Download Math 10th grade LEARNING OBJECT To solve problems involving

Math 10th grade
LEARNING UNIT
A world of relationships
using triangles!
S/K
LEARNING OBJECT
To solve problems involving the use of oblique
triangles
SKILL 1: Recognize the elements that make up the
scalene triangle.
SKILL 2: Relate the known data with the variable to
be found, using the law of sine and / or law of cosine.
Language
Socio cultural context of
the LO
Curricular axis
Standard competencies
Learning object
Basic Learning
Rights
Background knowledge
Vocabulary box
SKILL 3: Apply the laws of sine and cosine in problem
solving situations.
English
School, classroom
Spatial thinking and geometric systems
Solve and formulate problems using criteria of
congruence and similarity in triangles, never
forgetting to support answers.
To solve problems involving the use of oblique
triangles
Understand and use the law of sines and cosines to
solve math problems and other disciplines involving
non-right triangles
Measuring and comparing angles, functions and
trigonometric ratios, Pythagorean Theorem.
Hobby: Recreational activity. Leisure
Square root: Mathematical process to find the
number multiplied by itself.
Journey: Long path travelled
GPS: Global Positioning System
Aircraft: Aerial means of transportation. Airplane
Kites: Rhombus-shaped flying device.
English Review Topic
Counterweight: Weight used to create an opposite
force.
Prepositions – Place (Position and Direction)
NAME:
GRADE:
Introduction
Photography is a very common hobby that allows people to remember
lived experiences.
Phillipe is on the top of a mountain taking pictures. He can see two
towns: San Antonio and La Fuente (as seen below).
Mountain image
Town image
The distance from San Antonio to La Fuente is 10 Km. Is it possible to
find the altitude of Felipe’s position on the mountain? Can we find the
distance from San Antonio to Felipe’s position? Can we find the
mountain’s angle of inclination on San Antonio’s side?
Objectives




To find the sides and angles of an oblique triangle using the laws of
sine and cosine.
To identify the elements of an oblique triangle.
To recognize the four cases of oblique triangles.
To solve oblique triangles where three measurements are given by
means of the laws of sine or cosine.


To analyze problems using the laws of sine and cosine.
To explain results of calculations based on arguments.
ACTIVITY 1
Skill 1: Recognize the elements that make up the scalene triangle.
OBLIQUE TRIANGLES
Remember that!
To develop the following topic we will use the following notation: A is
the angle formed by sides b and c, B is the angle formed by sides a and
c, and C is the angle formed by sides b and a.
The oblique triangles have no right angles (90°), like right triangles do;
they can be solved using the laws of sines or cosines, as appropriate.
An oblique triangle is composed of three internal angles A, B, C and
three sides a, b, c as illustrated in Figure 3.
Figure. Example of an oblique triangle
Learning Activity
Circle all the oblique triangles.
SINE LAW
𝒂
𝒃
𝒄
=
=
𝒔𝒆𝒏 𝑨 𝒔𝒆𝒏 𝑩 𝒔𝒆𝒏 𝑪
ACTIVITY 2
FOUR CASES OF OBLIQUE TRIANGLES
Remember that a triangle has three sides (L) and three angles (A).
To solve problems involving oblique triangles we must know three
measurements, therefore we will take into account the following
considerations.
Figures.
SINE AND COSINE LAW
Skill 2: Relate the known data with the variable to be found, using the
law of sine and / or law of cosine.
SINE LAW
𝒂
𝒃
𝒄
=
=
𝒔𝒆𝒏 𝑨 𝒔𝒆𝒏 𝑩 𝒔𝒆𝒏 𝑪
Example: Solve for the following oblique triangle
C
b
45 cm
a
84°
35°
A
c
B
Solution:

Calculate b
45
𝑏
= 𝑠𝑒𝑛 84
𝑠𝑒𝑛 35
𝑏=

45∗𝑠𝑒𝑛84
𝑠𝑒𝑛 35
Replace the values of A, a, B in
the sine law equation
Solve for b and calculate the
value
𝑏 = 78.02 cm
Calculate C
𝐴+𝐵+𝐶 =0
35° + 84° + 𝐶 = 180°
35° + 84° + 𝐶 = 180°
Based on this relation…
Replace the values of A and B
Solve for C and calculate the
value
𝐶 = 180° − 35° − 84°
𝐶 = 61°

Calculate c
45
𝑐
𝑠𝑒𝑛 35
𝑏=
= 𝑠𝑒𝑛 61
45∗𝑠𝑒𝑛61
𝑠𝑒𝑛 35
Solve for c
Solve for c and calculate
the value
𝑏 = 68.6 cm
Did you know that...?
If you have the calculator in radians, the result is different than when
the calculator is in degrees. Keep the calculator in degrees when
performing the calculations proposed in the activities.
COSINE LAW
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠 𝐴
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 𝑐𝑜𝑠 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 𝑐𝑜𝑠 𝐶
Example: Solve for the following oblique triangle
Solution:

