Transcript Blended Schools

Lesson 8-6
The Sine and Cosine Ratios
(page 312)
The sine ratio and cosine ratio
relate the legs to
the hypotenuse .
How can trigonometric ratios be used
to find sides and angles of a triangle?
The tangent ratio is the ratio
of the lengths of the legs .
B
c
a
Review!
A
b
leg
C
leg
In relationship to angle A …
c
A
B
a
b
adjacent leg
C
opposite leg
opposite leg vs adjacent leg
In relationship to angle B …
c
A
B
a
b
opposite leg
C
adjacent leg
opposite leg vs adjacent leg
Definition of Tangent Ratio
B
c
a
Review!
A
b
tangent of ∠A = tan A
C
a
BC
=
=
AC
b
length of the leg opposite ÐA
tan A =
length of the leg adjacent to ÐA
Definition of Sine Ratio
B
c
A
a
b
sine of ∠A = sin A
C
a
BC
=
=
AB
c
length of the leg opposite ÐA
sin A =
length of the hypotenuse
Definition of Cosine Ratio
B
c
A
a
b
cosine of ∠A = cos A
C
b
AC
=
=
AB
c
length of the leg adjacent to ÐA
cos A =
length of the hypotenuse
B
c
a
A
C
b
length of the leg opposite ÐA
sin A =
length of the hypotenuse
length of the leg adjacent to ÐA
cos A =
length of the hypotenuse
length of the leg opposite ÐA
tan A =
length of the leg adjacent to ÐA
B
REMEMBER!
A
opposite
sin =
hypotenuse
c
a
b
adjacent
cos =
hypotenuse
DO NOT
FORGET!
C
opposite
tan =
adjacent
SOH CAH TOA
Example 1
Express the sine and cosine of ∠A & ∠B as ratios.
B
O
5
____
C
5
13
(a) sin A = ______
(c) sin B = ______
BC + 12 = 13
2
BC +144 =169
2
13
H
2
BC = 25
2
BC = 5
12
A
2
A
12
13
(b) cos A = ______
(d) cos B = ______
Example 1
Express the sine and cosine of ∠A & ∠B as ratios.
B
A
5
____
C
5
13
(a) sin A = ______
12
13
(c) sin B = ______
BC +12 =13
2
13
H
2
BC +144 =169
2
BC = 25
2
BC = 5
12
O
2
A
12
13
(b) cos A = ______
5
13
(d) cos B = ______
Example 2
Use the trigonometry table, then use a calculator to
find the approximate decimal values.
“≈” means “is approximately equal to”
0.3746
(a) sin 22º ≈ __________
0.1908
(b) cos 79º ≈ __________
Please note
that these are only
APPROXIMATIONS!
Now try this with a calculator!
To enter this in your calculator you will need
to use the SIN or COS function key.
Enter SIN(22) then press ENTER (=)
and round to 4 decimal places.
(a)
0.3746
sin 22º ≈ ____________
Enter COS(79) then press ENTER (=)
and round to 4 decimal places.
(b)
0.1908
cos 79º ≈ ____________
… Example 2
Use the trigonometry table to find the
approximate angle measures.
“≈” means “is approximately equal to”
61º ≈ 0.8746
(c) sin ______
39º ≈ 0.7771
(d) cos _______
Please note
that these are only
APPROXIMATIONS!
Now try this with a calculator!
To enter this in your calculator you will need to use
the inverse key or 2nd function key.
Enter SIN-1(.8746) then press ENTER (=)
and round to the nearest degree
61º ≈ 0.8746
(c) sin ______
Enter COS-1(.7771) then press ENTER (=)
and round to the nearest degree
39º ≈ 0.7771
(d) cos _______
What can you say about the values
for the sine or cosine of an angle?
The values for sine and cosine …
Think about it …
if the hypotenuse is the longest side
and it is the denominator of the ratios,
then it …
… will always be less than 1.
Enter SIN-1(1.5) then press ENTER (=) …
Example 3 (a)
Find the value of x and y to the nearest integer.
52
x ≈ ______
y ≈ ______
x
sin 38º =
84
84
38º
x
x = 84 × sin38º
x » 51.7
y
Example 3 (a)
Find the value of x and y to the nearest integer.
52
x ≈ ______
66
y ≈ ______
y
cos 38º =
84
84
38º
x
y
y = 84 × cos38º
y » 66.19
Example 3 (b)
Find the value of x and y to the nearest integer.
12
x ≈ ______
y ≈ ______
7
cos 55º =
1
x
x
55º
x
y
7
14
7
7
x=
1
cos55º
x »12.2
Example 3 (b)
Find the value of x and y to the nearest integer.
12
x ≈ ______
10
y ≈ ______
y
tan 55º =
7
x
55º
x
y
7
14
7
y = 7× tan55º
y » 9.997
Example 4 (a): Find “n” to the nearest degree.
44
n ≈ ______
14
sin nº =
20
nº
H
20
14 O
æ 14 ö
n = sin ç ÷
è 20 ø
-1
n » 44.4
Example 4 (b)
Find the measures of the 3 angles of a 3-4-5 ∆.
Example 4 (b)
Find the measures of the 3 angles of a 3-4-5 ∆.
5 ? 3 +4
2
5
3
4
2
2
25 ? 9 + 16
25 = 25
∴ a right triangle.
continue
Example 4 (b)
Find the measures of the 3 angles of a 3-4-5 ∆.
nº
5
3
4
sin nº =
5
æ 4ö
n = sin ç ÷
è 5ø
-1
4
90º - 53º = 37º
n » 53
∴ the angle measures are 37º, 53º, & 90º.
OPTIONAL Assignment
Written Exercises on pages 314 to 316
1 to 17 odd numbers
~ #20 is BONUS! ~
Trigonometry Worksheet #1
Lessons 8-5 & 8-6
The Sine, Cosine, and Tangent Ratios
How can trigonometric ratios be used
to find sides and angles of a triangle?