Name of School:
ROUTINE INFORMATION Inhlakanipho High School.
Student Surname and Name:
Maphumulo Mthokozisi
Student Number:
207503878
Grade:
10 and 11
Subject:
Mathematics
Topic:
Cosine Rule Number 1
Content /Concept Area:
Pythagoras Theorem, distance and interior
CAPS page number:
angles and Trigonometry. 13
Duration of Lesson:
45 minutes
Specific Aims: To develop fluency in computation skills using Geo-gebra and Desmos. To determine the general solution and / or specific solutions of trigonometric equations. To establish the sine, cosine and area rules To learners the ability to be methodical, to generalize, make conjectures and try to justify or prove geometry theory using their own deductive reasoning. To develop problem-solving and cognitive skills in geometry, trigonometry and algebra. Lesson Objectives:
Knowledge
Learners should gain
Skills Learners should be able to do
Value /Attitude Learners should acquire
knowledge and
following :
values and attitudes conducive to :
understanding of the following : Identify hypotenuse from the
Analyse, describe and solve for
Recognise
right angled tringle.
missing
properties
that
possible determine the value of
themselves
to
interior angles and size of the
trigonometry
heights and sides.
relations.
Explain Cosine rule
distance
where
geometric lends algebra,
and
cosine
Draw, illustrate, interpret and analyse cosine rule.
Acknowledge the development conjectures
for
establishing
calculations of sides , angles and
heights
using
the
application of sine rule Observe shapes , angles and
Determine the value of missing
drafts of Cosine rule
angles and sides from the
Acknowledge the integration
cosine rule
of cosine rule to applied maths involving trigonometry
algebra, and
cognitive
conjectures that are said to have developed from their previous grades.
Approaches / Teaching Strategies:
Role plays.
Case studies Class discussions. Questioning. Whole class-discussion Group Work Resources:
Textbook Calculators Chalkboard Interactive chart Worksheet
Lesson Phases:
Introduction:
During the introduction students are questioned to explain what Pythagoras’s theorem entails. The teacher is role playing through showing extra-large triangles and rulers and briefly outlie the importance of Cartesian plane and special angles. Students are expected to ask questions or respond from the challenges that will be asked concerning cosine rule. Students would also be motivated to try using Geo-gebra for their drawing to see if there is correlation between interior angles of triangles with sides. Basically cosine rule came about presenting solutions to problems for determining the size of angles of the triangle, the size of the sides in length and even areas. There are two forms of cosine rules used to determine sides, angles and areas. The first one is concerned with determining the value of sides of triangle where the right angle is missing. This one may be associated with scalene triangle, where all three sides are not equal and also all angles are not equal. In that type of scalene triangle there is no interior angle which is equal to 90 degree. (I other words it is impossible to make use of the Theorem of Pythagoras to determine the value of the longest side which is often known as hypotenuse. Mathemacian kept on questioning where the limitations for determining the value of the longest sides in the scalene triangle whereas it was possible to determine it in the right angled triangle. Cosine rule derive some beautiful calculations from the very same Theorem of Pythagoras which is used to determine the values of the longest side in the right angle triangles. Student often found themselves battling with the application of cosine rule because they have poor history of its development. The following is a picturesque its development.
The cosine rule of ∆ ABC is given by: a2 = b2 + c2 – 2bc Cos A. b2 = a2 + c2- 2ac Cos B. c2 = b2 + a2 – 2ba Cos C. Please notice that one can be able to determine the value of angles (interior angles of triangle) if all sides are given. Some students may decide to make Cos A the subject of a formula.
Therefore Cos A =
b2 +c 2−a 2 2bc
You should have question yourself why we are talking about ‘A’ and ‘a’ as if a triangle offer the value of a. This needs a careful development of ‘a’ on the triangle on which we shall be able to measure their sides and give their own interpretation. Could you be able to differentiate from ‘A’ and ‘a’, briefly outline: Development: Long ago, mathematician adopted a style for arranging alphabetic letters according to their order. This is very useful in pure applied mathematics because the arrangement of letters help us to develop meaningful observation during our theoretical development. In any triangle ABC small letter ‘a’ refers to the side which is opposite to the angle given by CosA. Similarly small letter ‘b’ refers to the side that is opposite to the angle Cos B Similarly small letter ‘c’ refers to the side that stands opposite to the angle Cos C
If you want to reflect carefully on this matter redraw the above triangle and place it in the Cartesian plane so that ‘a’, ‘b’, ‘c’ are taking their own sides to stand opposite to their angles as describe above. Therefore: in any triangle ABC, ‘a’ = side BC, opposite to angle A. (hint it the angle of its letter) Similarly: in any triangle ABC, ‘b’ = side AC, opposite to angle B. (hint it the angle of its letter) Similarly: in any triangle ABC, ‘c’ = side AB, opposite to angle C. letter and the angle of its letter).
(hint its own Capital
Required to prove Cosine rule which state that: a2 = b2 + c2 – 2bc Cos A. Steps to be followed: Step 1: you must begin by producing a line from the base to the apex angle, moving vertically and not horizontally. You want to make the height to remain perpendicular to the base line AB.
