Transcript Ratios in Right Triangles

Ratios in Right Triangles
Expectations:
1)
G1.3.1: Define and use sine, cosine and
tangent ratios to solve problems using
trigonometric ratios in right triangles.
2)
Determine the exact values of sine, cosine
and tangent for various angle measures.
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8-3: Ratios in Right Triangles
Opposite Legs
From an acute angle in a right triangle,
the leg opposite is the leg that lies in
the interior of the angle (except the
endpoints of the side).
B
C
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BC is the leg opposite
A
A
8-3: Ratios in Right Triangles
Opposite Legs
From an acute angle in a right triangle,
the leg opposite is the leg that lies in
the interior of the angle (except the
endpoints of the side).
B
C
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AC is the leg opposite
∠B
A
8-3: Ratios in Right Triangles
Adjacent Legs
The leg adjacent to an acute angle of a
right triangle is the leg that forms a side
of the acute angle.
B
C
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AC is the leg adjacent
∠A
A
8-3: Ratios in Right Triangles
Adjacent Legs
The leg adjacent to an acute angle of a
right triangle is the leg that forms a side
of the acute angle.
B
C
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BC is the leg adjacent
∠B
A
8-3: Ratios in Right Triangles
Sine Ratio
The sine ratio of an acute angle of a right
triangle compares the length of the leg
opposite the angle to the length of the
hypotenuse.
Sine is abbreviated sin, but it is still read
as “sine”.
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8-3: Ratios in Right Triangles
Sine Ratio
B
C
A
leg opposite
sin θ =
hypotenuse
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8-3: Ratios in Right Triangles
Sine Ratio
B
C
BC
sin A =
AB
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A
AC
sin B =
AB
8-3: Ratios in Right Triangles
Cosine Ratio
The cosine ratio of an acute angle in a
right triangle compares the length of
the leg adjacent the acute angle to the
length of the hypotenuse.
Cosine is abbreviated “cos” but is still
read as “cosine.”
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8-3: Ratios in Right Triangles
Cosine Ratio
B
C
A
leg adjacent
cos θ =
hypotenuse
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8-3: Ratios in Right Triangles
Cosine Ratio
B
C
AC
cos A =
AB
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A
BC
cos B =
AB
8-3: Ratios in Right Triangles
B
10
6
C
8
A
Give the sin and cos ratios for ∠A and ∠B.
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8-3: Ratios in Right Triangles
For the right triangle shown below,
what is the sin C?
a.
b.
c.
d.
e.
a/
b
a/
c
b/
a
c/
b
c/
a
B
b
a
A
c
C
Solve for x in the triangle
below.
24
35°
x
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8-3: Ratios in Right Triangles
Solve for x in the triangle
below.
75°
x
18
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If AC = 10 in the figure below, determine BD.
a.10 2
b.5 3
c.10 3
30°
45°
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d.5 2
e.5
Trig Basics
Tangent Ratio
The tangent ratio of an acute angle of a
right triangle compares the length of
the leg opposite the acute angle to the
length of the leg adjacent the acute
angle.
Tangent is abbreviated “tan” but is still
read as “tangent.”
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8-3: Ratios in Right Triangles
Tangent Ratio
B
C
A
leg opposite
tan θ =
leg adjacent
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8-3: Ratios in Right Triangles
Tangent Ratio
B
C
BC
tan A =
AC
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A
AC
tan B =
BC
8-3: Ratios in Right Triangles
Tangent Ratio
Solve for x in the triangle below.
15
65°
x
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8-3: Ratios in Right Triangles
Solve for x and y below.
y
12
x
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22
°
8-3: Ratios in Right Triangles
To guard against a fall, a ladder should form no more
than a 75° angle with the ground. What is the
maximum height that a 10 foot ladder can safely
reach?
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8-3: Ratios in Right Triangles
A kite is flying at the end of a 240-foot string which
makes a angle with the horizon. If the hand of the
person flying the kit is 3 feet above the ground, how
far above the ground is the kite?

b.120
c.240
d.240

3  3ft
3  3ft
3  3ft
a. 120 3  3 ft
e.123 ft
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Trig Basics
Arc functions
If you know the value of a trig function, you can work
backwards to determine the measure of the angle.
For example, say we know the cos A = .5, then we can
use the cos-1 (arc cosine or inverse of cosine)
function to determine that m∠A = 60°.
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8-3: Ratios in Right Triangles
To calculate angles from cos:
Use the 2nd (shift or inverse) key before
the cos key.
Ex: cos A = .8894
Type .8894 2nd cos .
This returns 27.20, so m∠ A = 27.2°
You may need to type 2nd cos .8894 =
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8-3: Ratios in Right Triangles
To calculate angles from sin:
Use the 2nd (shift or inverse) key before
the sin key.
Ex: sin A = .6
Type .6 2nd sin .
This returns 36.87, so m∠A = 36.87°
You may need to type 2nd sin .6 =
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8-3: Ratios in Right Triangles
To calculate angles from tan:
Use the 2nd (shift or inverse) key before
the tan key.
Ex: tan A = .2341
Type .2341 2nd tan .
This returns 13.17, so m∠ A = 13.17°
You may need to type 2nd tan .2341 =
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8-3: Ratios in Right Triangles
A patient is being treated with radiotherapy for a tumor that is
behind a vital organ. In order to prevent damage to the organ,
the doctor must angle the rays to the tumor. If the tumor is 6.3
cm below the skin and the rays enter the body 9.8 cm to the
right of the tumor, find the angle at which the rays should enter
the body to hit the tumor.
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8-3: Ratios in Right Triangles
The hypotenuse of the right triangle shown
below is 22 feet long. The cosine of angle L is
¾. How many feet long is the segment LM?
A.
B.
C.
D.
E.
18.4
16.5
11.0
6.7
4.7
L
22
M
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N
Assignment
Pages 416 – 419,
# 17-49 (odds), 50, 51, 53 – 65 (odds)
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8-3: Ratios in Right Triangles