Calculate a
Replace the values of
A, b, c in the cosine
law equation
𝑎2 = 72 + 132 − 2 ∗ 7 ∗ 13 𝑐𝑜𝑠 30°
Calculate the approximate value
of a2
2
𝑎 ≈ 60
Solve the square root
𝑎 = √60 ≈ 7.7

Calculate the smallest angle, in this case: B
7.7
7
= 𝑠𝑒𝑛 𝐵
𝑠𝑒𝑛 30
𝑠𝑒𝑛 𝐵 =
7∗𝑠𝑒𝑛30
7.7
= 0.45
𝐵 = 𝑠𝑒𝑛−1 (0.45) ≈ 27
Solve for SenB and calculate
Solve for B and
calculate sen-1 0.5

Calculate angle C by adding the angles
𝐴 + 𝐵 + 𝐶 = 180°
𝐶 = 180 − 27 − 30
𝐶 = 123
Replace the values of A
and B, solve for c
Calculate the value
Learning Activity:
1. The following is an oblique triangle
Using the sine law, find the approximate value.
B is __________
𝑎 is __________
c is ___________
2. Match the triangle to the missing values
𝐵 = 40°, 𝑎 = 13.5, 𝑐 = 15.3
𝐵 = 35°, 𝑎 = 25, 𝑐 = 13.8
𝐵 = 71°, 𝑎 = 13, 𝑐 = 20
ACTIVITY 3
APPLICATIONS
Skill 3: Apply the laws of sine and cosine in problem solving situations.
APPLICATIONS OF SINE LAW
Remember that…
We use the sine law when we know one side and two angles (Case 1
SAA), and when we know two sides and an angle (but not the angle
formed by the two sides (Case 2 SSA). Use the cosine law for Case 3
(SAS) and Case 4 SSS.
Example:
A car driving east observes the highest part of an antenna, with an
orientation of 65° E. When the car has traveled 2300 m, the orientation
of the upper part of the antenna is 48° east, as seen in the figure. The
car continued its journey. What is the height of the antenna?
Solution:
Step 1. Use the sum of the interior angles of a triangle to find C
180 = 𝐴 + 𝐵 + 𝐶
𝐶 = 180° − 135 − 25
𝐶 = 20
Step 2. Using the sine law, calculate a (the position of the car from the
top of the antenna).
𝑎
𝑐
=
𝑆𝑒𝑛 𝐴 𝑆𝑒𝑛 𝐶
𝑎=
𝑐 𝑠𝑒𝑛 𝐶 2300 ∗ 𝑠𝑒𝑛20
=
𝑠𝑒𝑛 𝐴
𝑠𝑒𝑛25
𝑎 ≈ 2842
Step 3. To find h, use the right triangle, therefore use the ratio:
𝑠𝑒𝑛 𝐻 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑎
, where h is the opposite side and a the hypotenuse
𝐻 = 180° − 135° = 45°
𝑠𝑒𝑛 45° =
ℎ
2842
ℎ = 2842 ∗ 𝑠𝑒𝑛 45
ℎ = 2010 𝑚
Conclusion The height of the antenna is 2010 m
Did you know that...?
Although aircraft have sophisticated equipment that provides them with
information about their orientation (such as GPS) and can determine the
position of the aircraft on the ground, planes deviate from their course
due to factors such as weather and winds; however trigonometry helps
aircraft not to lose their course.
APPLICATIONS OF COSINE LAW
The distance from Bogota to Barranquilla is 982.5 km. Due to weather
conditions a plane deviates from its path. If it has travelled a distance of
850 Km and has 180 Km left to reach Barranquilla. What is the angle at
which the pilot must turn to reach his destination?
The following figure shows a diagram of the situation.
Figure
Solution
Figure.
The plane is located at point C, use the cosine rule the find the angle at
which it must turn.
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 𝑐𝑜𝑠 𝐶
Replace the values in the formula
982.52 = 1802 + 8502 − 2 ∗ 180 ∗ 850 𝑐𝑜𝑠 𝐶
Solve for cos C
982.52 − 1802 − 8502
cos 𝐶 =
2 ∗ 180 ∗ 850
Calculate
cos 𝐶 = 0.813
𝐶 = 𝑐𝑜𝑠 −1 0.813 = 35°36"
We conclude that the pilot must turn with an angle of 35°36”when
located at point C
Learning Activity
1. From the example above use the law of cosine to calculate the
angle (A) at which the plane deviates and the angle at which it
reaches its destination (B).
2. A boy flies two kites at the same time. He has 380 feet of rope for
one kite and 420 feet for the other. The boy estimates the angle
between the two ropes is 30°.
Calculate
a. The distance between the two kites.
b. The magnitude of the angles formed by the ropes and the distance
between the kites.
Exercise retrieved from Stewart, J., Redli, L., Watson., Precálculo
Matemáticas para el cálculo, (5ta. Edición), México, CENGAGE
Learning.
SUMMARY
Oblique triangles
Suggested steps to find
the answer
Case I. SAA o ASA  Calculate the remaining
One side and
angle using the sum of
two
angles
the angles in a triangle.
𝐴 + 𝐵 + 𝐶 = 180°
are given
 Calculate the remaining
angles using the sine
law.
Case II. SSA
Ambiguous case
Two sides and one
angle (other than  Calculate the angle using
