OVERVIEW
Similarity Leads to Trigonometry
G.SRT.6
G.SRT.6
Understand that by similarity, side
ratios in right triangles are properties
of the angles in the triangle, leading to
definitions of trigonometric ratios for
acute angles.
(1) The student will be able to label a
triangle in relation to the reference
angle (opposite, adjacent &
hypotenuse).
(2) The student will be able to
determine the most appropriate
trigonometric ratio (sine, cosine, and
tangent) to use for a given problem
based on the information provided.
(3) The student will be able to solve
for sides and angles of right triangles
using trigonometry.
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Develop the foundational concepts of
trigonometry from similarity. AA
helps us know that all 30 degree right
triangles are similar and so they will
have proportional sides and fixed
ratios. This leads nicely into the
process to establish Sine, Cosine and
Tangent for all right triangles.
Trigonometry is similarity.
Proportional sides of similar triangles
provide equivalent ratios of those
sides.
This objective is really getting at the
conceptual understanding of
trigonometry and how it is established
from similarity and proportional sides.
1 – Use the special right triangle
relationships to develop the Sine,
Cosine and Tangent values and relate
them to a trigonometry table.
Trigonometry is a major topic and
probably one of the most used pieces
of mathematics outside of the
classroom. It has far reaching uses
because of its ability to relate angles
to sides.
2—Introduce trigonometry using a
table first and then a calculator. The
table allows students to see number
patterns and develop a more
conceptual understanding of what is
going on.
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NOTES
Similarity Leads to Trigonometry
G.SRT.6
CONCEPT 1 – Understand that by similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
In G.SRT.5 it was discussed that all 45° - 45° - 90° right isosceles triangles have the same relationships of its sides no
matter how large or small the triangle is. These ratio relationships are unchanging due to the similarity of the triangles.
Similarity preserves the angle measures and maintains the sides’ proportionality.
5 cm
7 cm
5 cm
7 cm
45°
7 2 cm
9 6 cm
45°
9 3 cm
5 2 cm
45°
45°
9 3 cm
It is this proportionality of all 45 - 45 - 90 that allows for the generalization of these relationships.
x
x
45°
45°
x 2
The ratios within the triangle, such as the leg to the hypotenuse or the leg to the leg, are fixed values because they
would remain proportional no matter the size of the triangle. Calculating them, it is found that:
The ratio of leg to hypotenuse:
The ratio of leg to leg:
leg
1

 0.7071
hypotenuse
2
leg
1
leg
This will be true
for ALL 45-45-90 triangles.
This will be true
for ALL 45-45-90 triangles.
When looking at ratios within the triangle name the sides so that there is no confusion on which side or leg that is being
referred to. When the sides are labeled it will always be based on a reference angle. In the previous example there was
no need to discuss this because the two reference angles were the same.
B
o
A
Adjacent Leg
Opposite
Leg
o
•
C
Opposite
Leg
B
A
A
B
Adjacent
Leg
•
C
Hypotenuse
If A is the reference angle, then
C
Hypotenuse
If C is the reference angle, then
BC is the opposite leg,
AB is the adjacent leg, and
AC is the hypotenuse.
AB is the opposite leg,
BC is the adjacent leg, and
AC is the hypotenuse.
The hypotenuse will always be opposite the right angle (the longest side).
The opposite leg will always be the side of the triangle that does not form the reference angle.
The adjacent leg will always be the non-hypotenuse side that forms the reference angle.
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NOTES
Similarity Leads to Trigonometry
G.SRT.6
TRIGONOMETRY (Triangle Measure) allows the linkage between angle size in right triangles to side proportionality. If
the reference angle in a right triangle is known, then there are three fixed ratios of its sides. The converse is also true, if
the ratio of two sides of a right triangle is known then there exists one reference angle that has that ratio. Trigonometry
connects angle measure to side measure, thus it is a very powerful piece of mathematics.
The three ratios established have names:
The Sine Ratio (sin)
Opposite
sin  
Hypotenuse
The Cosine Ratio (cos)
Adjacent
cos  
Hypotenuse
The Tangent Ratio (tan)
Opposite
tan  
Adjacent
Lets look at how trigonometry helps in the discovery of information about triangles.
A portion of the table has been clipped to demonstrate the power of these numbers.
Knowing Sides…. Get Angles
Knowing Angles… Get Sides…
B
2.62 cm
(opposite)
A
opposite
θ
2.20 cm
hypotenuse
54°
C
(adjacent)
Opposite
Adjacent
2.62
tan  
 1.191
2.20
tan  
tan   1.191
The angle must be approximately
50 because its tangent ratio is
1.191.
