Download 8.2A Trigonometric Ratios

8.2A 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.
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
32°
C
B
32°
N
Y
Since the triangles are similar . . .
32°
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.
B
c
a
C
b
A
Z
The sine of an angle is the ratio of the
length of the leg opposite the angle to
the length of the hypotenuse.
sin A 
Opposite Leg a

Hypotenuse c
sin B 
Opposite Leg b

Hypotenuse c
The cosine of an angle is the ratio of the
length of the leg adjacent to the angle to
the length of the hypotenuse.
cos A 
Adjacent Leg b

Hypotenuse c
cos B 
Adjacent Leg a

Hypotenuse c
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.
tan A 
Opposite Leg a

AdjacentLeg b
tan B 
Opposite Leg b

AdjacentLeg a
An often used method of remembering these is the word SOH-CAH-TOA.
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
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 X
5/13
0.38
13
5
c. tan Y
5/12
0.42
Y
Assessment:
Question Student pairs.
Independent Practice:
Text: pgs. 545 – 548 prob.
For a Grade:
Text: pgs. 545 – 548 prob.
12
Z
K