the angle formed by
the sine law.
the
sides)
are  Calculate the remaining
given.
angle using the sum of
the interior angles.
 Calculate the remaining
side using the sine law.
Case III. SAS
 Calculate
both
sides
Two sides and the
using the sine law.
angle formed by  Calculate the smallest of
them are given.
both angles using the
sine law.
 Calculate the remaining
angle using the sum of
the interior angles.
Case IV. SSS
 Use the cosine law to
All sides are given.
calculate
the
biggest
angle.
 Calculate any of the
remaining angles using
the cosine law.
 Calculate
the
last
remaining angle using
the sum of the interior
angles.
HOMEWORK
1. In groups of three, propose a situation where of the laws of sine and
cosine are applied.
2. Draw the situation and set values so that there is a real answer.
3. Present the situation and your results to your classmates.
EVALUATION
I.
WRITE TRUE (T) OR FALSE (F) WHERE IT
CORRESPONDS.
1. Oblique triangles are known to have three acute angles.
A. True
B. False
2. We have an oblique triangle where the measurement of
two angles (A, B) and a side (c) are given as illustrated in
the figure
To find the measurement of side b we must use the cosine
law.
A. True
B. False
3. The following is an oblique triangle
Suppose that the measurements of sides a and b are given,
also the measurement of angle B, therefore to calculate the
measurement of side b we use the laws of cosines.
A. True
B. False
II. Fill in the blanks
If we have all sides of an oblique triangle, we use the
_________ law to calculate the angles.
If we have two angles and any side of the triangle, we use
the ________ law to find the missing information.
III. In the next figure we see a crane with a counterweight.
USING THE PREVIOUS INFORMATION, ANSWER THE FOLLOWING
QUESTIONS:
6. To calculate the distance between the two ends of the crane
(a) it is best to
A. Use the sine law
B. Use the cosine law
C. Use the sine law and then cosine law
D. Use neither of the laws
7. The steps to solve the problem (finding side a and angles B
and C) are:
A. Sine law, cosine law, sum of interior angles.
B. Cosine law, sine law, sum of interior angles.
C. Sine law, sum of interior angles, cosine law.
D. Cosine law, sum of interior angles, sine law.
8. The approximate measurement of side a and angle B is:
A. 66 cm y 37°
B. 37 cm y 66°
C. 20 cm y 132°
D. 132 cm y 20°
Bibliography
Ayres, F., (1990), Trigonometría plana y esférica, Bogotá, McGraw – Hill
Latinoamericana, S.A.
Lial, M., Hornsby, J., Schneider, D., Dugopolski, M.,
Trigonometría, (8va. Edición), México, Pearson Educación.
(2006),
Stewart, J., Redli, L., Watson., Precálculo Matemáticas para el cálculo,
(5ta. Edición), México, CENGAGE Learning.
Realini, C.S. (21/05//2012). Teorema del seno y coseno. Plan Ceibal –
Creative Commons: Reconocimiento – No comercial –Compartir bajo la
misma
licencia.
http://www.ceibal.edu.uy/userfiles/P0001/ObjetoAprendizaje/HTML/Teor
ema%20del%20seno%20y%20el%20coseno_Silvana%20Realini.elp/fich
a.html. [Retrieved on: 02/04/2016].
RÍOS, Julio, "Problema con Ley de Senos" [video en línea], en: YouTube
http://www.youtube.com/], Colombia, 17 de marzo de 2011. Disponible
en
Internet:
https://www.youtube.com/watch?v=yizdJXO2yME&feature=related
[Retrieved on: 02/04/2016].
Glossary
Angle: Is formed by turning a ray around itself.
Ambiguous case: Corresponds to the Case II (SSA), it can be one, two
or no triangles.
Degree: The most common unit to measure angles. It is written as 1°
1
and it represents 360 of the rotation.
Radian: Unit used to measure angles. An angle whose vertex is at the
center of a circle intercepts an arc as long as the radius of the circle and
has a measure of 1 radian.
Vocabulary Box
Hobby: Recreational activity. Leisure
Square root: Mathematical process to find the number multiplied by
itself.
Journey: Long path travelled
GPS: Global Positioning System
Aircraft: Aerial means of transportation. Airplane
Kites: Rhombus-shaped flying device.
Counterweight: Weight used to create an opposite force.
English Review
Prepositions – Place (Position and Direction)