65° 48´ 45" a + b + c = 180° c = 180° - a - b c = 180° - 73° 47´ 7" - 65° 48´ 45" c = 40° 24´ 8" cC/sen c = B/sen b sen b = (B/C).sen c sen b = (601 m/1000 m).sen 95° 02´ 08" sen b = 0,601.0,9961 sen b = 0,5987 b = arcsen 0,5987 b = 36° 46´ 32" a + b + c = 180° a = 180° - b - c a = 180° - 36° 46´ 32" - 95° 02´ 08" a = 48° 11´ 20" A/sen a = C/sen c A = C.sen a/sen c A = (1000 m.sen 48° 11´ 20")/(sen 95° 02´ 08") A = (1000 m.0,7453)/0,9961 A = 748,23 m dB ² = A ² + C ² - 2.A.C.cos b B ² = (12,33 cm) ² + (24,05 cm) ² - 2.(12,33 cm).(24,05 cm).cos 76° 45´ 30" B ² = 152,0289 cm ² + 578,4025 cm ² - 593,073 cm ².0,2291 B ² = 594,582801 cm ² B =24,38 cm C/sen c = B/sen b sen c = (C/B).sen b sen c = (24,05 cm/24,38 cm).sen 76° 45´ 30") sen c = 0,9863.0,9734 sen c = 0,9601 c = arctg 0,9601 c = 0,7650 c = 43° 49´ 59" a + b + c = 180° a = 180° - c - b a = 180° - 43° 49´ 59" - 76° 45´ 30" a = 59° 24´ 31"
