Name of School:
ROUTINE INFORMATION Inhlakanipho High School.
Student Surname and Name:
Maphumulo Mthokozisi
Student Number:
207503878
Grade:
10
Subject:
Mathematics
Topic:
Trigonometric function and its graph.
Content /Concept Area:
Special angles and trigonometric graphs and
CAPS page number:
quadrants 101
Duration of Lesson:
45 minutes
Date :
06/05/2018
Specific Aims: To develop understanding of trigonometric function including sine, cosine and tangent function. To provide learners opportunity to explore four quadrants of Cartesian plan through the use of trigonometric functions highlighted above. To improve learners understanding of graph development.
Learners should gain
Knowledge
Skills Learners should be able to do
Value /Attitude Learners should acquire
knowledge and
following :
values and attitudes
understanding of the
conducive to :
following : Identify the relationship
Make a drawing to outlining
Recognise the definitions of
between radian (the angle)
relationship between degrees
trigonometric functions for a
subtended at the centre of a
and radians in the unit circle
unit circle from 1st to 4th
circle for which the arc length
quadrant.
is equal to radius of that circle.
Explain examples of
Describe
a
trigonometric functions of right
trigonometric
angles under one set of
angles
complete revolution.
circle .
Analyse the graphs of
Draw, explain and analyse
Recognise
trigonometric function.
three
between trig. Special angles
θ
main
difference
the
Describe the effect of most
function
of
commonly used angles in a
sets in a unit
types
trigonometric graphs.
Approaches:-
Role plays. Case studies Class discussions. Questioning. Whole class-discussion Group Work Resources:
Map of the earth structure Interactive chart Chalkboard Newspaper Worksheet
of
trigonometric function.
the
relationship
and the graphs. e.g. amplitude
Lesson Phases: Introduction: During the introduction students are requested to explain radius. Following then they would a fairly big Cartesian plane is drawn in the chalkboard so that kids will come to organise selected special angles using given parameters. Kids have never been exposed to trigonometric graphs before. This lesson is going to be tough because we have to alter our intention to redirect learners to development of special angles. So the solution to this came with understanding of special angles and Pythagoras theorem. We would allow learners to draft their own solutions to develop x- and y- co-ordinates from 1 st to 4th co-ordinates using the special angles. But also students must get a brief overview of properties of circle with a radius point in the centre. This is their introduction so they will learn to draw trigonometric graphs as time goes by. The main aim for doing this is to allow their own individual participation in determining conjectures for completing drawing and labels required in the unit circle as well as in the graph of a trig function. In order to manage this we have split the class into two separate groups with the one dealing with unit circle and special angles while the other one is dealing with special angles and the graphing of trigonometric function graphs. The circle has radius that makes up diameter. Diameter divides the circle into two equal halves namely semi- circles. A radian is defined as an angle subtended at the center of a circle for θ which the arc length is equal to the radius of that circle (see Fig.1).
θ
Fig. 1 radian angle
θ
Fig.2 The circle
Figure 3
Figure 4
The circumference of a circle is divided into four 2
π R, where R is radius of the circle.
Consequently, 360°=2π radians. (Remember circumference of a circle is perimeter of a circle: distance surround the whole circle in units). Therefore one radian = 3600/360) radians ≈
0.01745 radians
The Unit Circle:In mathematics, a unit circle is defined as a circle with a radius of 1. Often, especially in applications to trigonometry, the unit circle is centered at the origin (0, 0) in the coordinate plane. The general equation of the unit circle in the coordinate plane is x 2 + y2 = 1. Therefore the unit circle is taken to
be 360°, or 2π radians. We can divide the coordinate plane, and therefore, the unit circle, into 4 quadrants. The first quadrant is defined in terms of coordinates by x>0, y>0, or, in terms of angles, by 0°<θ<90°, or 0<θ<π/2. The second quadrant is defined by x<0, y>0, or 90°<θ<180°, or π/2<θ<π. The third quadrant is defined by x<0, y<0, or 180°<θ<270°, or π<θ<3π/2. Finally, the fourth quadrant is defined by x>0, y<0, or 270°<θ<360°, or 3π/2<θ<2π. Development: Trigonometric function: Using their instruments students must design a circle with a radius equal to I cm units. They must convert the scale so that 1 cm equal to 50 mm. This is done to avoid the elative small figures getting constricted in the small scale, since we need to label circle all right round. In order to achieve this students must move in an anti-clockwise direction of a compass. Protractor is used to measure angles. For the sake of time we shall force learners to use special angles only, (i.e. 30, 45, 60 and consecutive angles of 2nd, 3rd and 4th quadrant). Precautions: i)
Do not hold a compass point to point towards anyone or your eyes.
ii)
Only one student is allowed to handle a compass.
iii)
Do not slip your finger under a paper.
iv)
When finished put instruments inside their box neatly.
