Additional Topics In Trig

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Chapter 6

• Additional Topics In Trig

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• 6.1 Law of Sines for AAS, ASA, or SSA
• 6.2 Law of Cosines
• 6.3 Vectors in the Plane
• 6.4 Vectors and Dot Products
• 6.5 Trig Form of a Complex Number

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Remember from geometry
Sketch triangle. Label given information. Area
of triangle ½ bh
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Inequalities within one triangle

• If one angle of a triangle is larger than another
  angle in the triangle then the side opposite the
  larger angle will be longer than the side
  opposite the smaller angle.
• Converse is true

Large angle? long side Small angle ?shorter side
Long side ? Large angle Shorter side ?Small angle

• Right Triangles
• Pythagorean Theorem c² a²
  b²
• Special Right Triangles
• 45 45 - 90 30 60 -
  90
• s s s v 2 s
  sv3 2s
• Trig Ratios
• sin cos
  tan
• Angle of depression and elevation

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6.1 The Law
of SINES
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The Law of SINES Why?
For any triangle (right, acute or obtuse), you
may use the following formula to solve for
missing sides or angles Note A, B,C are
angles a, b, c are the sides opposite the
corresponding angle
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Use Law of SINES when ...
you have 3 dimensions of a triangle and you need
to find the other 3 dimensions - they cannot be
just ANY 3 dimensions though, or you wont have
enough info to solve the Law of Sines equation.
Use the Law of Sines if you are given

• AAS - 2 angles and 1 adjacent side
• ASA - 2 angles and their included side
• SSA (this is an ambiguous case)

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Example 1 (AAS)

• You are given a triangle, ABC, with angle A
  70, angle B 80 and side a 12 cm. Find the
  measures of angle C and sides b and c.
• In this section, angles are named with capital
  letters and the side opposite an angle is named
  with the same lower case letter

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Example 1 (AAS cont)
The angles in a ? total 180, so angle C
30. Set up the law of sines to find side b
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Example 1 (AAS cont)
Set up the law of sines to find side c
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Example 1 (AAS solution)
Angle C 30 Side b 12.6 cm Side c 6.4 cm
Note We used the given values of A and a in both
calculations, your answer is more accurate if you
do not used rounded values in calculations.
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Example 2 (ASA)

• You are given a triangle, ABC, with angle C
  115, angle B 30 and side a 30 cm. Find the
  measures of angle A and sides b and c.

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Example 2 (ASA cont)
To solve for the missing sides/angles, we must
have an angle/side opposite pair to set up the
first equation. We MUST find angle A first
because the only side given is side a. The
angles in a ? total 180, so angle A 35.
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Example 2 (ASA cont)
Set up the law of sines to find side b
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Example 2 (ASA cont)
Set up the law of sines to find side c
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Example 2 (ASA solution)
Angle A 35 Side b 26.2 cm Side c 47.4 cm
Note Use the law of sines whenever you are
given 2 angles and one side!
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The Ambiguous Case (SSA)

• When given SSA (two sides and an angle that is
  NOT the included angle) , the situation is
  ambiguous.
• The dimensions may not form a triangle, or there
  may be 1 or 2 triangles with those dimensions.
• We first go through a series of tests to
  determine how many (if any) solutions exist.

