Trigonometry

Trigonometry

Introduction
An important tool that is used in, for instance, criminology to investigate crimes at crime scenes, is trigonometry. The functions of trigonometry are helpful in estimating what is behind a collision in a car accident, how an object falls down
from somewhere, or the angle at which a bullet is shot. Trigonometry is also used in
navigation to pinpoint a location. It helps to estimate the direction in which to place
the compass so as to get a direct route to a particular destination.
For you to appreciate the different real-life applications of trigonometry, you need
to understand how to use the three basic trigonometric functions.

Trigonometric Functions
Activity 2.0: Understanding the trigonometric functions
Suggested materials: sticks, straws, sellotape, strings
Instructions:
(a)Explore your school environment and identify materials which you can use to make a model of a right-angled triangle.
b)Make a model of a right-angled triangle.
c)Identify and label all the angles and sides on your model.
d)From your model, identify the side of the triangle which is opposite the right angle. What name can you give to this side?
e)Describe how each angle relates with the sides of the triangle.
Explain your observations to the rest of the class.

2.1 Deriving Sine, Cosine and Tangent Functions from the Unit Circle
A unit circle is a circle of radius one unit. Using this unit circle, you will be guided on
how to derive the three basic trigonometric functions.
Deriving the sine function from the unit circle
Activity 2.1 (a) (Work in groups)
Suggested materials: a graph paper, a protractor, a pencil
Instructions:
(a) Draw a circle of radius (r) one unit (5 cm), centred at the origin on the Cartesian plane.
(b) How many quarters can you observe from your circle?
(c) Mark a reference point, A, on the circumference of the circle.
(d) Rotate point A through angle 0 = 00 about the origin.
(e) Measure and record the corresponding y-value.
(f) Repeat procedures (d)-(e) for the different angle values shown in the table.

Point A will rotate at different angles on the circumference, like 300 and 600
Deriving the cosine function from the unit circle
Activity 2.1 (b)
(Work in groups)
Suggested materials: a graph paper, a protractor, a pencil
Instructions:
(a)Draw a circle of radius (r) one unit (5 cm), centred at the origin on the Cartesian plane.
(b)How many quarters can you observe from your circle?
(c)Mark a reference point, A, on the circumference of the circle.
(d)Rotate point A through 9 = 00 about the origin.
(e)Measure and record the corresponding x value.
(f)Repeat procedures (bHe) for different angle values shown in the table.

(h) Deduce the expression for cos d
(i) Present your findings to the rest of the class.
Deriving the tangent function from the unit circle Activity 2.1 (c) (Work in groups)
(a) Use the values of x and y obtained in activities 2.1(a) and 2.1(b) to complete the following table.

2.2 Reading and Using Calculators to Find Values of Trigonometric Functions
The scientific calculator can be used to find values of trigonometric ratios, given the angle(s).
Activity 2.2(a)

(a)Identify any scientific calculator in your class.
(b)Compare the scientific calculator in your class with the scientific calculator model shown.
(c)How many trigonometric function buttons are on your calculator?

Using calculators to find the value of the sine function, give an angle
Activity 2.2(b) (Work in groups)
(a) Use your calculator(s) to find the following values of the sine function.
i) sin 00
ii) sin 150
iii)sin 300
iv) sin 450
v) sin 600
vi) sin -150
vii) sin -1800
viii) sin -450
ix) sin -1200
x) sin -300
xi) sin 900
xii) sin 1800
(b) Study and comment on your outputs.

Using calculators to find the value of the cosine function, given an angle
(Work in groups) Activity 2.2(c)

(a) Use your calculator (s) to find the following values of the cosine function.
i) cos 00
iv) cos 450
v) cos 600
ii) cos 150
vi) cos 900
iii) cos 300
(b) Study and comment on your outputs.
vii) cos 1800
viii) cos 2700
ix) cos -300
x) cos -2700
xi) cos 3600
Using calculators to find the value of the tangent function, given an angle
Activity 2.2(d) (Work in groups)
(a) Use your calculator(s) to find the following values of the tangent function.
ix) tan 3600
vii) tan -1800
v) tan 600
iii) tan 300
i) tan 00
x) tan 900
viii) tan -450
vi) tan -150
iv) tan 450
ii) tan 150
(b) Study and comment on your outputs.
Using calculators to find the values of inverse trigonometric functions
Angles will be obtained from any of the given angle’s trigonometric ratio.