Guidelines and explanations:
Arc length (s)
r 1
r
θ
2r
r
r 1 1
r 2r
Figure B
Steps to be followed: 1. Measure the angle in degrees. Try to measure. Start by measuring the angle of 360 degrees. This is called rotation angle or complete revolution. The arc shown has a length chosen to equal the radius; the angle is then 1 radian. In the general case, the length s, of an arbitrary arc, which subtends an angle θ , is r θ . The arc length s, is given by r × θ . Therefore arc
length s = r θ .
We know that the arc length for a full circle is the same as its circumference, 2r. In other words the perimeter of a circle is given by its circumference = 2 π r. Since the arc length = r θ . To determine the equivalent angles in degrees and in radians: hint equate the length of a full circle the arch length. ∴
2 π r=r θ
Divide by r both sides: therefore θ = 2 π In other words, when we are working in radians, the angle in a full circle is 2 π radians, in other words. 360◦ = 2 π radians
Classroom activity: 1 a) Test, evaluate and comment on the following radians in the full circle. : 360 0, 1800, 900, 450, 600, 300, b) Indicate one point of actual radian value. c) Indicate these points in the full circle. d) Evaluate cosine, sine and tan functions where radius is given by 1cm units (remember a scale is 1cm units = 50 mm). ( use the table method but … do not complete all values jump with few figures that follow each other e.g. : Degrees Radian angles Y= tanx
30 π 6
45 π 4
60 π 3
90 π 2
120 2π 3
135 3π 4
150 5π 6
180 π
So complete the function of y = tanx and plot the graph using radian method (for assessors: see memorandum) NB you may start from -360 to 360 or from 360 to 90 to minimize time.
NB):- Various conventions are used to denote radians. Some books use rads as in 2 rads. Some books use ‘c’ as in 2
c
.
Some others use no symbol at all and assume that radians are being
π used. When an
angle is expressed as a multiple of π , for example as in the expression sin
3π , it is 2
taken as read that the angle is being measured in radians.
e) Use your special angles to calculate draw a graph of y= tan x. (and where possible complete other graphs of sin- and cos x at your own convenient time). f) Express your amplitude and period in terms of radian measurements ( hint copy values from the unit circle and plot them where they belong to e.g. 180 degrees =
π )
Classroom Activity: 2
a) Produce a circle with a radius of a scale of 2cm (with a scale of 1cm = 30 mm to suite a drawing). Produce the following angles and label their unit measurements as well as their co-ordinate points: (30, 45, 60, 90, 120, 135, 150, 180, 2010, 225, 240, 270, 300, 315, 330 and 360) degrees. b) Use the very same angles to evaluate the values of y= sin x, y= tax and y= cos x on the same axes. Do not exclude special angles where Pythagoras application is made for sin, cos and tan function (but it not a serious threat not to include it in a lesson plan because it will be taught together with quadrants in the introduction). E.g. sin
θ =
opposite hypotenuse
(the next lesson
plan will deal with this in advance). c) Determine amplitude, d) Determine a period of each graph. Planned questions: What is radius? What is an arch? What are quadrants? List various types of angles that could develop within a circle What is trigonometric function? Make a right angled triangles and read sin θ , cos
θ and tan θ using their x, y and r values
Sin θ Cos
θ=¿
Tan θ Cos2i θ
= + sin2
θ=¿ ?
Complete the following: a) X2 + Y2 = r2 =?
(Hint evaluate x and y values: where oh 2).