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The Ambiguous Case (SSA)
In the following examples, the given angle will
always be angle A and the given sides will be
sides a and b. If you are given a different set
of variables, feel free to change them to
simulate the steps provided here.
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The Ambiguous Case (SSA)
Situation I Angle A is obtuse If angle A is
obtuse there are TWO possibilities
If a b, then a is too short to reach side c - a
triangle with these dimensions is impossible.
If a gt b, then there is ONE triangle with these
dimensions.
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The Ambiguous Case (SSA)
Situation I Angle A is obtuse - EXAMPLE
Given a triangle with angle A 120, side a 22
cm and side b 15 cm, find the other
dimensions.
Since a gt b, these dimensions are possible. To
find the missing dimensions, use the law of sines
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The Ambiguous Case (SSA)
Situation I Angle A is obtuse - EXAMPLE
Angle C 180 - 120 - 36.2 23.8 Use law of
sines to find side c
Solution angle B 36.2, angle C 23.8, side
c 10.3 cm
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The Ambiguous Case (SSA)
Situation II Angle A is acute If angle A is
acute there are SEVERAL possibilities.
Side a may or may not be long enough to reach
side c. We calculate the height of the
altitude from angle C to side c to compare it
with side a.
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The Ambiguous Case (SSA)
Situation II Angle A is acute
First, use SOH-CAH-TOA to find h
Then, compare h to sides a and b . . .
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If a lt h, then NO triangle exists with these
dimensions.
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If h lt a lt b, then TWO triangles exist with these
dimensions
If we open side a to the outside of h, angle B
is acute.
If we open side a to the inside of h, angle B
is obtuse.
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If h lt b lt a, then ONE triangle exists with these
dimensions
Since side a is greater than side b, side a
cannot open to the inside of h, it can only open
to the outside, so there is only 1 triangle
possible!
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The Ambiguous Case (SSA)
Situation II Angle A is acute
If h a, then ONE triangle exists with these
dimensions
If a h, then angle B must be a right angle and
there is only one possible triangle with these
dimensions.
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
Given a triangle with angle A 40, side a 12
cm and side b 15 cm, find the other
dimensions.
Find the height
Since a gt h, but lt b, there are 2 solutions and
we must find BOTH.
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
FIRST SOLUTION Angle B is acute - this is the
solution you get when you use the Law of Sines!
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
SECOND SOLUTION Angle B is obtuse - use the
first solution to find this solution.
In the second set of possible dimensions, angle B
is obtuse, because side a is the same in both
solutions, the acute soln for angle B the
obtuse soln for angle B are supplementary. Angle
B 180 - 53.5 126.5
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
SECOND SOLUTION Angle B is obtuse
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EX. 1 (Summary)
Angle B 126.5 Angle C 13.5 Side c 4.4
Angle B 53.5 Angle C 86.5 Side c 18.6
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 2
Given a triangle with angle A 40, side a 12
cm and side b 10 cm, find the other
dimensions.
Since a gt b, and h is always less than b, we know
this triangle has just ONE possible solution -
side aopens to the outside of h.
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The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 2
Using the Law of Sines will give us the ONE
possible solution
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The Ambiguous Case - Summary
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The Ambiguous Case - Summary
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The Ambiguous Case

• Determine the of triangles possible in each of
  the following cases.
• A 62, a 10, b 12 0 triangles
• A 98, a 10, b 3 1 triangle
• A 54, a 7, b 10 0 triangles
• A 58, a 4.5, b 12.8 0 triangles
• A 58, a 11.4, b 12.8 2 solutions

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6.1 The Law of Sines
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6.1 The Law of Sines

• Homework p.398 3 15 m3, 16-20

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Area of Oblique Triangles
Formulas Area 1/2 bc sin A Area 1/2 ab sin
C Area 1/2 ac sin B

• PROOF
•   Draw a triangle on a rectangular coordinate
  system, and draw in its altitude.   Label the
  base AC   with the origin being A,  and the upper
  vertex B. B has     coordinates (x,y).
• Area 1/2 by
• sin A y/c
• Therefore, y c sin A
• so Area 1/2 bc sin A
•   The other three formulas can be obtained in a
  similar way.

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Area of Triangles
Example Find the area of the triangle with A
40º, b 10 cm, and c 14 cm. Area 1/2
(10)(14) sin 40º Area 44.995 sq cm  
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6.2 The Law
of COSINES
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6.2 The Law of COSINES WHY?
For any triangle (right, acute or obtuse), you
may use the following formula to solve for
missing sides or angles
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Use Law of COSINES when ...
you have 3 dimensions of a triangle and you need
to find the other 3 dimensions - they cannot be
just ANY 3 dimensions though, or you wont have
enough information to solve the Law of Cosines
equations. Use the Law of Cosines if you are
given

