Trigonometry

TEACHER’S GUIDE

MATHEMATICS

UNIT 1 : TRIGONOMETRY

PART 1 - THE SINE OF AN ANGLE

TARGET GROUP: S 3

BRIEF DESCRIPTION OF THE UNIT:

Error: Reference source not found

TIME REQUIRED: Minimum:40 Mins Maximum:80 Mins

MAIN CONTENT AND CONCEPTS TO EMPHASISE:

By the end of this sub-topic learners should be able to: Identify and name sides of a right – angled triangle. Find the sine of an acute angle Apply sine of an angle in daily life applications

THE TEACHING/LEARNING MATERIALS:

Learners must have covered the following areas Types of angles and triangles Measuring of angles Ratios and fractions TASK 1 Procedure Draw accurately a right angled triangle ABC, where AB = 10 cm, angle BAC = 30o Mark on AB points D, F, and H at a distance of 9cm, 8cm and 7 cm from A and draw perpendicular lines to meet AC at E, G and F respectively C E G F 30o A H F D B Ask the students to measure and record the opposite side and the Hypotenuse for each of the triangles above in the table below: Column 1 Column 2 Column 3 OPPOSITE SIDE HYPOTENUSE OPPOSITE SIDE HYPOTENUSE BC = AC = BC = AC DE = AE = FG AG = HF = AF = (Give your answers for column 3 to 4 decimal places) Ask learners to comment on the results in column 3. Ask learners to open mathematical tables (Natural sines). (ICT designer to insert a page of natural sines 00 t0 900 see ahead) Help the students to see the angle 30o and its corresponding sine (expected answer is 0.5000) Let the learners compare their answers with the exact answer in the mathematical tables. TASK 2 Draw accurately a right angled triangle ABC, where AB = 10cm, produce the side BC and on it mark points D, E and F where BC= 5cm, BD= 6cm, BE= 7cm BF = 9cm F E D C A 10cm B Measure and record the following sides of the triangle Column 1 Column 2 Column 3 Column 4 Opposite side Hypotenuse Opposite side Hypotenuse (4 decimal paces) Angle BC= AC= BC = AC BAC BD= AD= BD = AD BAD BE= AE= BE = AE BAE BF= AF= BF = AF BAF Ask learners to check for the sines of angles in column 4 Use the sine tables to show learners the relationship between column 3 and column 4. Different sines give different angles. Sines decrease. The teacher by this stage should emphasize the formula Sine of an acute angle = opposite side Hypotenuse Help learners to appreciate how to read the trigonometric tables. Use such angles as 30.8o, 40.5o WORKED EXAMPLES Example 1 Calculate the length a in the triangle below 70o a 10 cm Solution Opposite side = a Hypotenuse = 10 cm Using sine x = Opposite side Hypotenuse Sin 700 = a 10 a = 10 sin 700 ( not sin 700 x 10) = 10 x 0.9397 = 9.397 ~ 9.4 cm (1 dp) Cal Example 2 A builder has been asked to repair a ventilator on a house. The ventilator is 5.0m up a vertical wall above the ground. He will need to use a ladder and, for safety, the ladder will need to be placed at an angle of about 600 to the ground. What is the minimum length of ladder he should hire for the job? Solution Ventilator Ladder L 5m 60o Sin 600 = Opposite Hypotenuse Let L = length of the ladder Sin 600 = 5 L L = 5 Sin 600 = 5 0.8666 = 5.77m APPLICATION This topic can be used in areas of Construction engineering Surveying Astronomy Flights Navigation References: School Mathematics Uganda (SMU) Bk 2 Ch.14 Secondary School Mathematics (SSM) BK 3 Ch.7 National Curriculum Development Center (NCDC)

