Transcript Section 10.1

Section 10.1
Tangent Ratios
Tangent Ratios
• For a given acute angle / A with a measure of
θ°, the tangent of / A, or tan θ, is the ratio of
the length of the leg opposite / A to the
length of the leg adjacent to / A in any right
triangle having A as one vertex, or
• tan θ = opposite/adjacent
Tangent Ratio Examples
• Find the tan θ.
A
D
θ
adj.
5
B
13
hyp.
4.5
hyp.
C
opp. 12
tan θ = opp./adj.
tan θ = 12/5 ≈ 2.4
E
θ
2.7 adj.
F
opp. 3.6
tan θ = opp./adj.
tan θ = 3.6/2.7 ≈ 1.333
Finding Angles Using Tangent Ratios
• Find the indicated angle.
X
W
6
12
R
22.57
T
8
Find / Y.
tan Y = 6/8
/ Y = tan⁻¹(6/8)
/ Y = 36.87°
Y
P
Find / W.
tan W = 22.57/12
/ W = tan⁻¹(22.57/12)
/ W = 62°
Finding Side Measurements Using
Tangent Ratios
• Find the indicated side.
M
N
B
75°
12
x
x
37°
D
18
tan 37 = x/18
18tan37 = x
13.56 ≈ x
H
G
tan 75 = x/12
12tan75 = x
44.78 ≈ x
Finding Side Measurements Using
Tangent Ratios
• Find the indicated side.
M
N
B
53°
5
x
22
42°
D
x
tan 42 = 5/x
5/tan42 = x
5.55 ≈ x
H
G
tan 53 = 22/x
22/tan53 = x
16.58 ≈ x
Section 10.2
Sines and Cosines
Sine and Cosine Ratios
• For a given angle / A with a measure of θ°, the sine
of / A, or sin θ, is the ratio of the length of the leg
opposite A to the length of the hypotenuse in a
right triangle with A as one vertex, or
• sin θ = opposite/hypotenuse
• The cosine of / A, or cos θ, is the ratio of the length
of the leg adjacent to A to the length of the
hypotenuse, or
opp.
• cos θ = adjacent/hypotenuse
adj θ°
hyp.
Sine and Cosine Ratio Examples
• Find the sin θ and cos θ.
A
D
θ
adj.
5
B
13
hyp.
4.5
hyp.
C
opp. 12
sin θ = opp./hyp. cos θ = adj./hyp.
sin θ = 12/13
cos θ = 5/13
sin θ ≈ 0.92
cos θ ≈ 0.38
E
θ
2.7 adj.
F
opp. 3.6
sin θ = opp./hyp. cos θ = adj./hyp.
sin θ = 3.6/4.5
cos θ = 2.7/4.5
sin θ ≈ 0.8
cos θ ≈ 0.6
Finding Angles Using Sine and Cosine
• Find the indicated angle.
X
W
6
10
T
8
25.56
Y
12
R
22.57
P
Find / Y.
Find / W.
sin Y = 6/10
cos Y = 8/10
sin W = 22.57/25.56 cos W = 12/25.56
/ Y = sin⁻¹(6/10) / Y = cos⁻¹(8/10) / W = sin⁻¹(22.57/25.56) / W = cos⁻¹(12/25.56)
/ Y ≈ 36.87°
/ Y ≈ 36.87°
/ W = 62°
/ W = 62°
Finding Side Measurements Using
Tangent Ratios
• Find the indicated side.
M
N
45
75°
x
25
34°
D
sin 34 = x/25
25sin34 = x
13.98 ≈ x
x
H
G
cos 75 = x/45
45cos75 = x
9.36 ≈ x
B
Two Trigonometric Identities
• tan θ = sin θ/cos θ
(sin θ)² + (cos θ)² = 1
Section 10.3
Extending the Trigonometric Ratios
Extending Angle Measure
• Imagine a ray with its endpoint at the origin of
a coordinate plane and extending along the
positive x-axis. Then imagine the ray rotating
a certain number of degrees, say θ,
counterclockwise about the origin. θ can be
any number of degrees, including numbers
greater than 360°. A figure formed by a
rotating ray and a stationary reference ray,
such as the positive x-axis, is called an angle of
rotation.
The Unit Circle
• The unit circle is a circle with its center at the
origin and a radius of 1.
• In the language of transformations, it consists
of all the rotation images of the point P(1, 0)
about the origin.
P(1, 0)