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Without knowing anything about
the opposite and the hypotenuse
other than the reference angle,
the value of ratio between them is
known.
sin 54 
opposite
hypotenuse
opposite
 0.8090
hypotenuse
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Degrees
46
47
48
49
50
51
52
53
54
Sine
(sin)
0.7193
0.7314
0.7431
0.7547
0.7660
0.7771
0.7880
0.7986
0.8090
Cosine
(cos)
0.6947
0.6820
0.6691
0.6561
0.6428
0.6293
0.6157
0.6018
0.5878
Tangent
(tan)
1.0355
1.0724
1.1106
1.1504
1.1918
1.2349
1.2799
1.3270
1.3764
The chart allows users to find ratios of
sides if angle measures are known or to
find the reference angle if
the ratio of sides is given.
The chart is a nice way to learn the
concept but calculators are much more
powerful because they can handle angle
sizes other than whole numbers.
2/8/2016
ASSESSMENT
Similarity Leads to Trigonometry
G.SRT.6
1. The opposite side of A is:
A) AB
C
B) BC
C) CA
D) Depends
B
A
2. The Sine ratio (opposite/hypotenuse) in a 30 right triangle is:
1
3
A)
B)
1
2
C)
1
2
D)
3
2
3. The Sine ratio (opposite/hypotenuse) in a 45 right triangle is:
1
2
A)
B)
2
1
C) 1
D)
1
2
4. If C is the reference angle, then the tangent ratio would be:
C
17 cm
B
A)
15
17
5. The ratio
B)
8
15
C)
8
17
D)
15
8
A
A
5
represent the which relationship:
13
A) sin B
15 cm
8 cm
12 cm
B) tan B
C) tan C
D) cos B
C
5 cm
B
13 cm
6. The value of Ɵ is approximately:
A) 18.43
B) 19.47
C) 65.41
D) 70.53
7. The value of x is approximately:
x
30°
10 cm
A) 5.000 cm
C) 6.428 cm
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B) 5.774 cm
θ°
5 cm
15 cm
Degrees
Sine
(sin)
Cosine
(cos)
Tangent
(tan)
30
0.5000
0.8660
0.5774
40
0.6428
0.7660
0.8391
50
0.7660
0.6428
1.1918
D) 8.660 cm
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ASSESSMENT
Similarity Leads to Trigonometry
8. The value of Ɵ is approximately:
G.SRT.6
4.5 cm
10 cm
Degrees
Sine
(sin)
Cosine
(cos)
Tangent
(tan)
25
0.4226
0.9063
0.4663
35
0.5736
0.8192
0.7002
45
0.7071
0.7071
1.0000
55
0.8192
0.5736
1.4281
θ°
A) 25
B) 35
C) 45
D) 55
A
9. Jeremy looks at the given triangle and says “The measure of angle C is 30.” How
could Jeremy determine this from only measurements on diagram and not any other
tools?
5 cm
C
B
10 cm
10. When looking at a trigonometry table Sarah notices that the Sine and Cosine ratios are the same value for
the 45 right triangle. Explain why that happened.
11. When looking at the trigonometry table Jessica notices that the Sine ratio for all angles between 0 and
90 are values between 0 and 1. Why is this an appropriate range of values for the Sine ratio?
12. Solve for the missing information. (Round all final answers to 2 decimals places)
a)
b)
12 cm
c)
d)
8 cm
11 cm
9 cm
θ°
36°
x ≈ ____________
e)
10 cm
63°
x
Ɵ ≈ ____________
f)
θ°
x
x ≈ ____________
g)
Ɵ ≈ ____________
h)
12 cm
58°
9 cm
12 cm
θ°
Ɵ ≈ ____________
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x
3 cm
x
13 cm
41°
x ≈ ____________
21 cm
x ≈ ____________
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θ°
14 cm
Ɵ ≈ ____________
2/8/2016
ASSESSMENT
Similarity Leads to Trigonometry
G.SRT.6
Answers:
1)
2)
3)
4)
5)
6)
7)
8)
9)
B
C
A
B
D
D
B
A
The ONLY right triangle that has a ratio of opposite : hypotenuse that is 1 : 2 is a 30- 60- 90 . Thus
mC = 30 because then the ratio of opposite : hypotenuse is 1 : 2.
10) A 45 right triangle is an isosceles triangle with congruent legs. These two congruent legs are the
opposite and the adjacent for the 45 angle so
opposite
adjacent
is the same as
.
hypotenuse
hypotenuse
opposite
. The hypotenuse is the longest side, so when we
hypotenuse
compare an opposite side to the hypotenuse we will always have value between 0 and 1.
11) The sine ratio is the comparison of the
12) a) 8.72 cm
b) 43.34
c) 10.10 cm
d) 73.30
e) 41.41
f) 15.33 cm
g) 15.85 cm
h) 59.00
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