LESSON PLAN Date:….. Class: S.3 Period: 2 No. of Students: . Topic: Trigonometry Sub-topic: The Sine of an Angle Lesson Objectives: (i) Identify and name sides of a right – angled triangle. (ii) Find the sine of an acute angle (iii) Apply sine of an angle in daily life applications Teaching Methods: (i) Guided discovery; (ii) Demonstrations; and (iii) Question and Answer Teaching Aids: Chalk board; Mathematical instruments, mathematical tables, manila papers References: SSM Bk 3 (NCDC), p. 70; SMU Bk 3, (Fountain) p. 65 STEPS TIME CONTENT STUDENTS ACTIVITY TEACHER’S ACTIVITY I (10 Mins) Review Ratios, Angles, measurement of angles Respond as being asked prior knowledge Asks for definition of background terms II (10 Mins) Introduction Description of Sine Write down the description Writes down on the black III (25 Mins) Stimulus activity of measuring different lengths with constant angle Participate after observing demonstration Demonstrates with black board instrument IV (25 Mins) Measuring difference in constant length Participate after observing demonstration Demonstrates with blackboard instruments V (10 Mins) Exercise Conclusion of Lesson Attempt desk work Moves around and give assignments Black Board Plan Date Topic Notes New Words Self-Evaluation:………………………………………………………………. MODEL QUESTION AND MARKING SCHEME QUESTION ANSWER MARK COMMENT a) Calculate the length a in the triangle below 10 cm 700 a Opposite = a Hypotenuse = 10 cm θ =700 Using sin θ = Opp Hyp Sin 700 = a 10 a = 10 sin 700 a = 9.4 cm B1 B1 M1 A1 Identifying sides Formula Substitute /use Accuracy Calculate the angle b in the triangle below b 3 cm 2 cm Opp = 2 cm, hyp = 3cm Sin b = 2 = 0.6667 3 b = 41 .80 B1 M1 A1 Identifying Substitute Accuracy 2) A ladder is placed on the wall of a house at an angle of 600. How long must the ladder be to reach a ventilator 5.0 m up the wall L 5.00m 600 Sin 600 = 5 L L = 5 Sin 600 L = 5 = 5.77m 0.866 B1 B1 M1 B1 A1 Diagram Formula Substitute / use Sin 600 and Accuracy

PART 3 - THE TANGENT OF AN ANGLE

TARGET GROUP: S3

BRIEF DESCRIPTION OF UNIT:

Error: Reference source not found

TIME REQUIRED: Minimum:1 hr Maximum:1¹/₃ hrs

By the end of this sub-topic the Learners should be able to : Define the tangent of an angle Use the formula for the tangent to find tangents of angles Find the relationship between sine, cosine and tangent of an angle Read the tangents of acute angles from tables Use a calculator to find the values of tangents of angles Apply trigonometric ratios to solve daily life problems

MATERIALS REQUIRED

• mathematical sets
• chalkboard construction instruments
• school mathematics e.g. East Africa book 2 and 3
• mathematical tables
• scientific calculator

THE TEACHING/LEARNING MATERIALS:

Learners must have covered the following areas sine and cosine of angles know how to read mathematical tables know how to construct a circle and name the four quadrants construct a right angled triangle TASK 1 Procedure 1 Draw a circle with a radius of 5 cm on the chalkboard as shown in the figure below. y Y P x 2. Using a protractor, measure an angle of 300 from the horizontal line OX, in the anticlockwise direction. Draw the line OP such that angle XOP is 300 3. Measure the side PX and record it in the table below Column 1 Column 2 Column 3 Column 4 Angle size Opposite side (PX) Adjacent side (OX) Ratio PX OX 300 450 600 900 4. Repeat using the angles 450 , 600 and 900. Again measure the lengths PX and OX, then calculate the ratio PX and fill it in the fourth column of the table OX 5. Continue with a few more values of other angles, until you have several ratio values in column 4. 6. Use the results in column 4 to show learners that when the angle column 1 is changed, the ratio in column 4 also changes. Explain why this is so to the learners in respect to changes in length of the sides PX and OX Introduce the ratio, Tangent of an angle = Side opposite to angle Side adjacent to angle 8. Point out to the learners that the values in the fourth column are the tangents of the angles in the first column. 9. Confirm these values and results using tables for tangents in the mathematical tables. WORKED EXAMPLE The distance between two hilltops of the same height is 0.6 km. On one hilltop there is a television mast and on the other hilltop there is a cathedral. A man replacing a bulb on the mast sees the base of the cathedral at an angle of 850. He wants to use this information to estimate the height of the mast? Do the calculation for him. SOLUTION. 850 h mast cathedral 0.6 km Since we do not know the hypotenuse we cannot use the sine or cosine ratio. The only possible ratio is the tangent of the 85o angle By definition of the tangent of an angle = Opposite side to angle Adjacent side to angle Tangent 850 = 0.6 km h h = 0.6 km Tan 850 = 0.6km 11.430 = 0.052 km Help learners to get tangent value from calculators. TASK 2 The relationship between the sine, cosine and tangent of angles. Using a mathematical table fill in the table below: Column 1 Column 2 Column 3 Column 4 Column 5 θ0 sin θ cos θ sin θ cos θ tan θ 350 450 600 57.20 820 Comment on the values in column 4 and column 5 Draw the conclusion that Sin θ = tan θ Cos θ Activity 3: Reading of tables of tangents Read and record the tangents of the following angles tan 100 tan 42.40 tan 450 tan 63.50 Draw the learners’ attention to the change in the whole numbers in bold type in table values tan 63.9 tan 64.3 tan 71.50 tan 71.60 tan 88.80 tan 900

