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UNIT 36 FUNCTIONS OF ANY ANGLE, OBLIQUE TRIANGLES OBLIQUE TRIANGLES An oblique triangle is a triangle that does not have a right angle An oblique triangle may be either acute or obtuse In an acute triangle, each of the three angles is acute or less than 90° In an obtuse triangle, one of the angles is obtuse or greater than 90° 2 LAW OF SINES In any triangle the sides are proportional to the sines of their opposite angles or, stated as a formula: a sin A b sin B c sin C C b A a B c Pay attention to how A and a are across from each other! 3 LAW OF SINES The Law of Sines is used to solve the following two kinds of oblique triangle problems: 1. Problems where any two angles and any side of an oblique triangle are known 2. Problems where any two sides and an angle opposite one of the given sides of an oblique triangle are known 4 SOLVING TRIANGLES GIVEN TWO ANGLES AND A SIDE Solve the oblique triangle shown below for side x: – Set up the proportion using the Law of Sines: x 6 . 7 cm 49° 47° sin 49 x sin 47 – Solve the proportion: 6.7(sin 47°) = (sin 49°)x x = 6.49 cm Ans 5 AMBIGUOUS CASE It is possible to have two different triangles with the same two sides and the same angle opposite one of the given sides. A situation of this kind is called an ambiguous case The only conditions under which a problem can have two solutions is when the given angle is acute and the given side opposite the given angle is smaller than the other given side Since most problems do not involve two solutions, this situation will not be discussed in any more detail 6 SOLVING TRIANGLES GIVEN TWO SIDES AND AN OPPOSITE ANGLE Solve for angle C in an oblique triangle given that A is 101, side a is 25 inches, and side c is 14 inches – First, draw the triangle and label the given parts – Now set up the proportion: A 14" B 101° 25" b 25 " sin 101 14 " sin C C – Now, solving for angle C yields: C = 33.3 Ans 7 LAW OF COSINES In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of these two sides multiplied by the cosine of their included angle a2 = b2 + c2 – 2bc(cos A) b2 = a2 + c2 – 2ac(cos B) c2 = a2 + b2 – 2ab(cos C) The Law of Cosines is used to solve oblique triangle problems where two sides and the included angle are known or when all three sides are known 8 SOLVING PROBLEMS GIVEN TWO SIDES AND THE INCLUDED ANGLE Solve for side c in the oblique triangle shown below using the Law of Cosines: 42 mm – Use the appropriate law: c c a b 2 ab (cos C ) 2 2 2 – Now, substitute the given values: 49 mm c 42 49 2 ( 42 )( 49 )(cos 87 ) 2 2 2 – Solving for c yields: c = 62.85 mm Ans 9 LAW OF COSINE PROBLEMS GIVEN THREE SIDES OF A TRIANGLE Find angle A in the triangle shown below: A – Use the appropriate law in order to find the largest angle first: a b c 2 bc (cos A ) or 2 29" 2 cos A 2 29 2 15 2 16 2 ( 2 )( 15 )( 16 ) – Solving for angle A yields: A = 138.6 Ans 10 PRACTICE PROBLEMS Solve problems 1–3 using the Law of Sines. Round all angles to the nearest tenth of a degree and all sides to two decimal places. 1. Find side x in the figure below: 89° x 48° 29 cm 11 PRACTICE PROBLEMS (CONT) 2. Given an oblique triangle with C = 101, side c = 35 inches, and side a = 10 inches, solve for A, B, and side b. 3. Given an oblique triangle with A = 115, C = 23, and side a = 47 cm, find side c, side b, and angle B. 12 PRACTICE PROBLEMS (CONT) Solve problem 4 using the Law of Cosines. Round all angles to the nearest tenth of a degree and all sides to two decimal places. 4. Completely solve the triangle shown below: a C B 4.9" 98° 4.7" A 13 PRACTICE PROBLEMS (CONT) Solve problem 5 using the Law of Cosines. Round all angles to the nearest tenth of a degree and all sides to two decimal places. 5. 6. Determine all the angles of an oblique triangle given that side a is 3.5 feet, side b is 2.7 feet, and side c is 1.2 feet. A triangular patio has sides of 12 feet, 8 feet, and 7.5 feet. Determine its largest angle. 14 PROBLEM ANSWER KEY 1. 2. 3. 4. 5. 6. x = 21.55 cm A = 16.3, B = 62.7, and b = 31.68 inches B = 42, c = 20.26 cm, and b = 34.70 cm B = 42, C = 40, and a = 7.25 inches A = 122.9, B = 40.4, and C = 16.7 101.4° 15