Day 2:
The Trigonometric Ratios to Solve for Sides
Problem:
Read the instructions and answer the following questions:
A
B
C
E
E
D
E
F
G
H
I
1. Four nested right triangles are drawn on the grid paper. The four triangles are
____________________.
a) What angle is common to all four triangles?
b) State the following lengths.
DF = 3 units
EF =
CG =
EG =
BH =
EH =
AI =
EI =
c) Determine the lengths of DE, CE, BE, and AE using the Pythagorean Theorem.
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d) Complete the following chart. Put the ratios in lowest terms.
Triangle
Ratio 1
Ratio 2
Ratio 3
EDF
DF

EF
DF

DE
EF

DE
ECG
CG

EG
CG

CE
EG

CE
EBH
EAI
BH

EH
BH

BE
EH

BE
AI

EI
AI

AE
EI

AE
e) How are the ratio 1’s related? Why is this the case?
f) This ratio is called the tangent ratio for angle E. Explain the meaning of the
tangent ratio by describing the positions of the two sides in relation to the
common angle, E.
g) How are the ratio 2’s related? Why is this the case?
h) This ratio is called the sine ratio for angle E. Explain the meaning of the sine
ratio by describing the positions of the two sides in relation to the common angle.
i) How are the ratio 3’s related? Why is this the case?
j) This ratio is called the cosine ratio for angle E. Explain the meaning of the cosine
ratio by describing the positions of the two sides in relation to the common angle.
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Rule:
For an acute angle A in a right triangle, the trigonometric ratios are
defined as follows:
B
hypotenuse
opposite
C
A
adjacent
length of side opposite to A
tan A 
length of side adjacent to A
tan A 
opposite
adjacent
sin A 
length of side opposite to A
length of hypotenuse
sin A 
opposite
hypotenuse
cos A 
length of side adjacent to A
length of hypotenuse
cos A 
adjacent
hypotenuse
Note: The trigonometric ratios depend only on the size of the angle, not on the size of
the right triangle.
For trigonometric calculations, your calculator MUST be in DEGREE mode!!
Example 1:
Evaluate each ratio to the nearest thousandth.
a)
tan 48
b)
cos 65
c)
sin 89
d)
cos11
e)
sin 17
f)
tan 73
Example 2:
M
In ΔMNO, determine the length of MN, to the nearest tenth.
9cm
O
38˚
N
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Rule: To solve a triangle means to find the lengths of all the unknown sides and
measures of all unknown angles.
Example 3:
Solve ΔABC. Round side lengths to one decimal place and angles to the
nearest degree.
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