Transcript Angles, Degrees, and Special Triangles

Law of Sines
Trigonometry
MATH 103
S. Rook
Overview
• Sections 7.1 & 7.2 in the textbook:
– Law of Sines: AAS/ASA Case
– Law of Sines: SSA Case
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Law of Sines: AAS/ASA Case
Oblique Triangles
• Oblique Triangle: a triangle containing no
right angles
• All of the triangles we have studied thus far
have been right triangles
– We can apply SOHCAHTOA or the Pythagorean
Theorem only to right triangles
• Naturally most triangles will not be right
triangles
– Thus we need a method to apply to find side
lengths and angles of other types of triangles
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Four Cases for Oblique Triangles
• By definition a triangle has three sides and three
angles for a total of 6 components
– We can find the measure of all sides and all angles if
we know AT LEAST 3 of these components
• Broken down into four cases:
– AAS or ASA
• Measure of two angles and the length of any side
– SSA
• Length of two sides and the measure of the angle
opposite one of two known sides
• Known as the ambiguous case because none, one, or
two triangles could be possible
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Four Cases for Oblique Triangles
(Continued)
– SSS
• Length of all three sides
– SAS
• Length of two sides and the measure of the angle
opposite the third (possibly unknown) side
– AAA is NOT a case because there are an infinite
number of triangles that can be drawn
• Recall that the largest side is opposite the largest and
angle, but there is no limitation on the length of the
side!
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Law of Sines
• The first two cases (AAS/ASA and SSA) are covered by
the Law of Sines: a
b
c
sin A

sin B

sin C
– i.e. the ratio of the measure of any side of a triangle to
its corresponding angle yields the same constant value
• This constant value is different for each triangle
– The Law of Sines can be proved by dropping an
altitude from an oblique triangle and using
trigonometric functions with the right angle
• See page 339
• ALWAYS draw the triangle!
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Law of Sines: AAS/ASA (Example)
Ex 1: Use the Law of Sines to solve the triangle
– round answers to two decimal places:
a) A = 102.4°, C = 16.7°, a = 21.6
b) A = 55°, B = 42°, c = ¾
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Law of Sines: SSA Case
SSA – the Ambiguous Case
• Occurs when we know the length of two sides and
the measure of the angle opposite one of two
known sides
– e.g. a, b, A and b, c, C are SSA cases
– e.g. a, b, C and b, c, A are NOT (they are SAS cases)
• To solve the SSA case:
– Use the Law of Sines to calculate the missing angle
across from one of the known sides
– There are three possible cases:
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SSA – the Ambiguous Case
(Continued)
• The known angle corresponds to one of the given sides
of the triangle (e.g. In a, b, A, the known angle is A)
– i.e. the side which both the angle and length are given
is the side with the known angle
• The missing angle corresponds to the second given side
(e.g. In a, c, C, the missing angle is A)
• Case I: sin missing > 1
– e.g. sin missing = 1.3511
– Recall the domain for the inverse sine: -1 ≤ x ≤ 1
– NO triangle exists
• If -1 ≤ missing ≤ 1, at least one triangle is guaranteed to
exist
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• Inverse sine will give the value of missing in Q I
SSA – the Ambiguous Case
(Continued)
• An angle of a triangle can have a measure of up to 180°
• Sine is also positive in Q II (90° < θ < 180°) meaning that
it is POSSIBLE for missing to assume a value in Q II
• Find this second possible value using reference angles
• Case II: second + known ≥ 180°
– i.e. the measure of the possible second angle yields a
second triangle with contradictory dimensions
– ONE triangle exists
• Case III: second + known < 180°
– i.e. the measure of the possible second angle yields a
second triangle with feasible dimensions
– known is the same in BOTH triangles
– TWO triangles exist
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SSA – the Ambiguous Case
(Example)
Ex 2: Use the Law of Sines to solve for all
solutions – round to two decimal places:
a)
b)
c)
d)
A = 54°, a = 7, b = 10
A = 98°, a = 10, b = 3
C = 27.83°, c = 347, b = 425
B = 58°, b = 11.4, c = 12.8
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Law of Sines Application (Example)
Ex 3: A man standing near a radio station
antenna observes that the angle of elevation
to the top of the antenna is 64°. He then
walks 100 feet further away and observes that
the angle of elevation to the top of the
antenna is 46° (see page 345). Find the height
of the antenna to the nearest foot.
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Summary
• After studying these slides, you should be able
to:
– Apply the Law of Sines in solving for the components of a
triangle or in an application problem
– Differentiate between the AAS/ASA and SSA cases
• Additional Practice
– See the list of suggested problems for 7.1 & 7.2
• Next lesson
– Law of Cosines (Section 7.3)
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