Download Trigonometry - Ms. Albarico`s e

Math 10
Ms. Albarico
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Introduction to Trigonometry
Review on Types of Triangles
Labelling Triangles
Trigonometric Ratios
Solving Word Problems Related to Trigonometry
Introduction
Triangles Around Us
For this activity, you'll need:
• a piece of paper, pencil
• a ruler, protractor
• some coloured pencils or pens
What To Do:
1) Using your pencil and ruler, draw some straight lines on
your piece of paper to make an interesting pattern. You can
draw as many or as few as you like.
• 2) Using your coloured pencils or pens, decorate all the threesided shapes in some way. You could colour them all in using
a particular colour or you could cover them with a special
design or pattern
When you finish your work, you have to explain your
design by answering this questions:
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•
•
What is your pattern all about?
Can you describe what you see in your own pattern?
Can you find any shapes which have three sides?
How about any with four sides?
Which shape or shapes have the least sides?
• What are the measures of their angles?
• What is the total sum of their angles?
Students are expected to:
• M04.01 Explain the relationships between similar right
triangles and the definitions of the primary trigonometric
ratios.
• M04.02 Identify the hypotenuse of a right triangle and the
opposite and adjacent sides for a given a cute angle in the
triangle.
• M04.03 Solve right triangles, with or without technology.
• M04.04 Solve a problem that involves one or more right
triangles by applying the primary trigonometric ratios or
the Pythagorean theorem.
• M04.05 Solve a problem that involves indirect and direct
measurement, using the trigonometric ratios, the
Pythagorean theorem, and measurement instruments such
as a clinometer or metre stick.
Vocabulary
perpendicular
parallel
sides
angle
triangle
congruent
similar
dilate
sail
navigate
approach
Key Terms
Primary Trigometric Ratios
Sine Ratio
Cosine Ratio
Tangent Ratio
Pythagorean Theorem
Angle of Inclination
Angle of Elevation
Angle of Depression
What is a TRIANGLE
Triangles classified
according to their Sides:
• Scalene Triangle. A scalene triangle that
has no equal sides.
• Isosceles Triangle. An isosceles triangle
is a triangle that has two equal sides.
• Equilateral Triangle. An equilateral
triangle is a triangle that has three equal
sides.
Identify the following triangles:
Triangle B
Triangle A
Triangle C
Triangles classified
according to their Angles:
• Right Triangle. A right triangle has a 900
angle.
• Obtuse Triangle. An obtuse triangle has
one angle that has bigger than 900 angle.
• Acute Triangle. In an acute triangle, all
angles has less than 900 angle.
Identify the following triangles:
Triangle B
Triangle A
Triangle C
Names of the sides:
– the HYPOTENUSE
– the OPPOSITE side
– The ADJACENT side
• A REFERENCE ANGLE must be
selected before you start labeling the
sides of the right angle
• It can be either of the two acute angles.
• The HYPOTENUSE is the longest side
of the right triangle.
• The OPPOSITE SIDE is the side across
the 900 angle.
• Based from our drawing, which is our
opposite side?
• The ADJACENT SIDE is the side next
to our reference angle.
• The ADJACENT SIDE is the side next
to our reference angle.
• Given the same triangle, how would the
sides be labeled if we choose the other
acute angle as our reference angle?
• Will there be any difference?
• Which side does not change?
• Primary Trigonometric Ratios – constant values based
on the ratios of sides for particular angles in right-angled
triangles. Sine, Cosine, and Tangent are called primary
trigonometric ratios.
• tan X – a constant value based on the ratio of the length of
the side to a chosen angle X in a right triangle.
• sin X – a constant value based on the ratio of the length of
the side opposite to a chosen angle X to the length of the
hypotenuse in a right triangle.
• cos X – a constant value based on the ratio of the length of
the side adjacent to a chosen angle X to the length of the
hypotenuse in a right triangle.
5
3
x
4
3
5
x
4
5
3
x
4
opp
sin x 
hyp
o
adj
cos x 
hyp
Take the first letter of each word.
o
opp
tan x 
adj
o
S O H C A H T O A
Note:
Given
Ratio of sides
Angle, side
Looking for
Use
Angle measure
SIN-1
COS-1
TAN-1
Missing side
SIN, COS, TAN
Calculator Commands Reminder
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Calculator Commands
Set your calculator to ‘Degree’…..
MODE (next to 2nd button)
Degree (third line down… highlight it)
2nd
Quit
Write the ratio for sin A
B
Sin A = a
c
c
Write the ratio for cos A
a
C
b
A
Cos A = b
c
Write the ratio for tan A
Let’s switch angles!
Find the sin, cos and tan for
Angle B.
Tan A = a
b
3
5
x
4
To solve for Angles:
tan x
o
opp

adj
C
hyp’
Now we need to look at
the two ratios involving
the hypotenuse:
sin xo =
Opposite
Hypotenuse
cos xo =
Adjacent
Hypotenuse
Opp’
xo
A
Adj’
B
• For Trigonometric Inverse Functions:
Press 2nd, use
SIN for SIN-1
COS for COS-1
TAN for TAN-1
Name
“say”
Abbreviation
Abbrev.
Ratio of an angle
measure
Sine
Cosine
Tangent
Sin
Cos
Tan
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
• What are the three sides of a right
angle triangle?
• In your calculator, what function will
we use to find angle measures?
• How can you remember easily the
trigonometric ratios?
Open your textbook to page 70.
Find an angle that has a
tangent (ratio) of 3
C
3cm
B
4
Round your answer to the
nearest hundredth degree.
4cm
A
Process:
I want to find an ANGLE.
I was given the sides (ratio).
Tangent = opposite
adjacent
Solution:
TAN-1(3/4) = 36.87°
CLASS BOARD WORK
INDIVIDUAL CLASS WORK
• #10-14 ON page 76.
HOMEWORK
• #15-19 ON page 75-76.
ASSIGNMENT 2
ASSIGNMENT 2
ASSIGNMENT 2
CLASS BOARD WORK
INDIVIDUAL CLASS WORK
• #3-8 ON page 82.
HOMEWORK
• #10-16 ON page 83.
ASSIGNMENT 3
•
-
By partner, bring the following:
Piece of string
Heavy object (stopper)
Needle
Drinking straw (must be at least 15cm long)
Read pages 85-86.
How are you going to measure an
inaccessible height?
ASSIGNMENT 3
1) Students will create their own clinometer
by pair.
2) Teacher will assign which inaccessible
height to measure.
3) Students answer E-G on page 86 and
Assess Your Understanding #1-3
Resources:
• (PEARSON CANADA) Foundations and
Pre-Calculus Math 10
• (NELSON CANADA) Mathematical
Modelling Book 1
• Polyhedron figures
For Tutorial
• http://www.sd43.bc.ca/Resources/ParentRes
ources/math/10/Pages/Trigonometry.aspx
• https://www.mathsisfun.com/rightangle.htm
l
References:
• https://www.mathsisfun.com/geometry/trian
gles-interactive.html
• http://www.math10.ca/lessons/measurement
/trigonometryOne/trigonometryOne.php
• mathworld.wolfram.com