Similar Triangles, Right Triangles, And The Definition Of The Sine, Cosine And Tangent Functions Of Angles Of A Right Triangle Worksheets With Answer Key Page 7
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43Case I:
One side and two angles are given.
Case II:
Two sides and one opposite angle are given.
Case III: Two sides and the included angle are given.
Case IV: Three sides are given.
The Law of Sines is used in Cases I and II. The Law of Cosines is used in Cases III and IV.
LAW OF SINES
For any triangle ABC,
a
b
c
=
=
sin A
sin B
sin C
Figure T16
Example: Case I: One side and two angles given. Solve ΔBQK where a = 12, Q = 110° and
K = 45°.
Solution: In this case, the angle B is immediately found as
=
°
−
+
=
°
−
°
=
°
B
180
(
Q
K
)
180
155
25
.
We can solve for q by the Laws of Sines:
q
12
=
°
°
sin
110
sin
25
so
Figure T17
12
=
⋅ °
=
q
sin
110
26
.
68
.
°
sin
25
We can solve for k by the Law of Sines:
k
12
=
°
°
sin
45
sin
25
so
12
=
⋅ °
=
k
sin
45
20
.
08
.
°
sin
25
Problems:
20. Find side b in ∆ABC if a = 11, A = 80 and B = 65°.
21. Solve ∆PDQ where q = 72.1, P = 27° and Q = 25°.
Case II: (Given two sides and one opposite angle) may also be solved by the Law of Sines ––
provided a solution exists. Case II is called the ambiguous case because there may be two solutions,
or just one solution, or no solution at all. Use your review references to study the various possibilities.
The figures on the next page show some of the possibilities for solutions (or no solution) given sides a
and c, and angle A.
- 7 -
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