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Lesson #3: Cosine Law The Sine Law allows you to solve for triangles where: • 2 sides and a corresponding angle are known (SSA or side-side-angle) • 2 angles and a corresponding side (AAS or angle-angle-side) The Cosine Law allows you to solve for: • The 3rd side of the triangle if you know 2 sides of a triangle and the angle that is formed between these two sides. (SAS or side – angle – side) • An angle if you know the three side lengths of the triangle. Page 1 of 8 The Cosine Law formula: To Find A Side: a2 = b2 + c2 – 2bc cos A To Find An Angle: cos A = b2 + c2 – a2 (2bc) Now that you know another trig formula, remember… When solving triangles: Check for right angles (900). Use basic trigonometric ratio’s (SOH CAH TOA) Check for Sine Law Ratios (a side and an opposite angle). Use Sine Law. If none of the above possibilities exist: Use Cosine Law. Example 1: Write the cosine formula for the missing side R of the following triangle PQR. To Find A Side: a2 = b2 + c2 – 2bc cos A Page 2 of 8 Example 2: Write the cosine formula for the missing angle Q of the following triangle PQR. To Find An Angle: cos A = b2 + c2 – a2 (2bc) Example 3: Find the measure of angle Y. X z = 15 Y Note: It’s a SSS triangle y = 17 x = 20 To Find An Angle: Z cos A = b2 + c2 – a2 (2bc) Find angle Y Page 3 of 8 Example 4: Find the measure of side a, angle B, and angle C. First check… 1. Right angle triangle? 2. Sine-Law ratio’s? (opposites?) 3. If no to both, use Cosine Law A 55° 18 cm Note: It’s a SAS triangle 14 cm C B To Find A Side: a2 = b2 + c2 – 2bc cos A First: Find side a Second: Find angle B Third: Find angle C Page 4 of 8 Assignment #3: Cosine Law 1. Given ∆ABC. Solve for side a. A c = 350 68° b= 475 B C 2. Given ∆ABC. Solve for ∠A. A c = 55 b = 75 B a = 70 C 3. From a lighthouse, a cruise ship can be seen 8.3 km away and a freighter can also be seen 12.5 km away. How far away is the cruise ship from the freighter if the angle between the lines of observation are 68°? Lighthouse 8.3 Cruise Ship 68° 12.5 ? Freighter Page 5 of 8 4. Solve for all the interior angles. A c = 18 cm B b = 20 cm a = 19 cm C NOTE: For the following questions, make sure to draw a diagram to help you with the question. 5. At a provincial park, there is a sign, a reception area, and a picnic area. The reception area is 350 m away from the picnic area, the picnic area is 475 m away from the sign. From the picnic area, the angle between the 2 lines of sight for the reception area and the sign is 64°. How far apart is the sign from the reception area? Page 6 of 8 6. An Art Gallery is in the shape of a triangle. Two of the walls are 114 m and 61 m in length. The angle between these 2 walls is 72°. a. How long is the 3rd wall? b. What are the angles of the other 2 corners of the triangle? Page 7 of 8 7. Construction has been started on a building as shown by the diagram. 12 ft Pier 10 ft Braces a. What is the length of each brace? b. What is the angle between both braces? Page 8 of 8