Consolidation / Lesson Conclusion The trigonometric function are function of an angle because they relate the angle of a triangle to the tenths of its sides. Studying triangles of functions and their hypotenuse is becoming important for trigonometric graph drawing and periodic modelling. For example it is easy to demonstrate a trigonometric graph function using the special angles because they overlap very smooth with amplitude, range and period of a graph. In a plain circle we only know one complete revolution to be a complete revolution of angle that is subtended by angles in the circumference of circles. The trigonometric function tare sin, cosine and tangent functions. Unit circle is a circle that is made with a
line starting from the origin of the centre to the circumference. The unit circle has four main important points known as quadrants, similar Cartesian plane. These four main points occupy fixed angle s including o0, 900, 1800, 2700 and 3600. These four main important points are used together with radian angle to base and support development of special angles surround the point. The radian angle is symbolised by different positions in relations to the point of an angle at centre. For example radian = 180 degrees,
π 2
π
= 90 degrees and 360degrees = 2
π
. We make use of these few basic
angles together with special angles to examine the function of tangent, cosine and sin angles. For example, the tangent function is always increasing (goes up as we move to the right). Like all other graphs a function of a tangent has asymptote, amplitude, range and sometimes wavelength and period. . It is not difficult to deal with a function of a tan graph on a simple unit circle. The function takes the value −1 when x = −
/4 (−45°) and 0 when x = 0. It then takes the value 1 in x =
rapidly as x gets near to the value x = itself at intervals of
/4 (45°) and rises
/2 (90°) where there is another asymptote. This curve repeats
or 180° which is the period of the function. The degree of an arch is
measurement of an angle defined so that a full rotation is equal to 360 degrees. This implies that one full rotation of 360 degree is equivalent to 2 π
. In addition to that one degree is equivalent to
π . The CAST diagram is used to separate four (4) quadrants. Then in a Cartesian plane we 180 make use of a right angles triangle constructed from set of parallel line produced against the point at circumference of radian and angle at centre to express cosine, sin and tan angles. The fact that sin2 θ + cos2 θ = 1, is a true reflection of the power of a radius when it is drawn against its radian and right angle in the circle produced. 1) X2 +Y2 = r2 Theorem of Pythagoras 2)
Divide all sides by a radius: (r2) this becomes :-
3) Therefore Cos2 θ +Sin2
θ =1
(
x r
)2 + (
y r
r )2 = r )2 ¿
The end
References: Daria E, (2015) ‘Trigonometric Functions: Derivatives of Trigonometric and Inverse Trigonometric Functions’, (nd), GNU Free Documentation License, Version 1.2 Nicholas, J. and Adamson, P. (2006) ‘Introduction to Trigonometric Functions: Mathematics Learning Centre’, University of Sydney NSW 2006 pp( 01- 26)
The memorandum: Equivalent angles in degrees and in radians:2 π r=r θ
θ
=2 π
3600 = 2 π radians
∴
180 degrees =
∴
90 degrees =
∴
45 degrees =
π
π 2 π 4
radians
radians
radians
∴ 60 degrees =
π 3
radians
∴ 30 degrees =
π 6
radians
It is not at the interest of grade 10 syllabus to determine the length of an arc, when the angle is given in degrees. Therefore, learners would skip that exercise, begin the function of sine, cos, and tan function. (MC-TY-Radians -2009-1).
Classroom activity 1: answers.
a) 360 degrees lies in the common point as 0 degrees. 180 degrees is equal to a value of a pie of a circle because 2πr = rθ. Hence the formula for calculating all values of a unit circle in terms of radian angle is derived from 2πr = rθ. We have to cross multiply to get any remaining formula. The answers are also indicated in the unit circle above. Students must highlight that special angles differ across different tangent: they are reflecting CAST diagram. It is a subject to an assessor to get what is stated correctly.
b) Since 180 degrees = π radians Or π radians = 180 degrees π radians π
=
180 degrees π
Radian = 57, 2960. c) See all points in the graph.
d) See answers below ( hint use cross multiplication as illustrated in the class )
Degrees Radian angles Y= tanx
30 π 6 1 √3
45 π 4
60 π 3
90 π 2
√3
1
120 2π 3
135 3π 4
√3
Undefine
150 5π 6 1 √3
-1
d
180 π 0
The graph of y= tan x
Y= tan x
Classroom Activity 2: answers Three graphs all together: sin- , cos- and tan- x
Degrees y=six
y= cosx
y=tanx
30
45
60
90
1 2
1 √3
√ 31 2
√3 2
√2 2
√ 30 2
1 1 √3
√ 3 U/ D
120
√3 2 -
13
18
5
0
210
225
27
300
315
330
0
√2 0 2
−1 2
−1 1 √2
−√ 32
1 2 −√ 3 −1 0 1 2
240
1 1 √3
−√ 2 −√ 3 -1 2 2 -
1 √2
0
1 2 −√ 3 UD 1
−√ 3 −1 2 √2 1 2 −√ 3 -1
1 √2
36
0
0 0
0
1 2 √3 1 2 −1 0 √3
1
0
The graph of y= sin x, y= cosx and y= tanx in terms of radian angles.
Y= sin x Y= cos x Y= tan x
e) answer differ as students are confused , some come with sin graph , some come with tan , while other dealt with cos x graph . But here were are looking for a period of tan = Ď&#x20AC;
. The clue was also given to students so they ought to mention that.
f) A period is
Ď&#x20AC; , amplitude is 1 since the radius of a graph is 1 units cm. Scale and
magnitude does not count.