Given two intersecting lines, let L 2 be the line with the larger angle of inclination ϕ 2 , and let L 1 be the line with the smaller angle of inclination ϕ 1 . We define the angle θ between L 1 and L 2 by θ = ϕ 2 − ϕ 1 . (See the accompanying figure.) (a) Prove: If L 1 and L 2 are not perpendicular, then tan θ = m 2 − m 1 1 + m 1 m 2 where L 1 and L 2 have slopes m 1 and m 2 . (b) Prove Theorem 10.4.5. [Hint: Introduce coordinates so that the equation x 2 / a 2 + y 2 / b 2 = 1 describes the ellipse, and use part (a).] (c) Prove Theorem 10.4.6. [Hint: Introduce coordinates so that the equation x 2 / a 2 − y 2 / b 2 = 1 describes the hyperbola, and use part (a).]
Given two intersecting lines, let L 2 be the line with the larger angle of inclination ϕ 2 , and let L 1 be the line with the smaller angle of inclination ϕ 1 . We define the angle θ between L 1 and L 2 by θ = ϕ 2 − ϕ 1 . (See the accompanying figure.) (a) Prove: If L 1 and L 2 are not perpendicular, then tan θ = m 2 − m 1 1 + m 1 m 2 where L 1 and L 2 have slopes m 1 and m 2 . (b) Prove Theorem 10.4.5. [Hint: Introduce coordinates so that the equation x 2 / a 2 + y 2 / b 2 = 1 describes the ellipse, and use part (a).] (c) Prove Theorem 10.4.6. [Hint: Introduce coordinates so that the equation x 2 / a 2 − y 2 / b 2 = 1 describes the hyperbola, and use part (a).]
Given two intersecting lines, let
L
2
be the line with the larger angle of inclination
ϕ
2
,
and let
L
1
be the line with the smaller angle of inclination
ϕ
1
.
We define the angle
θ
between
L
1
and
L
2
by
θ
=
ϕ
2
−
ϕ
1
.
(See the accompanying figure.)
(a) Prove: If
L
1
and
L
2
are not perpendicular, then
tan
θ
=
m
2
−
m
1
1
+
m
1
m
2
where
L
1
and
L
2
have slopes
m
1
and
m
2
.
(b) Prove Theorem 10.4.5. [Hint: Introduce coordinates so that the equation
x
2
/
a
2
+
y
2
/
b
2
=
1
describes the ellipse, and use part (a).]
(c) Prove Theorem 10.4.6. [Hint: Introduce coordinates so that the equation
x
2
/
a
2
−
y
2
/
b
2
=
1
describes the hyperbola, and use part (a).]
(c) A third tower is located at Heights Barn Hill. Let DEF represent the points on the map for Cleggswood Hill, Hollingworth Hill and Heights Barn Hill respectively. On the map, DE = 3.5 cm and EF = 5.5 cm and ZDEF = 105°.
(i) Is ZDEF on the map greater than, less than, or the same as the angle between the horizontal line between Cleggswood Hill and Hollingworth Hill and the horizontal line between Hollingworth Hill and Heights Barn Hill in real life? Explain your answer.
(ii) Find the length DF.
(iii) Find the EFD.
(iv) Find the area of triangle DEF.
(c) A third tower is located at Heights Barn Hill. Let DEF represent the
points on the map for Cleggswood Hill, Hollingworth Hill and Heights
Barn Hill respectively. On the map, DE = 3.5 cm and EF = 5.5 cm and
ZDEF = 105°.
(i)
Is ZDEF on the map greater than, less than, or the same as the
angle between the horizontal line between Cleggswood Hill and
Hollingworth Hill and the horizontal line between Hollingworth Hill
and Heights Barn Hill in real life? Explain your answer.
(ii) Find the length DF.
(iii) Find the ZEFD.
Find a · b if |a| =
14, |b| = 9, and the angle between a and bis T/2.
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