The Trigonometric Ratios
Math 10 Plus Notes
Date:
Warm up: Similar Triangles
Similar
Triangles
Similar triangles have the same _______________ but can be
different _______________. This means the angles in similar
triangles are the _____________. The side lengths may be different
but the ratio between corresponding sides is _________________.
Recall
Triangles are similar if they have _______________________________
in common (and AAA can be proven by showing that
corresponding sides share the same ratio).
Example
Show that the triangles below are similar.
1)
2)
Background Information: Solving Algebraic Expressions
Practice
Re-arrange the equation below for ‘b’. What did you do?
a
b
c
Re-arrange the equation for ‘c’. What did you do?
a
b
c
1
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Labelling Right-Angle Triangles
Right
Triangle
A right triangle is any triangle that has a right _______________, or
an angle that measures ________ degrees. A ________ degree angle
is shown by labelling that angle with a ________________.
Hypotenuse
The hypotenuse is the side in a right triangle that is ___________
_______ (or opposite) the right angle. It is often written as ‘hyp’.
Theta ( θ )
Theta is a Greek letter written as _______. It is used to represent
or label an ________________ angle.
90° angle
hyp
θ
The remaining two sides are labelled in relation to the angle
we are interested in. In the case above, theta is in the bottom
right corner, so we label the triangle like so:
opposite
Adjacent
Side
Opposite
Side
Note
hypotenuse
θ
adjacent
The adjacent side of a right-angle triangle is the side that is
adjacent (or ______________________________) to the specified angle
(often labelled with theta or a capital letter).
The opposite side of a right-angle triangle is the side that is
opposite (or ______________________________) the specified angle.
By convention, a side is named after the angle opposite it. So, if
theta above is called ‘B’, the opposite side would be called ‘b’.
We use capital letters for angles, lower-case letters for sides.
2
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Example
Label the triangle.
θ
Practice
Label the right angle, hypotenuse, opposite and adjacent for
the following triangles.
θ
θ
Practice
Label the missing sides and angles.
A
b
c
Using a Protractor
Protractor
Protractors measure angles. To Use, line up the origin of the
protractor with the vertex of the angle. Make sure that the
horizontal line of the protractor lines up with the bottom line
of the angle. Read the measurement that the top line of the
angle passes through.
origin
vertex
The angle is 35°
3
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Deriving the Tangent Ratio
Activity
In the triangles below, label the opposite and adjacent sides.
Using a protractor, measure and record the size of the angle,
theta. Using a ruler, measure and record the lengths of the
opposite and adjacent sides.
Triangle #
Angle Size
Length of
Opp. Side
Length of
Adj. Side
Ratio
(Opp/Adj)
4
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Think
What do you notice about the angle size of each triangle? What
do you notice about the ratios of opposite divided by adjacent
for each triangle?
Try
This
Draw a right-triangle that has a 40° angle. Find the ratio of
opposite divided by adjacent. Compare your ratio to your
group members’ answers. What do you notice? What can you
conclude?
Tangent
Ratio
The tangent ratio is the name for the constant ______________
created by dividing the opposite side by the adjacent side for
each angle.
tan  
opp
adj
The ancient Greeks noted this relationship between angles
and the ratio of the opposite and adjacent sides. They created
tables to record these values. We now have technology that
can store the table and retrieve the data for us. If you want to
know the tangent ratio for an angle, hit the ‘tan’ button on
your calculator and then enter the value of the angle.
tan
5
0
=
1.19
5
The Trigonometric Ratios
Math 10 Plus Notes
Example
Date:
Label the sides of the triangle below and calculate the tangent
ratio for the unknown angle (theta) and find the length of the
hypotenuse.
7
5
θ
Example
Find the tangent ratio for angle 80°, label the triangle sides.
80°
Practice
Calculate the tangent ratio for θ.
9.4 yds.
θ
3 m.
12.2 yds.
θ
5 m.
Challenge
Find the missing side of the triangle below.
7
70°
x
6
The Trigonometric Ratios
Date:
Math 10 Plus Notes
Using the Tan Ratio to Find Missing Lengths
Recall
tan  
opp
adj
The tangent ratio of each angle is a ____________________ for any
triangle. For example, if an angle has a measure of 50°, the
ratio of the opposite over the adjacent (in relation to angle 50)
will always be 1.19 no matter the size of the triangle. This
means that if we know the size of the angle and one side
length, we can solve for an ________________________________________.
Example
Solve for side length x.
42°
2 in.
9 in.
27°
x
Practice
x
Solve for side length x.
7
The Trigonometric Ratios
Math 10 Plus Notes
Practice
Date:
Solve for the unknown lengths.
