Download 4º eso – e. unit 7: trigonometry

CURSO ACADÉMICO 2011/12
UNIDAD DIDÁCTICA DNL
“TRIGONOMETRY”
MATEMÁTICAS 4º E.S.O.-OPCIÓN B
PROFESOR: Juan Luis Fernández López
1. INTRODUCTION
Trigonometry is a branch of Mathematics that studies triangles and the relationships
between their sides and the angles between these sides.
This branch of Mathematics has lots of applications consisting of the measure of
inaccessible objects such as mountains, buildings,…
We will study this unit with students from 4ºESO during eight sessions.
The previous knowledge involves similarity and sexagesimal system.
2. OBJECTIVES

Identify what trigonometric ratio you have to use depending on the situation.

Apply trigonometric ratios in order to solve right-angled triangles.

Use the protractor in order to measure angles.

Use the calculator in order to get the angle that corresponds to a trigonometric
ratio.

Represent situations related to real problems.

Apply trigonometry in order to solve real problems involving the measure of heights
and distances.

Master the trigonometric vocabulary and use it to explain the related situations.
3. CONTENTS
S1: TRIGONOMETRIC RATIOS OF AN ACUTE ANGLE
Let’s consider  an acute angle. We can draw a right-angled triangle in such a
way that  is one of its angles.
The hypotenuse is the side opposite to the 90º degree angle in a right-angled
triangle. The adjacent cathetus is the other side that is adjacent to the angle  . The
opposite cathetus is the side that is opposite to the angle  .
The trigonometric ratios are:



opposite cathetus
hypotenuse
adjacent cathetus
Cosine of  =
hypotenuse
opposite cathetus
Tangent of  =
adjacent cathetus
Sine of  =
BC
AB
AC
cos  
AB
BC
tan  
AC
sin  
Note that these definitions don’t depend on the triangle measures because all
right-angled triangles being  one of its acute angles are similar and so, the ratios are
equal.
S2: STANDARD TRIGONOMETRIC IDENTITIES
Certain equations involving trigonometric functions are true for all angles and
are known as standard trigonometric identities.
The following trigonometric identities prove that trigonometric ratios of an angle
 are not independent. If you know the value of one of the ratios, you can calculate
the value of the others.
These are:
sin    cos   1
2
2
and
tan  
sin 
cos
Let us prove these identities. Look at the triangle on the previous page and
apply Pythagoras’ theorem. Then, divide the whole expression by the squared
hypotenuse and you will get the first identity.
AC 2 BC 2
AC  BC  AB 

1
AB 2 AB 2
2
2
2
2
2
 AC   BC 
2
2

 
  1  cos    sin    1
 AB   AB 
tan  
BC
BC
AB  sin 

AC
AC
cos 
AB
Now, we are going to calculate the value of the trigonometric ratios of the most
common angles, i.e. 30º, 45º and 60º.
Let us consider the following triangles:
Looking at the first triangle we get:

sin 
cos 
tan 
30º
1
2
3
2
3
3
45º
2
2
2
2
1
60º
3
2
1
2
3
S3: USING THE CALCULATOR
When dealing with angles different from 30º, 45º and 60º we will be forced to
use the calculator.
The calculator is also useful if, given a trigonometric ratio, we want to calculate
the angle.
S4: SOLVING RIGHT-ANGLED TRIANGLES
Solving a right-angled triangle consists of, given some data (sides or angles),
calculate the rest of sides or angles.
Let us consider two different cases:
A. Given one side and an acute angle.
1. Calculate the value of the other acute angle using the fact that both
acute angles in a right-angled triangle add up 90º.
2. Write a fraction with one of the unknown sides and the known one
3. Name the fraction as a trigonometric ratio of the known angle.
4. Isolate the unknown side.
5. Repeat 2-5 twice.
B. Given two sides.
1. Calculate the other side using Pythagoras’ theorem.
2. Each of the acute angles must be calculated using the trigonometric ratio
that relates the known sides.
S5: SOLVING OBLIQUE TRIANGLES
An oblique triangle is a one that contains no right angles. See that you can
turn any oblique triangle into two right-angled triangles by drawing in an altitude:
The “altitude strategy” consists of drawing the suitable altitude in such a way
that the resulting right-angled triangles can be solved separately or together.
If they cannot be solved separately, you will have to relate the results by means of
systems of equations.
S6: EXTENDING TRIGONOMETRIC RATIOS
The goniometric circumference is the one whose centre is at the origin of
coordinates and whose radius is of 1. We can represent any angle in such a way that:



