12
Chapter 1
•
Right Triangle Trigonometry
§1.2
Example1.10
radicallow
3
30
◦
A
B
C
D
E
F
radicalBig
3
2
radicalBig
3
2
2
1
45
◦
Find the sine, cosine, and tangent of 75
◦
.
Solution:
Since 75
◦
=
45
◦
+
30
◦
, place a 30
−
60
−
90 right triangle
△
ADB
with legs of length
radicallow
3 and 1 on top of the hypotenuse
of a 45
−
45
−
90 right triangle
△
ABC
whose hypotenuse has
length
radicallow
3, as in the figure on the right. From Figure 1.2.2(a) we
know that the length of each leg of
△
ABC
is the length of the
hypotenuse divided by
radicallow
2. So
AC
=
BC
=
radicallow
3
radicallow
2
=
radicalBig
3
2
. Draw
DE
perpendicular to
AC
, so that
△
ADE
is a right triangle. Since
∠
BAC
=
45
◦
and
∠
DAB
=
30
◦
, we see that
∠
DAE
=
75
◦
since
it is the sum of those two angles. Thus, we need to find the sine,
cosine, and tangent of
∠
DAE
.
Notice that
∠
ADE
=
15
◦
, since it is the complement of
∠
DAE
.
And
∠
ADB
=
60
◦
, since it is the complement of
∠
DAB
. Draw
BF
perpendicular to
DE
, so that
△
DFB
is a right triangle.
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 Fall '11
 Dr.Cheun
 Calculus, Trigonometry, Right triangle

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