• SAS - 2 side and their included angle
• SSS

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Example 1 Given SAS
Find all the missing dimensions of triangle, ABC,
given that angle B 98, side a 13 and side c
20.
Use the cosines equation that uses a, c and B to
find side, b
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Example 1 Given SAS
Now that we know B and b, we can use the law of
sines to find one of the missing angles
Solution b 25.3, C 51.5, A 30.5
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Example 2 Given SAS
Find all the missing dimensions of triangle, ABC,
given that angle A 39, side b 20 and side c
15.
Use the cosines equation that uses b, c and A to
find side, a
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Example 2 Given SAS
Use the law of sines to find one of the missing
angles
Important Notice that we used the sine equation
to find angle C rather than angle B. The SINE
equation will never produce an obtuse angle. If
we had used the SINE equation to find angle B we
would have gotten 87.5, which is not correct, it
is the reference angle for the correct answer,
92.5. If an angle might be obtuse, never use
the sine equation to find it.
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Example 3 Given SSS
Find all the missing dimensions of triangle, ABC,
given that side a 30, side b 20 and side c
15.
We can use any of the cosine equations, filling
in a, b c and solving for one angle. Once we
have an angle, we can either use another cosine
equation to find another angle, or use the law of
sines to find another angle.
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Example 3 Given SSS
Important The law of SINES will never produce
an obtuse angle. If an angle might be obtuse,
never use the law of sines to find it. For this
reason, we will use the law of cosines to find
the largest angle first (in case it happens to be
obtuse).
Angle A is largest because side a is largest
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Example 3 Given SSS
Use law of sines to find angle B or C (its safe
because they cannot be obtuse)
Solution A 117.3 B 36.3 C 26.4
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The Law of Cosines
Important The law of SINES will never produce
an obtuse angle. If an angle might be obtuse,
never use the law of sines to find it.
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Heron's Area Formula

• K is Area         
• where  s a b c              
                       2

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Heron's Area Formula

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Why We Need a Hero The area of a triangle is one
half base times height. We all remember that. But
what do you do if you don't know the height?
Euclid showed us 23 centuries ago that a triangle
is completely determined by the lengths of its
three sides. So if you know those lengths, you
ought to be able to determine the triangle's
area, right? But unless the triangle has a right
angle in it, none of those three sides are its
height. Then how do you find the area? Hero to
the Rescue                                    
                                               
                              Here is a diagram
of a triangle with sides of length a, b, and c.
The height is shown as length, h, and the point
on the bottom where the height segment intersects
the base at right angles is shown as point, O. In
this case we have side a as the base, and more
importantly, we have chosen the longest side as
the base. Whether we know the height or not, by
the formula we learned in grade school, we still
have for area, A A ½ ah
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Drop the altitude from angle C and label it d and
label where it intersects its base D.   Then sin
56o d/30.  Solve for d. Find angle A, using sin
A d/25. Find angle C, using C 180 - (A
56o). K 1/2 ab sin C.
240.143 sq m
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Draw a circle with radius 1.  Draw a regular
hexagon inside the circle.  Connect the center of
the circle to each vertex of the hexagon.  The
length of each of these lines is 1. Pick one
triangle and find its area.  The central angle of
the triangle is 360/6 60o.
(3/2)v2 sq units
2.598 sq units
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Draw a parallelogram.   Label the horizontal
sides 12.5 and the other two sides 8.  One of
the acute angles 40o.   Draw the diagonal that
is opposite the 40o angle. Find the area of one
of the triangles and multiply it by two to get
the area of the parallelogram
64.279 sq cm
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K \/27(27 - 10)(27 - 20)(27 - 24)    9
\/119    98.178 sq in
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6.2 The Law of Cosines

• Homework p.398 21, 25
• p. 405 3, 5, 7, 11, 21

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6.2 The Law of Cosines

• Homework p.398 21, 25
• p. 405 3, 5, 7, 11, 21

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6.1/6.2 The Law of Sines/Cosines

• Word Problems p.399 27, 29, 33, 35, 37
• p. 405 27, 30, 32, 36, 37

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• 6.1 Law of Sines for AAS or ASA and ambiguous
  case SSA
• 6.2 Law of Cosines for SAS or SSS
• 6.3 Vectors in the Plane
• 6.4 Vectors and Dot Products
• 6.5 Trig Form of a Complex Number

Formulas for Area of Triangle Standard formula A
½ b h Obliques A ½ bc sin A ½
ab sin C ½ ac sin B Herons A
vs(s-a)(s-b)(s-c) where s
(a b c)/2
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