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MCR3U – Trig Functions 493720309 Date: __________________________ Triangle Trigonometry What does SOHCAHTOA mean to you? For A : For C sin A = sin C = cos A = cos C = tan A = tan C = Example: Find the missing side length x along with the missing angle. Example: Find the missing angles. Page 1 of 6 MCR3U – Trig Functions 493720309 SOHCAHTOA requires that we have a right angle triangle but what do we do when the triangle does not have a right angle? We need to use the Sine Law. From the following triangle: sin A sin B sin C a b c or a b c sinA sinB sinC Example: Solve the following triangle. Determine all missing sides and angles. Page 2 of 6 MCR3U – Trig Functions 493720309 Can the "c" be solved using Sine Law? ______________ Why or why not? _______________________________________________________ Since we can’t use Sine Law, we have to use Cosine Law. a 2 b 2 c 2 2bccos A b 2 a 2 c 2 2accos B c 2 a 2 b 2 2abcos C Example: Find the measure of angle A. Page 3 of 6 MCR3U – Trig Functions 493720309 Example: Given the ΔABC with a = 3, b = 5, A = 30°, determine the measure of angle B. The Sine Law – The Ambiguous Case 1. Evaluate the following correct to 4 decimal places: a) sin 65° sin 115° b) sin 100° sin 80° c) sin 50° sin 130° d) sin 170° sin 10° e) sin 70° sin 110° f) sin 91° sin 89° 2. What do you notice about the sum of each pair of angles? ____________________ 3. What do you notice about the value of each pair? Why? (hint - see diagram below) ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ Page 4 of 6 MCR3U – Trig Functions 493720309 4. Find the values of theta, , between 0° and 180°, correct to 1 decimal place. a) sin = 0.5664 _______________ b) sin = 0.2598 _______________ c) sin = 0.8750 _______________ d) sin = 0.9060 _______________ 5. How many possible angles are there? ________________ Examples: Solve these triangles - you will either get two triangles, one triangle or no triangles Triangle 1 A = 42° a = 30 b = 25 Triangle 2 B = 27° b = 25 a = 30 Triangle 4 D = 30° d = 50 a = 25 Triangle 3 C = 37.3° c = 85 b = 90 Triangle 5 E = 38.7° e = 10 b = 25 Triangle 1 Triangle 2 Triangle 3 Triangle 4 Triangle 5 1 case 2 cases 2 cases 1 case No solutions Page 5 of 6 MCR3U – Trig Functions 493720309 How do you recognize when you need to consider the Ambiguous Case? To solve an oblique triangle, you need to know the measure of at least one side and any two other parts of the triangle. There are four cases in which this can happen. Given Information What Can Be Found Law Required Two angles and any side (AAS or ASA) Side sine law Two sides and the contained angle (SAS) Side cosine law Three sides (SSS) Angle cosine law Two sides and an angle opposite one of them (SSA) Angle sine law Case 4 is called the Ambiguous Case because sometimes it is possible to draw more than one triangle for the given information. Hints for recognizing the ambiguous case: Swing side a as a pendulum fixed at vertex C. Each time line a touches c, you create a new triangle. Given A, if a > b then there is only one triangle created. Given A, if a < b then there are three possible outcomes (Note: h = b sin A) Homework: Page 318 Questions 1, 4 Page 325 Questions 1, 3 Page 6 of 6