Right Triangles

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Right Triangles

• The Trig Ratios

Brought to you by Moody Mathematics
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Lets review some vocabulary.
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A
Hypotenuse
B
C
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Opposite Leg
A
B
C
Opposite Leg to A
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Opposite Leg
Opposite Leg to B
B
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Adjacent Leg
A
Adjacent Leg to A
B
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Adjacent Leg
B
Adjacent Leg to B
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Consider the right triangles in this next slide
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What can you say about them?
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They are similar
By AA
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They have the same right angle
They have the same acute angle
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All right triangles having one acute angle the
same are similar.
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For example, all 45-45-90 triangles are similar.
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The legs of a 45-45-90 triangle are in a 1 to 1
ratio.
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In a 45-45-90 triangle the ratio leg
hypotenuse
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Also, all 30-60-90 triangles are similar.
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In a 30-60-90 triangle, the ratio leg
opposite the 30o hypotenuse
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The ratio leg opposite the 600 hypotenuse
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We have names for the 3 most common ratios that
we will form in right triangles.
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The names are the Sine Ratio, the Cosine
Ratio, the Tangent Ratio.
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Sin A
Cos A
Tan A
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S O H C A H T O A
Some Old Hippy Caught Another Hippy Tripping On
Antacid
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in
SOH
pposite
ypotenuse
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os
CAH
djacent
ypotenuse
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an
TOA
pposite
djacent
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Sin A
A
Hypotenuse
C
B
Opposite Leg to A
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Cos A
A
Hypotenuse
Adjacent Leg to A
B
C
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A
Tan A
Adjacent Leg to A
B
Opposite Leg to A
C
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Now lets set up the three ratios for angle B.
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Sin B
A
Hypotenuse
Opposite Leg to B
B
C
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Cos B
A
Hypotenuse
B
C
Adjacent Leg to B
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Tan B
A
Opposite Leg to B
B
Adjacent Leg to B
C
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Now lets use a ratio to solve for a missing side
of a right triangle
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Lets estimate the value of x before we start
a. Xgt12 b.
6ltxlt12 c. Xlt6
A
C
B
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a. Xgt12 b. 6ltxlt12 c. Xlt6
Its not (a) because A leg cant be longer than
the hypotenuse.
A
C
B
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b. 6ltxlt12 c. Xlt6
If B were 30o then x would be 6 exactly. Since B
is smaller than 30o xlt6.
A
C
B
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Look at the parts involved and decide which ratio
fits best.
A
C
B
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Where are the given and missing sides in relation
to the known angle?
A
C
B
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X is the opposite leg to B and 12 is the
hypotenuse.
A
C
B
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A
C
B
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Now lets use another ratio to solve for a
missing side of a right triangle
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Lets estimate the value of x before we start
a. Xgt16 b.
8ltxlt16 c. Xlt8
A
B
C
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Its not (a) because A leg cant be longer than
the hypotenuse.
a. Xgt16 b. 8ltxlt16 c. Xlt8
A
B
C
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If A were 30o then x would be 8. Since ltA 55o
is bigger than30o, xgt8.
b. 8ltxlt16 c. Xlt8
A
B
C
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Look at the parts involved and decide which ratio
fits best.
A
B
C
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Where are the given and missing sides in relation
to the known angle?
A
B
C
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X is the adjacent leg to B and 16 is the
hypotenuse.
A
B
C
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A
B
C
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Now lets solve another ratio to find a missing
side of a right triangle, but this time x is on
the bottom.
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Look at the parts involved and decide which ratio
fits best.
A
B
C
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Notice that only the legs are involved, not the
hypotenuse.
A
B
C
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A
B
C
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Now lets solve a ratio to find a missing angle
of a right triangle.
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Lets estimate the value of x before we start
a. Xlt45o b. Xgt
45o
A
B
C
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If x were 45o then both legs would be
(which is between 8 and 9).

• Xlt45o
• b. Xgt 45o

A
B
C
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Look at the parts involved and decide which ratio
fits best.
A
B
C
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9.1 is the opposite leg to x and 12 is the
hypotenuse.
A
B
C
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Hit the 2nd key then sin key
A
B
C
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Now lets solve another ratio to find a missing
angle of a right triangle.
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Look at the parts involved and decide which ratio
fits best.
A
B
C
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15.8 is the adjacent leg to x and 17 is the
hypotenuse.
A
B
C
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Hit the 2nd key then cos key
A
B
C
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The End

• Now go practice!

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