Download Lesson plan for limit of small angles

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