Download 8.2 Trigonometric Ratios

8.2 Trigonometric Ratios
Objectives:
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.
For the Board: You will be able to find the sine, the cosine, and the tangent of an acute angle.
Bell Work:
Write each fraction as a decimal rounded to the nearest hundredth.
1. 2/3
2. 7/24
Solve each equation.
5.8
x
3. 0.8 
4. 0.94 
x
8.5
Anticipatory Set:
By AA Similarity, a right triangle with a given acute angle, such as 32°, is similar to every other right
triangle with that same acute angle measure.
L
X
A
ΔABC ~ ΔLMN ~ ΔXYZ
M
C
32°
B
32°
N
Y
32°
Since the triangles are similar . . .
The ratio of the short leg to the hypotenuse in each of these triangles will be the same
AC LN XZ
.


AB LM XY
The ratio of the long leg to the hypotenuse in each of these triangles will be the same
BC MN YZ
.


AB LM XY
The ratio of the short leg to the long leg in each of these triangles will be the same
AC LN XZ
.


BC MN YZ
These ratios are called trigonometric ratios.
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as
sin, cos, and tan.
Trigonometric Ratios
Let ABC be a right triangle.
The sine, the cosine, and the tangent of the acute angles are defined as follows.
Z
The sine of an angle is the ratio of the
length of the leg opposite the angle to
the length of the hypotenuse.
The cosine of an angle is the ratio of the
length of the leg adjacent to the angle
to the length of the hypotenuse.
The tangent of an angle is the ratio of
the length of the leg opposite the angle
to the length of the leg adjacent to the
angle.
sin A 
Opposite Leg a

Hypotenuse c
sin B 
Opposite Leg b

Hypotenuse c
B
Adjacent Leg b
cos A 

Hypotenuse c
cos B 
Adjacent Leg a

Hypotenuse c
tan A 
Opposite Leg a

AdjacentLeg b
tan B 
Opposite Leg b

AdjacentLeg a
c
a
C
A
b
An often used method of remembering these is the work SOHCAHTOA.
Open the book to page 541 and read example 1.
Example: Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest
hundredth.
J
a. sin J
60/61
0.98
61
11
b. cos J
11/61
0.18
c. tan K
11/60
0.18
L
60
d. tan J
e. sin K
f. cos K
White Board Activity:
Additional Practice: Write each trigonometric ratio as a fraction and as a decimal rounded to the
nearest hundredth.
X
a. sin Y
5/13
0.38
b. cos Y
12/13
0.92
13
5
c. tan Y
5/12
0.42
d. sin X
12/13
0.92
Y
Z
12
e. cos X
5/13
0.38
f. tan X
12/5
2.4
K
The trigonometric ratios of 30°, 45°, and 60° can be found using the special right triangle formulas.
Thus they are NEVER written as rounded off decimals.
Use any number you like for a.
Let a = __3____
Determine the measures of the sides
of the triangles.
45
2
a
a
60
a
2a
30
45
a 3
a
3, 3, 3 2
3, 3 2 , 6
3
Sin 45 =
3 2
Cos 45 =
3
3 2

1

1
2

2
2
sin 30 = 3/6 = ½

2
2
cos 30 =
2
Tan 45 = 3/3 = 1
tan 30 =
sin 60 =
3 3
3

6
2
3
3 3

3 3
3

6
2
cos 60 = 3/6 = ½
1
3

3
3
tan 60 =
3 3
 3
3
To find the sin, cos, and/or tan of all other angles a calculator is necessary.
Before you can do any evaluation you must make sure your calculator is in degree mode.
For Graphing Calculators or calculators which have a MODE key:
push MODE, use the arrow key to move down to Radian Degree, use the arrow key to move
right till Degree is highlighted, push ENTER or =.
For Non-graphing Calculators:
look for a button labeled DRG, push repeatedly till you see a small D in the display.
There are two types of calculators which can do trigonometric functions:
algebra logic, and non-algebra logic.
Algebra Logic Key Strokes
Sin
Cos
Tan
(
Degree
Non-Algebra Logic Key Strokes
)
=
(
Degree
)
Sin
Cos
Tan
Open the book to page 541 and read example 3.
Example: Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
a. sin 52°
0.79
b. cos 19°
0.95
c. tan 65°
2.14
White board Activity:
Practice: Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
a. sin 65°
0.91
b. cos 74°
0.28
c. tan 27°
0.51
Trigonometric ratios can be used to find a missing side on a right triangle given an acute angle and a side.
Open the book to page 542 and read example 4.
Example: Find each length. Round to the nearest hundredth.
(Do problem (a) as an example.)
a. BC
b. QR
A
c. FD
D
Q
10.2 ft
15°
C
R
B
12.9 cm
63° P
F 39°
20 m
Use x to represent BC.
sin 63° = QR/12.9
x is adjacent to <B
QR = 12.9 sin 63°
10.2 is opposite <B
QR = 11.49 c m
Use tan to solve for x.
Tan 15° = 10.2/x
BC = 10.2/tan 15° = 38.07 ft.
White Board Activity:
Practice: Find each length. Round to the nearest hundredth.
a. AB
b. RQ
Q
A
17 m
C
51°
sin 51 = 17/AB
AB = 21.87 m
B
R
9.5 in
42° P
sin 42 = RQ/9.5
RQ = 6.36 in
E
cos 39° = 20/FD
FD = 20/cos 39°
FD = 25.74 m
D
c. DE
F 27°13.6 m E
tan 27 = DE/13.6
DE = 6.93 m
Open the book to page 543 and read example 5.
Example: The Pilatusbahn in Switzerland is the world’s steepest railway. Its steepest section makes
an angle of about 25.6° with the horizontal and rises about 0.9 km. To the nearest
hundredth of a kilometer, how long is this section of the railway track?
sin 25.6° = 0.9/D
D
D = 0.9/sin 25.6°
0.9 km
D = 2.08 km
25.6°
White Board Activity:
Practice: A contractor is building a wheel chair ramp for a doorway that is 1.2 ft above the ground. The
ramp will make an angle of 4.8° with the ground. Find the length of the ramp to the nearest
hundredth of a foot.
Sin 4.8 = 1.2/D
D
1.2 ft
D = 14.34 ft
4.8°
Assessment:
Student pairs will complete “CHECK IT OUT” prob. 1 – 5 from this section.
Independent Practice:
Text: pgs. 545-548 prob. 4-8 even, 9-11, 12-16 even, 18-20, 22-26 even, 28-30, 32-42 even, 43.