X
C-x AB = c
Step 2: Recall that side AB = c of a small letter. Purposefully name or let AD = X. Now determine line DB in terms of X. African students often battle with this concept because normally they are not used to start by questioning a value of something from the vacuum. Inform the clearly and leave no any signs of ambiguity. Line AD + Line DB = Line AB Given Line AD = X, determine DB in terms of X X+ DB = AB
- remember that AB = c (small capital letter).
Therefore: X + DB = c Make DB the subject of a formula or simple solve for DB. Therefore: X + DB = c : X –X + DB = c –X 0+ DB = c-X DB = c-X
Step 3: Replace the new values of AD and DB in the triangle as they are … do not change anything.
Pause and ask if student would predict how Pythagoras would Theorem is going to be used in this classroom exercise. It is lawful to give student their own momentum as a teacher. This make them to remain accountable to what you are teaching them. A teacher must not move alone, where possible give them a chance to illustrate a Theorem of Pythagoras. Step 4: Basically Pythagoras’s Theorem state that the square of the length of the right angled triangle is equal to two sides standing against the right angle. There are limited cases where you will see one remind you to make use of Cosine rule to calculate the length of the longest side whose angles interior angles opposite to the longest side does not count to 90 degrees. But there are many chances that you will be required to solve for sides, angles and area calculations using Cosine rule(s). Therefore learning this theory appropriately develops learner’s independent cognitive thinking for being able to determine missing sides, angles and area values relying on cosine rule.
But the question is why we often state that: a2 = b2 + c2 – 2bc Cos A. Proof of the Cosine Rule:
X
C-x
AB = c
In ∆
CAD
Cos A ¿
X b
, therefore x = b Cos A ------------------ ( 1st statement) or just (1)
And b2 = x2 + h2 Make h2 the subject of a formula: implies h2 = b2 - x2 h2 = b2 - x2
----------------------(2nd statement) or just (2)
In ∆ BCD : And a2 = (c-x)2 + h2 = (c-x)(c-x) + h2 = c2 -cx-cx + x2 + h2 = c2 -2cx + x2 + h2
Substitute (1) and (2) into (3)
------------------------------ (
3rd statement) or just (3)
--------------------- simultaneously
a2 = c2 -2cx + x2 + h2
Make no mistake here...Students are clueless … you will need to start afresh, you will rarely get students create a cognition of what you are substituting without telling them about trigonometry and Pythagoras theorem. Be patient, role play and demonstrate carefully do not rush into practical exercise. Where possible take an angle of depression and elevation. (Whatever it takes).
You may choose to arrange the above equation into the RTP form (required to prove formula) But you may choose to move ahead as it constitute les significant difference in mathematics. None arranged form goes as follows:
a2 = c2 -2cx + x2 + h2
Non arranged
Arranged
a2 = c2 -2cx + x2 + h2
a2 = h2 + c2 -2cx + x2
substitution : (1) and (2) where (1) is x = b Cos A
substitution (1) and (2) where (1) is x = b Cos A
and where (2) is h2 = b2 - x2
and where (2) is h2 = b2 - x2
for
a2 = c2 -2cx + x2 + h2
Student task: participate, share ideas and take notes.
a2 = c2 -2cx + x2 + h2
Educator
: facilitates and moderate
Caution Role play. NB: seek for x in the position for x and not x2. NB: seek for h2 in position for h2 and not h In a2 = c2 -2cx + x2 + h2 a2 = c2 -2c(bCosA) + x2 + (b2 - x2) = c2 -2bcCosA + x2 + b2 - x2 –group terms = c2 -2bcCosA + b2 a2 = b2 + c2 -2bcCosA
Classwork a) In a2 = b2 + c2 -2bc CosA make Cos A the subject of a formula. (hint solve for CosA) b) In the figure drawn bellow, Line QR represents a proposed channel. ‘Q’ and ‘R’ are visible from a point P. The three points are in the same plane. Question: Given: QR = 100m; and PR = 60 m And angle QPR = 1100. Determine the value of QR? (NB QR is the length of a tunnel).
Planned Questions: What is a scalene triangle? Why it is impossible to determine the length of a longest side of scalene triangle using the Theorem of Pythagoras application. ? (What was solution to this practical problem through mathematician intervention?) What do you think you will get if you calculate Cos A using the Cosine rule
Consolidation and Conclusion: Students must recall the fact that there are triangles without right angles, whose longest side cannot be determined through the use of Pythagoras’s Theorem. There are two main formulae used for Cosine rule but for this exercise we have been dealing with a 2 = b2 + c2 -2bc CosA . This formula is useful for calculation of the sides of a triangle where you are given an angle, the two sides and statement leading you to recall a Cosine rule application. Sometimes this formula is used to determine the angle of a triangle provided you were given all three sides. It is important to present triangle ABC and recall its sides in terms of a, b and c to recall how the Cosine rule works. Revision of algebra and factorization is advisable to students who wish to master this topic .