Showing posts with label Trigonometery. Show all posts
Showing posts with label Trigonometery. Show all posts
Tuesday, 25 October 2011
Sunday, 23 October 2011
Sunday, 16 October 2011
Properties of an Isosceles Triangle
Definition: A triangle is isosceles if two of its sides are equal.
Theorem: Let ABC be an isosceles triangle with AB = AC. Let M denote the midpoint of BC (i.e., M is the point on BC for which MB = MC). Then
Ray AM is the angle bisector of angle BAC.
Line AM is the altitude of triangle ABC through A.
Line AM is the perpendicular bisector of B
Segment AM is the median of triangle ABC through A
If |AB| is not equal to |AC| then D is not the midpoint of BC so this method doesn't work. In this case you can use trigonometry. The sine of the angle at B is |AD|/|AB| and thus http://www.k6-geometric-shapes.com/isosceles-triangle.html
|AD| = |AB| sin(B) 
Theorem: Let ABC be an isosceles triangle with AB = AC. Let M denote the midpoint of BC (i.e., M is the point on BC for which MB = MC). Then
If |AB| is not equal to |AC| then D is not the midpoint of BC so this method doesn't work. In this case you can use trigonometry. The sine of the angle at B is |AD|/|AB| and thus http://www.k6-geometric-shapes.com/isosceles-triangle.html
Wednesday, 12 October 2011
Trigonometry: Oblique Triangles - The Ambiguous Case
http://algebralab.org/studyaids/studyaid.aspx?file=Trigonometry_LawSines2.xml
The Law of Sines is used to find angle and side measurements for triangles where the givens fit in the cases of AAS or ASA.
The Law of Sines can also be used in the SSA case, however, additional work is needed to verify the number of possible triangles that can result from being given this combination.
The Law of Sines is used to find angle and side measurements for triangles where the givens fit in the cases of AAS or ASA.
The Law of Sines can also be used in the SSA case, however, additional work is needed to verify the number of possible triangles that can result from being given this combination.
AAS ASA SSA 
2 angles & non-included side.
2 angles & included side.
2 sides & non-included angle.
The Law of Sines:
The "Ambiguous" Cases of Two Triangles http://users.rcn.com/mwhitney.massed/law_of_sines/law_of_sines.html Given a triangle with sides a and b of certain lengths and height h, there are four possible situations, depending on the lengths of a, b, and h:
Case 1: If a < h there are no possible triangles.
Case 2: 2. If a = h there is exactly one triangle (a right triangle).
Case 3: If h < a < b there are two possible triangles.
one "large" triangle if B is an acute angle … OR one "small" triangle if B an obtuse angle …
Case 4: If a > b or if a = b there are two possible triangles.
one "large" triangle if B is an acute angle … OR one "small" triangle if B an obtuse angle …
An Example of Case 3:
If h < a < b, where there are two possible triangles.
Given a triangle with the following measurements, solve the triangle for the missing measurements.
a = 4.361 cm
b = 4.824 cm
a = 60° 
First note that two triangles can be formed.
In this case we see that h < a < b. The two possible triangles are configured as shown below:
To solve the triangle if angle B is acute, we proceed as follows:
To find 
:
To find 
:
To find 
, we use the Law of Sines:
To solve the "skinny" triangle (when angle B is obtuse), we need to find 
by a clever trick:
is isosceles, by the fact that it has two sides of equal length. Thus, the angles opposite these sides are equal.
Thus, 
.
is the supplement of 
because they form a straight angle.
Therefore,
To find 
:
To find c2:
An Example of Case 4:An Obtuse Angle
... and ... a Case When the Law of Cosines is more efficient
To solve the triangle ABC as shown below using the Law of Sines would require a great deal of extra work. In this case, the Law of Cosines is much more applicable.
Here, the angle C is known, but A and B are to be found. Also, sides a and b are known, but side c has yet to be found.
First, we see that the Law of Sines cannot be used with the information currently available.
Each of the following pairings involve two variables, thus none can be solved independently.
If the Law of Sines were to be used, the key is to extend segment CB into a ray and then find the height (h or segment AD) of the triangle. Once that is done, however, there are still four steps that must be completed before the missing parts of the triangle can be found. [While the Law of Cosines will solve the triangle with less effort.]
1. Find the height, h, with 
.
2. Find length of segment CD, w, with 
.
3. Find length of BD, v, by using a + v = w.
4. Find angle ABD with 
.
5. Find the original triangle's angle B, angle ABC, by it being supplementary with angle ABD.
6. Find the original triangle's angle A, angle BAC, by the sum of the interior angles of a triangle being 180 degrees.
7. Finally, find the length of side c using (1) the Law of Sines or (2) the Pythagorean Theorem or (3) a trigonometric ratio with sides h or v.
Why is the Law of Cosines more efficient?
By setting up each variation of the Law of Cosines, we see that while the first two versions of the Law of Cosines will not succeed independently (more than one variable in each remain), the third version works like a charm.
With side c found using the Law of Cosines, we can then turn to the Law of Sines to find the other missing information. Thus, the problem is greatly streamlined using the Law of Cosines.
Case 1: If a < h there are no possible triangles.
Case 2: 2. If a = h there is exactly one triangle (a right triangle).
Case 3: If h < a < b there are two possible triangles.
Case 4: If a > b or if a = b there are two possible triangles.
An Example of Case 3:
If h < a < b, where there are two possible triangles.
Given a triangle with the following measurements, solve the triangle for the missing measurements.
| a = 4.361 cm b = 4.824 cm a = 60° | |
First note that two triangles can be formed.
In this case we see that h < a < b. The two possible triangles are configured as shown below:
To solve the triangle if angle B is acute, we proceed as follows:
To find
To find
To find
To solve the "skinny" triangle (when angle B is obtuse), we need to find
Thus,
Therefore,
To find
To find c2:
An Example of Case 4:An Obtuse Angle
... and ... a Case When the Law of Cosines is more efficient
To solve the triangle ABC as shown below using the Law of Sines would require a great deal of extra work. In this case, the Law of Cosines is much more applicable.
Here, the angle C is known, but A and B are to be found. Also, sides a and b are known, but side c has yet to be found.
First, we see that the Law of Sines cannot be used with the information currently available.
Each of the following pairings involve two variables, thus none can be solved independently.
If the Law of Sines were to be used, the key is to extend segment CB into a ray and then find the height (h or segment AD) of the triangle. Once that is done, however, there are still four steps that must be completed before the missing parts of the triangle can be found. [While the Law of Cosines will solve the triangle with less effort.]
1. Find the height, h, with
2. Find length of segment CD, w, with
3. Find length of BD, v, by using a + v = w.
4. Find angle ABD with
5. Find the original triangle's angle B, angle ABC, by it being supplementary with angle ABD.
6. Find the original triangle's angle A, angle BAC, by the sum of the interior angles of a triangle being 180 degrees.
7. Finally, find the length of side c using (1) the Law of Sines or (2) the Pythagorean Theorem or (3) a trigonometric ratio with sides h or v.
Why is the Law of Cosines more efficient?
By setting up each variation of the Law of Cosines, we see that while the first two versions of the Law of Cosines will not succeed independently (more than one variable in each remain), the third version works like a charm.
| |
| |
With side c found using the Law of Cosines, we can then turn to the Law of Sines to find the other missing information. Thus, the problem is greatly streamlined using the Law of Cosines.
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