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Class: Extension 1 Date 1/Nov/2010 Topic Finding the limits of the sin of small angles. Syllabus reference Period No: 1 Period Start: 12.00 Period finish: 12.40 13.5E Outcomes Finding the limit of sin x / x Link with previous learning Concepts of radian measure Concepts of limit of a function Trigonometric ratios Resources Ruler, compass, scientific calculator Assessment Students will be asked to solve a question and a member will be asked to explain it on the board. Activity Stage 1 Minut es 3 Timing Introduction Refresh students on how radian measure works. 12.00 – 12.03 Ask the students use calculators to work on the following. Remind them to switch to radian mode. Sin 0.0023 (should come up with 0.0023) Page 1 of 6 Sin 0.0000000078 (should come up with 0. 0000000078) Tan 0.0023 (should come up with 0.0023) Tan 0.00000057 (should come up with 0. 00000057) Cos 0.0023 (should come up with a value close to 1) Cos 0.000000043 (should come up with a value close to 1) 2 10 12.03 to 12.13 Derivation Hand out a geogebra printout in worksheet 1. Ask students to fill up the table. (Using Geogebra, Ramil to create worksheets showing a circle and with x radians getting smaller and smaller). Ask the students to manually measure the ratios as x becomes smaller and smaller. As x becomes smaller and smaller, show the students that Sin x tends to become x Cos x tends to become 1 Tan x tends to become x As a result, Lim of sin x /x = 1 Lim of cos x / x = x Lim of tan x / x = 1 3 10 Demonstrate Show the following on the board: 12.13 to 12.23 Page 2 of 6 lim sin 7x / 7x lim sin 4x / x lim sin x / 5x 4 10 Group Work Ask the students to solve worded questions similar to below: 12.23 to 12.33 Find the diameter of the sun to the nearest kilometre if its distance from the earth is 149 000 000 km and it subtends an angle of 31’ at the earth. 5 5 12.33 To 12.38 Challenge Give each group a challenging question similar to below: Find limit of (1 – cos x) / x2 (Ramil to find relevant exercises on this) Check that sin x < x < tan x for 0 < x < π/2 by using your calculator. (x in radians). Does this work on x > π/2? 6 2 12.38 to 12.40 Recap Students in their own words should state the following rules: Lim of sin x /x = 1 Page 3 of 6 Lim of cos x / x = x Lim of tan x / x = 1 And what to do in scenarios where the argument is not the same as the denominator: Sin 4x / x etc Self How did the lesson go? evaluation of lesson Have student learnt what was expected? Effective learning environment? Were students interested and motivated? Was use of resources (maps, overheads) successful? What could have been changed? Page 4 of 6 Worksheet 1 1. Using a protractor, measure the angle ABC. 2. Complete the table and supply the required values. Ramil to include the Geogebra drawing here. X Sin X Sin X / X Cos X Cos X / X Tan X Tan X / X Page 5 of 6 Worksheet 2 Evaluate the following: 1. Lim sin x / 4x 2. Lim tan (t/3) / x Ramil to produce more exercises Page 6 of 6