Its vertex is at the centre of the circumference.
One of its sides is the positive x-semi-axis.
The other side is placed where it corresponds and it opens anticlockwise.
In this case, the trigonometric ratios are the lengths of the sides of a related
right-angled triangle. The sign of them depends on the quadrant where the angle is
situated.
sin   AB , cos  OB
and tan   ST .
Note that the trigonometric ratios of an angle greater than 360º are equal to
those of an angle less than 360º (first turn).
E.g: 765º = 2·360º +45º. So ratios of 765º are equal to those of 45º.
4. SESSIONS AND ACTIVITIES
We will spend on studying this lesson 8 sessions. Five of them will be devoted
to explaining the related theory and during the rest we will apply the theory in the
solution of real problems like the ones bellow.
An angle of elevation refers to the acute angle a line makes with a horizontal
line, when measured above the horizontal.
An angle of depression refers to the acute angle a line makes with a
horizontal line, when measured below the horizontal.
1. Michael, whose eyes are six feet off the ground, is standing 36 feet away from the
base of a building, and he looks up at a 50° angle of elevation to a point on the
edge of building’s roof. To the nearest foot, how tall is the building?
2. A pilot is travelling at a height of 35,000 feet above level ground. According to her
GPS, she is 40 miles away from the airport runway, as measured along the
ground.
At what angle of depression will she need to look down to spot the runway ahead?
(1 mile = 5280 ft)
3. A pilot is travelling at a height of 30,000 feet above level ground. He looks down at
an angle of depression of 6° and spots the runway. As measured along the
ground, how many miles away is he from the runway?
4. A dog, who is 8 meters from the base of a tree, spots a squirrel in the tree at an
angle of elevation of 40°. What is the direct-line distance between the dog and the
squirrel?
5. A ship is on the surface of the water, and its radar detects a submarine at a
distance of 238 feet, at an angle of depression of 23°. How deep underwater is the
submarine?
6. The sun is at an angle of elevation of 58°. A tree casts a shadow 20 meters long
on the ground. How tall is the tree?
7. Two observers on the ground are looking up at the top of the same tree from two
different points on the horizontal ground. The first observer, who is 83 feet away
from the base of the tree, looks up at an angle of elevation of 58°. The second
observer is standing only 46 feet from the base of the tree. (Note: you may ignore
the heights of the observers and assume their measurements are made directly
from the ground.)
a) How tall is the tree, to the nearest foot?
b) At what angle of elevation must the second observer look up to see the top
of the tree?
8. The arms of a compass are 12 cm long and form an angle of 50º. What’s the
radius of the circumference that can be drawn?
9. Calculate the angle of elevation of a
road where the following signal has
been installed.
10. An observer on the ground looks up to the top of a building at an angle of elevation
of 30°. After moving 50 feet closer, the angle of elevation is now 40°. Determine
the value of x and the height of the building.
11. A person starts walking 17 miles far from the base of a tall mountain, and looks up
at a 4° angle of elevation to the top of the mountain. When they move 12 miles
closer to the base of the mountain, what will be their angle of elevation when they
look to the top? Answer to the nearest degree.
12. A pilot maintains an altitude of 25,000 feet over level ground. The pilot observes a
crater on the ground at an angle of depression of 5°. If the plane continues for 16
more miles, what will be the angle of depression to the crater? Answer to the
nearest degree.
13. Two observers (located at points A and B in the diagram) are watching a climber
on the opposite face of a chasm. The chasm is 81 feet wide. When observer A
looks down to the bottom of the opposite wall of the chasm, he must look down at
an angle of depression of 51°. However, observer A sees the climber at an angle
of depression of 20°. Observer B will see the climber at what angle of elevation?
14. Calculate the width of the river.
15. Calculate the distances d and D.
5. EDUCATIONAL RESOURCES
At the beginning of the lesson, the students are provided with the following
material:

A sheet with all the theory. On this sheet, concepts are perfectly
explained.

A worksheet with activities that students have to do at home. These
activities are checked by the teacher once they have been worked by
the students.
6. ASSESMENT
ASSESMENT CRITERIA

Calculate trigonometric ratios of angles less than 90 degrees.

Represent real situations by means of geometrical shapes.

Be able to use the calculator in order to calculate angles and
trigonometric ratios.

Solve word problems involving the measure of a distance by means of
trigonometric ratios.

Describe the process and use the specific vocabulary in order to explain
the situation.

Understand simple texts about trigonometry and its history.

Participate in conversations about trigonometry.
Students will be assessed by means of the everyday observation and a final
test like the one bellow.
The test represent 70% of the total mark and the everyday situation the
remaining 30%.
Name:
23/03/12
4º E.S.O. – E. UNIT 7: TRIGONOMETRY
3
. You cannot use
4
your calculator and results must be given as a rationalized fraction.
Use your calculator to get  . Express the result in degrees, minutes and
seconds.
1. (1.5 p) Calculate cos  and tan  knowing that sin  
2. (2 p) Get the volume of a cone trunk if its altitude is 10 cm and the radius of

its bases measure 6 cm and 21 cm. Hint: VCONE  R 2 h .
3
3. (1.5 p) Calculate (answer to the nearest tenth):The distance from observer A
to the top of the building.
a. The height of the building.
b. The distance from the observer and the top of the building.
You are not allowed to use Pythagoras’ Theorem.
4. (2 p) Rachel and Joey want to measure the height of a statue. They decide
to stand at both sides of it. Rachel looks up to the top of the statue at an
angle of elevation of 50 º while Joey looks up to the top at angle of elevation
of 40º. Fill the sketch and calculate the height of the statue if the distance
between Rachel and Joey is 15 metres.