Activity 2.2(e) (Work in groups)

(a) Identify on your scientific calculator(s) the inverse trigonometric function buttons.
(b) Use them to find the values of the angles whose trigonometric ratios are given.
i) (0.9347) ii) sin 9=0.25 iii) cos e = 0.8665 iv) tan 9=0.7524
2.3 Using Sine, Cosine and Tangent Functions in Calculating
lengths of Sides and Angles of Right-angled Triangles
Triangles can be solved with the help of trigonometric functions, since the sines, cosines, and tangents are related to the lengths of sides and angles of the right-angled triangle.
Activity 2.3 (a) (Work in groups)
(a) Identify any set square from a mathematical set.
(b)Trace out the set square on a piece of paper.
(b)Identify the polygon formed.
(c)Use trigonometric functions to determine all the interior angles of the polygon formed.
(e)Share your findings with the rest of the class.

Activity 2.3 (b)
(a) Identify any other set square from the mathematical set.
(b) Trace out the set square on a piece of paper.
(c) Measure all the interior angles and the length of any one side of the polygon formed.
(d) Use trigonometric functions to determine the lengths of the other sides of the polygon.
(e) Compare your results with others’ in the class.

Project Work:
As a Senior Three learner, you are required to make a wall clock craft model for science exhibition in your district.
(a) Identify the materials which you will use to make the wall clock craft model.
(b) Explain to the chief guest how she / he can tell the angle between the arms.

2.4 Finding Angles of Elevation and Depression Activity 2.4(a) (Work in groups)
AB and CD are two vertical poles on a horizontal ground. In groups, copy the diagram on a piece of paper and draw on it the angle of elevation of D from B and the angle of depression of C from B. c

Activity 2.4(b) (Work in groups)
(a)Trace out any set square in your mathematical set.
(b)Complete the shape in (a) to form a rectangle or a square.
(c)Identify and label the angles of elevation and depression in the rectangle or square in (b).
(d)Find the size of each angle.
(e)Describe where angles of elevation and depression can be identified in real-life situations in your society.

Exercise 2.3
1) From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression
of 400. If the tower is 45 feet high, how far is the partner from the base of the tower, to the nearest
tenth of a foot?

2) Find the length of the shadow cast by a 10-feet lamp post when the angle of elevation of the sun is 580.
3) The distance of a car from the foot of a building is 46 m. If the building is 65 m high, find the angle of depression of the car.
4) Aman is 170 cm tall and his shadow is 320 cm long. Find the angle of elevation to the sun.
5) The angle of elevation of a water tank from a man of height 1.8 m is 350• If he is 6 m from the base of
6)the water tank, how far is the to of the tank from the ground The angles of elevation from points
D and C, on the ground level, to the top of a flag pole are 300 and 600, respectively. Points D and C are on the same side of the flag pole. If the distance between D and C is 20 m, find the height of the flag pole. The figure below shows the end wall of a classroom. Calculate the height of A from the floor.
280
4.2

ICT Activity In groups:
(a) Having learnt trigonometry, find out how this topic is useful in real life
(b) Using Microsoft PowerPoint or any other presentation software, prepare a brief report on your findings and present it to the whole class.
Revision Questions:
1) The angles of elevation of an artificial earth satellite, measured from two earth stations which are situated on the same side of the satellite, are found to be 300 and 600, respectively. The two earth stations and the satellite are in the same vertical plane. If the distance between the earth stations is 4,000 krn, find the vertical distance between the satellite and the earth. (Take V 3 = 1.7321)
2) The angles of depression from the top of a tower of height 60 m to the top and
bottom of a building are 300 and 600, respectively. How high is the building? ‘