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LAW OF COSINES WORKSHEET Law Of Cosines The law of cosines is used to solve oblique triangles. Use the law of cosines to compute the angles when the lengths of all three sides are known. The law of cosines may also be used when the length of two sides and the measure of their included angle are known. The formula states that : And can be rewritten as: The law of cosines can be used in conjunction with the law of sines to find all angles and sides of any triangle. The cases in which the law of cosine can be used : Solve for x in each triangle. Round to the nearest tenth. 1. 2. X x 17m 39mm 47mm 42° 22m 35mm 3. 4. X 23m x 13cm 9.4cm 47° 7cm 20m 55cm 5. 6. 61° x 4.9m 9.1m 50cm 8.3m X Solve for all missing sides and angles in each triangle. Round to the nearest tenth 7. XYZ : x 29m, y 15m, Z 122 X = __________ 8. Y = __________ z = __________ GHI : g 13cm, h 8cm, i 15cm G = __________ H = __________ I = __________ 9. MNO : n 31m, o 28m, M 62 N = __________ O = __________ m = __________ Area of a triangle using trigonometry : The general formula for the area of a triangle is well known. While the formula shows the letters band h, it is actually the pattern of the formula that is important. The area of a triangle equals ½ the length of one side times the height drawn to that side (or an extension of that side). The area of ΔABC can be expressed as: where a represents the side (base) and h represents the height drawn to that side. Using trigonometry, let's take another look at this diagram. In the right triangle CDA, we can state that: The height, h, of the triangle can be expressed as b sin C. General Formula for Area of Triangle: b = length of a side (base) h = height draw to that side Substituting this new expression for the height, h, into the general formula for the area of a triangle gives: where a and b can be any two sides and C is the included angle. The area of a triangle can be expressed using the lengths of two sides and the sine of the included angle. AreaΔ = ½ ab sin C. You may see this referred to as the SAS formula for the area of a triangle. Example : Given the triangle at the right, find its area. Express the answer to the nearest hundredth of a square unit. Let a = ST, b = RT, and C = ∠RTS. Your turn: Given the parallelogram shown at the right, find its area to the nearest square unit. Let a = PS, b - RS, and C =∠PSR. Applications : 1. A farmer has a triangular field with sides 120 yards, 170 yards, and 220 yards. Find the area of the field in square yards. 2. Two surveyors are determining the distance to a tower located between them but across the river. The first one determines that the line of sight to the tower makes an angle of 89° with the bank of the river. 495 m downstream another surveyor determines his line of sight to the tower is 55° to the tower with the river. How far is each surveyor to the tower? 3. To determine the length of an airplane, a person walks 300 ft from point A, the nose of the aircraft to point B, the end of the wing and then turns 80° and walks 250 ft to the tail of the airplane. Find the length of the airplane.