Using the Tan Ratio to Solve for an Unknown Angle
If we know the lengths of the opposite and unknown sides, we
can find the ratio. Because this ratio is a constant for each
angle, it allows us to solve for the ____________________.
Before calculators, we would have referred to a table.
However, calculators can now tell us what angle gives us a
certain ratio. Taking the “tan inverse” of a side ratio will tell us
the angle size.
Example
Find the unknown angle, theta.
opp
adj
10
tan  
 0.8333
12
  tan 1 (0.8333)
tan  
θ
12
10
Tan-1
0.833
=
*
44.4
*tan-1 is 2nd tan
So, the measure of θ is 44.4°
8
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Practice
Label the triangles and find the measure of the unknown
angle.
Think
What does the value of tan (40) mean?
Think
If tanθ is less than 1, is the opposite or adjacent side longer?
Challenge
What do you know about the angles of a triangle if tanθ = 1?
9
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Deriving the Sine and Cosine Ratios
Activity
Use the table on page 4 to fill in the first three columns of the
table below. On page 4, measure the length of the hypotenuse
for each triangle and complete the rest of the table.
Angle
Size
Length of
Opposite
Side
Length of
Adjacent
Side
Length of
Hypotenuse
Ratio of
Lengths of
Opposite Side
and
Hypotenuse
Ratio of
Lengths of
Adjacent Side
and
Hypotenuse
Ratio of Lengths
of Opposite
Side and
Adjacent Side.
Think
What do you notice about the ratio of the opposite and
hypotenuse side? What do you notice about the ratio of the
adjacent and hypotenuse side?
Think
What do you think this means?
Challenge
Find the unknown side length.
10
15
20°
20
x
20°
10
The Trigonometric Ratios
Math 10 Plus Notes
Sin Ratio
Date:
The Sine, or Sin, ratio is the name for the constant ratio that
occurs for each angle when the opposite side length is divided
by the hypotenuse side length.
opp
sin  
hyp
Cos Ratio
The Cosine, or Cos, ratio is the name for the constant ratio that
occurs for each angle when the adjacent side length is divided
by the hypotenuse side length.
adj
cos  
hyp
Example
Find the Sin and Cos ratios for the unknown angles below.
7
θ
sinθ =
sinσ =
4
σ
cosθ =
Example
cosσ =
Find the measure of angle θ and angle σ for the triangle above.
11
The Trigonometric Ratios
Math 10 Plus Notes
Practice
Date:
Find the value of the sin ratio for the triangle below and solve
for theta.
9 cm
θ
12 cm
Practice
12.5 cm
Use the cos ratio to solve for the unknown side length.
134.5
θ
231.2
267.5
Trigonometric
Ratios
Trigonometric ratios are the ratios of two sides of a ___________
triangle and a related ________________. Trigonometric ratios are
used to find _________________ lengths or angles in a right
triangle. We will use the three most common trig. ratios: ______
________________________________.
Note
To help you decide which trig. ratio to use, remember:
SOH
Sin, Opp, Hyp
sin  
opp
hyp
CAH
Cos, Adj, Hyp
cos  
TOA
Tan, Opp, Adj
adj
opp
tan=
hyp
adj
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The Trigonometric Ratios
Math 10 Plus Notes
Date:
Steps to Solving a Right-Angle Triangle Problem
Reference
1) Find the right angle, then label the hypotenuse.
2) Find the indicated angle, then label the opposite and
adjacent accordingly.
3) Decide which trig. ratio to use based on the given
information.
4) Solve the trigonometric ratio.
Example
Solve for the unknown side length, x.
Practice
Decide which trigonometric ratio to use and solve for the
unknown side length in the triangles below.
13
The Trigonometric Ratios
Math 10 Plus Notes
Date:
Practice
Decide which trigonometric ratio to use and solve for the
unknown angle in the triangles below.
Example
Sin M for the right-angled triangle is 0.8. What do we know
about the relationship between the side lengths ‘m’ and ‘h’?
M
h
t
T
H
m
Example
Find and correct the error in the calculation below.
opp
adj
x
tan20=
10
10 tan 20  x
tan=
3.64  x
14
The Trigonometric Ratios
Math 10 Plus Notes
Challenge
Date:
Prove that the following is true.
tan=
sin 
cos 
Challenge
Using a ruler and protractor, draw (to scale) two triangles
where cosθ = 0.25.
Think
What are the greatest possible value for Sin and Cos?
Explain.
Think
Omar argues that there is no need to know trigonometry – we
can always use a ruler or protractor to measure a missing side
length or angle. Do you agree? Why or why not?
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