MATERIALS REQUIRED

• mathematical sets
• chalkboard construction instruments
• school mathematics e.g. East Africa book 2 and 3
• mathematical tables
• scientific calculator

USEFUL WEB SITES

www.tpub.com/math1

www.themathlab.com/geometry/section13/trigtest

www.acts.tinet.ie/trig.ratiosforacuteang_668

SCHEME OF WORK

School………………………….. Year……………

Term……………………………

Class S.3

No. Of periods per week: 6 No. of students…………….

Week	No of periods	Topic	Sub-topic (content)	Aims/objectives	Teaching methods	Teaching aids	References	Comments
(Dates)	2	Trigonometry	The sine of an angle	By the end of this subtopic students should be able to: i) name the sides of a right angled triangle ii) find the sine of an acute angle iii) apply sine in daily life situations	Guided discovery Demonstrations Question and answer	Chalk board Mathematical instruments, mathematical tables, manila papers	SSM Bk 3 (NCDC) pg 70 SMU Bk 3 (fountain) pg 65
	2	Trigonometry	The cosine of an angle	By the end of this sub topic students should be able to: i)name he sides of a right angled triangle ii) find the cosine of an acute angle iii) apply the cosine in daily life situations	Guided discovery, Demonstrations, Question and answer	Chalk board Mathematical instruments, Mathematical tables, Manila papers.	SSM Bk 3 (NCDC) pg 70 SM<U Bk 3 (Fountain) pg 70
	2	Trigonometry	The tangent of an angle	By the end of this sub topic the learners should be able to: i)define the tangent of an angle ii) use the formula for the tangent to find tangents of angles iii) find the relationship between sine, cosine and tangent of an angle iv) read the tangents of acute angles from tables. v) use a calculator to find the value of tangents of angles vi) apply trigonometrical ratios to solve daily life problems	Guided discovery Demonstration Question and answer	Chalk board, Mathematical instruments, Mathematical tables Manila, scientific calculators	SSM Bk 3 (NCDC) pg 64 SMU Bk 3 (fountain) pg 70

TRIGONOMETRY

Learners’ Activities

Activity One

Group Activity

1. In a group of 5 people, choose a secretary.
2. Write down names of people you know around the school or in your home area who are applying trigonometry as they do their jobs.

Say, Surveyors: e.g. Mrs. Brown, --------------, -----------, -----------,

Construction engineers: --------------,------------------------

Flight engineers: ------------‘----------

Astronomers: ----------------, ------------, ------------------

3. Agree on which of these persons you should invite to give a talk to your class on a topic

“The use of trigonometry in our daily life”

4. Present the name you have chosen to your teacher such that she/he lists down the chosen speaker for each group

5. As a class you must choose one person one from those listed in No.3. So, one member of your group must convince the whole class that the guest you suggested is the best choice. Prepare a summary of points for the group representative.

1. The whole class has finally chosen the name of the guest speaker that was presented by your group. Write a letter requesting this person to deliver a 2-hour talk about the topic in (3) above on a suitable date you have communicated to your teacher of mathematics.

2. Write another letter that you will present to the Head teacher /principal to allow you host the guest speaker and request him to fund /facilitate this talk. In this letter show clearly how your class will benefit from this talk.