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Math 112 Elementary Functions Chapter 7 – Applications of Trigonometry Section 1 The Law of Sines Solving Right Triangles – Revisited! Solving Triangles? Using given information, determine the lengths of the sides and measures of the angles. What must be known to solve a right triangle? Lengths of two sides. Length of a side and the measure of an acute angle. How do you solve the triangle? sin = opp/hyp cos = adj/hyp tan = opp/adj Solving Oblique Triangles – Five Cases Given 1 side and 2 angles 20 AAS and ASA 40° 60° 40° 60° 25 Solving Oblique Triangles – Five Cases Given 1 side and 2 angles AAS and ASA Given 2 sides and 1 angle 20 60° 25 SSA and SAS 20 40° 25 Solving Oblique Triangles – Five Cases Given 1 side and 2 angles AAS and ASA Given 2 sides and 1 angle SSA and SAS Given 3 sides SSS 20 13 25 Solving Oblique Triangles – Five Cases Law of Sines (this section) Used to solve AAS, ASA, and SSA triangles. 20 20 40° 40° 60° 60° 60° 25 25 Law of Cosines (next section) Used to solve SAS and SSS triangles. 20 40° 25 20 13 25 The Law of Sines – Acute Triangle C b h A c h sin A b h b sin A h a h a sin B a sin B B b sin A a sin B a b sin A sin B The Law of Sines – Obtuse Triangle C h sin A b b a A c h h sin 180 B a B b sin A a sin B a b sin A sin B h b sin A h a sin B The Law of Sines C b A a c a b c sin A sin B sin C B Solving Oblique Triangles – AAS w/ the Law of Sines 1. Find the third angle. = 180° - (60° + 40°) = 80° 2. Use the law of sines to find a second side. x/sin 40° = 20/sin 60° x 14.8 3. Use the law of sines to find the third side. y/sin 80° = 20/sin 60° y 22.7 20 x 40° 60° y NOTE: Always use EXACT values if possible. Solving Oblique Triangles – ASA w/ the Law of Sines 1. Find the third angle. = 180° - (60° + 40°) = 80° 2. Use the law of sines to find a second side. x/sin 40° = 25/sin 80° x 16.3 3. Use the law of sines to find the third side. y/sin 60° = 25/sin 80° y 22.0 y x 40° 60° 25 NOTE: Always use EXACT values if possible. Solving Oblique Triangles – SSA w/ the Law of Sines With AAS and ASA, the given data will determine a unique triangle. With SSA, the given data could determine … no triangle one triangle two triangles ? 60° 20 25 Solving Oblique Triangles – SSA w/ the Law of Sines Case 1: No Solution The side opposite the given angle is not long enough to reach the other side of the angle. 22 8 sin B sin 40 C 8 22 40° A B? 22 sin 40 sin B 1.77 8 B sin 1 1.77 No Solution! Solving Oblique Triangles – SSA w/ the Law of Sines Case 2a: One Solution The side opposite the given angle is just barely long enough to reach the other side of the angle. C 22 11 22 11 sin B sin 30 22 sin 30 sin B 1 11 B sin 1 1 90 Right Triangle! 30° A B? NOTE: Angle C and side c still need to be determined. Solving Oblique Triangles – SSA w/ the Law of Sines Case 3: Two Solutions The side opposite the given angle is more than long enough to reach the other side of the angle but is shorter than the other given side. 22 20 sin B sin 40 C 22 20 20 22 sin 40 sin B 0.707 20 B sin 1 0.707 B 45 or 135 40° A B? B? NOTE: Angle C and side c still need to be determined for BOTH solutions. Solving Oblique Triangles – SSA w/ the Law of Sines Case 2b: One Solution The side opposite the given angle is more than long enough to reach the other side of the angle but is longer than the other given side. C 22 22 25 sin B sin 40 25 22 sin 40 sin B 0.566 25 B sin 1 0.566 B 34.4 or 145.6 40° A NOTE: Angle C and side c still need to be determined. B? Since 145.6 + 40 180, this solution is invalid. The Area of a Triangle area = ½bh sin C = h / a h = a sin C a h Therefore, … area = ½ ab sin C b C In general, the area of a triangle is half the product of two sides times the sine of the included angle.