Chapter Three: Right Triangle Trigonometry
 The Tangent Ratio (3.1)
In the last lesson we talked about the ratios:
B
opp
sinA = opp
hyp
cos A = adj
hyp
Quad II
Quad I
Quad III
Quad IV
hyp
C
adj
A
Is there another ratio we could make?
opp/adj
tangent ratio: for an acute angle in a right triangle, the ratio of the length of the
𝒐𝒑𝒑
opposite side to the length of the adjacent. tan A =
𝒂𝒅𝒋
primary trigonometric ratios: are the three ratios, sine, cosine and tangent defined in a
right triangle.
SOH
CAH
TOA
eg #1 Determine the ratio to four decimal places and list the quadrant where the angle is
found.
a. tan 23°= 0.4245
and it lies in Quadrant
I .
b.
tan 98°= -7.1154
and it lies in Quadrant
II .
c.
tan 123° = -1.5399 and it lies in Quadrant
II .
d.
tan 224°= 0.9657
III .
e.
tan 330°= -0.5774 and it lies in Quadrant
and it lies in Quadrant
IV .
eg #2 What is the measure of the angle to the nearest degree if the following ratios are
known?
a. tan Ө = 1.3270
Ө = 53
b.
Ө = 25
tan Ө = 0.4663
c. tan Ө = 2.1445
Ө = 65
eg #3 What are the measures of the unknown sides and angles, (solve the triangle).
**Three angles of a triangle = 180 Pythagorean Theorem: c2 = a2 + b2
a.
b.
12 mm


5 mm
c2 = a2 + b2
c2 = 52 + 122
c = 13 mm
13 ft
c2 = a2 + b2
c2 = 132 + 82
c = 15.2 ft
8 ft
tan  =
𝒐𝒑𝒑
𝒂𝒅𝒋
tan  =
𝒐𝒑𝒑
𝒂𝒅𝒋
tan  =
𝒐𝒑𝒑
𝒂𝒅𝒋
tan  =
𝒐𝒑𝒑
𝒂𝒅𝒋
tan  =
𝟓
𝟏𝟐
tan  =
𝟏𝟐
𝟓
tan  =
𝟖
𝟏𝟑
tan  =
𝟏𝟑
𝟖
  = 23
  = 67
  = 32
  = 58
eg #4 From the top of a building 35 m tall, the angle of elevation to the top of a taller
building is 35. The distance between the buildings is 43 m. What is the height of the
taller building?
x
35
43 m
35 m
tan  =
𝒐𝒑𝒑
𝒂𝒅𝒋
tan 35 =
𝒙
𝟒𝟑
x = tan 35 x 43
x = 30.1 m
Assign: p.108-110 #3-9, 11, 13
height of taller building:
35 m + 30.1 m = 65.1 m