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Law of Sines Trigonometry MATH 103 S. Rook Overview • Sections 7.1 & 7.2 in the textbook: – Law of Sines: AAS/ASA Case – Law of Sines: SSA Case 2 Law of Sines: AAS/ASA Case Oblique Triangles • Oblique Triangle: a triangle containing no right angles • All of the triangles we have studied thus far have been right triangles – We can apply SOHCAHTOA or the Pythagorean Theorem only to right triangles • Naturally most triangles will not be right triangles – Thus we need a method to apply to find side lengths and angles of other types of triangles 4 Four Cases for Oblique Triangles • By definition a triangle has three sides and three angles for a total of 6 components – We can find the measure of all sides and all angles if we know AT LEAST 3 of these components • Broken down into four cases: – AAS or ASA • Measure of two angles and the length of any side – SSA • Length of two sides and the measure of the angle opposite one of two known sides • Known as the ambiguous case because none, one, or two triangles could be possible 5 Four Cases for Oblique Triangles (Continued) – SSS • Length of all three sides – SAS • Length of two sides and the measure of the angle opposite the third (possibly unknown) side – AAA is NOT a case because there are an infinite number of triangles that can be drawn • Recall that the largest side is opposite the largest and angle, but there is no limitation on the length of the side! 6 Law of Sines • The first two cases (AAS/ASA and SSA) are covered by the Law of Sines: a b c sin A sin B sin C – i.e. the ratio of the measure of any side of a triangle to its corresponding angle yields the same constant value • This constant value is different for each triangle – The Law of Sines can be proved by dropping an altitude from an oblique triangle and using trigonometric functions with the right angle • See page 339 • ALWAYS draw the triangle! 7 Law of Sines: AAS/ASA (Example) Ex 1: Use the Law of Sines to solve the triangle – round answers to two decimal places: a) A = 102.4°, C = 16.7°, a = 21.6 b) A = 55°, B = 42°, c = ¾ 8 Law of Sines: SSA Case SSA – the Ambiguous Case • Occurs when we know the length of two sides and the measure of the angle opposite one of two known sides – e.g. a, b, A and b, c, C are SSA cases – e.g. a, b, C and b, c, A are NOT (they are SAS cases) • To solve the SSA case: – Use the Law of Sines to calculate the missing angle across from one of the known sides – There are three possible cases: 10 SSA – the Ambiguous Case (Continued) • The known angle corresponds to one of the given sides of the triangle (e.g. In a, b, A, the known angle is A) – i.e. the side which both the angle and length are given is the side with the known angle • The missing angle corresponds to the second given side (e.g. In a, c, C, the missing angle is A) • Case I: sin missing > 1 – e.g. sin missing = 1.3511 – Recall the domain for the inverse sine: -1 ≤ x ≤ 1 – NO triangle exists • If -1 ≤ missing ≤ 1, at least one triangle is guaranteed to exist 11 • Inverse sine will give the value of missing in Q I SSA – the Ambiguous Case (Continued) • An angle of a triangle can have a measure of up to 180° • Sine is also positive in Q II (90° < θ < 180°) meaning that it is POSSIBLE for missing to assume a value in Q II • Find this second possible value using reference angles • Case II: second + known ≥ 180° – i.e. the measure of the possible second angle yields a second triangle with contradictory dimensions – ONE triangle exists • Case III: second + known < 180° – i.e. the measure of the possible second angle yields a second triangle with feasible dimensions – known is the same in BOTH triangles – TWO triangles exist 12 SSA – the Ambiguous Case (Example) Ex 2: Use the Law of Sines to solve for all solutions – round to two decimal places: a) b) c) d) A = 54°, a = 7, b = 10 A = 98°, a = 10, b = 3 C = 27.83°, c = 347, b = 425 B = 58°, b = 11.4, c = 12.8 13 Law of Sines Application (Example) Ex 3: A man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is 64°. He then walks 100 feet further away and observes that the angle of elevation to the top of the antenna is 46° (see page 345). Find the height of the antenna to the nearest foot. 14 Summary • After studying these slides, you should be able to: – Apply the Law of Sines in solving for the components of a triangle or in an application problem – Differentiate between the AAS/ASA and SSA cases • Additional Practice – See the list of suggested problems for 7.1 & 7.2 • Next lesson – Law of Cosines (Section 7.3) 15