MATH 5200/7200, Fall 2008
Homework 14, due Monday, October 27
This assignment is a review of the following three trig formulas: the area formula, the law of cosines, and the law of sines.
Part I. For each of these formulas:
(a) Check it using GSP. (Turn in your GSP file.)
(b) Prove it using the definitions, axioms, and theorems we have discussed in this course.
Note: The definitions of the sine and cosine functions have been posted on the course web site.
Part II. Choose one of the three formulas, find an application or a generalization, and discuss your application or generalization.
The formulas refer to the triangle ABC, with sides a = BC, b = AC, c = AB. Following the usual shorthand notation, angles will be labelled by their vertices.
1. Area formula:
Area = 1/2 ab sin(C)
Consider both the case when C is acute and the case when C is obtuse.
Note that when C is a right angle, sin(C) = 1, so this formula says Area = 1/2 ab, or Area = 1/2 base times height.
A possible application is to find a formula for the area of a quadrilateral.
2. Law of cosines:
c2 = a2 + b2 - 2ab cos(C)
Consider both the case when C is acute and the case when C is obtuse.
Hint for the proof: Make some right triangles.
Note that when C is a right angle, then cos(C) = 0, so this formula says c2 = a2 + b2, which is just the Pythagorean theorem.
A possible topic for discussion is to interpret this formula geometrically, generalizing the interpretation of the Pythagorean theorem using squares on the three sides of a right triangle.
3. Law of sines:
a/sin(A) = b/sin(B) = c/sin(C)
Consider both acute and obtuse triangles.
Note that when C is a right angle, then these equations just say that a/sin(A) = b/sin(B) = c, or sin(A) = a/c and sin(B) = b/c, which are just the definitions of sin(A) and sin(B) for a right triangle.
A possible topic for discussion is to investigate the relation of these numbers to the circumradius of the triangle ABC.
Email your homework to Dr. McCrory by 9 am Monday morning.
Homework 15, due Monday, November 3
This homework consists of two practical trigonometry problems and three problems that are more theoretical.
1. Smoke from a forest fire is spotted simultaneously by rangers at two fire towers. Tower B is ten miles southeast of tower A. Viewed from tower A, the fire has compass bearing North 50 degrees East. