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MPM2D1 7.3: Opening Activity: The Trigonometric Ratios/ Tangent, Sine and Cosine Read the instructions and answer the following questions: 1. Four nested right triangles are drawn on the grid. A B a) What angle is common to all four triangles? C b) State the following lengths: DF = 3 units EF = CG = D EG = E BH = EH = AI = EI = F G H I c) Determine the lengths of DE, CE, BE, and AE using the Pythagorean Theorem. d) Complete the following chart. Put the ratios in lowest terms. Triangle EDF ECG EBH EAI Ratio Type 1 DF EF CG EG BH EH AI EI Ratio Type 2 DF DE CG CE BH BE AI AE Ratio Type 3 EF DE EG CE EH BE EI AE e) How are the Type 1 Ratios related? f) This ratio is called the tangent ratio for angle E. How would you clearly and accurately explain to someone which sides of the triangles you used to form the tangent ratio? Use a sketch as well if necessary. 1 Note: The trigonometric ratios depend only on the size of the angle, not on the size of the triangle! Example 1. Find the Tangent Ratio from Given Sides Find tan ø for each triangle, expressed as a fraction and as a decimal correct to four decimal places. 6.8 b. θ 13 5 a. 4.5 8.2 12 Example 2 Find the Tangent of an Angle Evaluate each of the following with a calculator. Record your answer rounded to four decimal places. a) tan 300 b) tan 500 c) tan 750 Example 3 Find an Angle Using the Tangent Ratio Flora is building a skateboarding ramp. She wants the ramp to rise 1 m in a horizontal distance of 2 m . At what acute angles should she cut the wood, rounded to the nearest degree? 2 Example 4 Find a Side length using the Tangent Ratio Find the length x, in the diagram, rounded to the nearest of a centimeter. 9.5 x 250 Example 5 Solve a Multi-Step Problem Using the Tangent Ratio A radio transmitter is to be supported with a guy wire. The wire is to form a 650 angle with the ground and reach 30 m up the transmitter. The wire can be ordered in whole- number lengths of metres. How much wire should be ordered? Draw the diagram first. p. 361CYU #3 Practice #1ace, 34ace, 6-7ac, 9, 10, 11,13,14,15,17 3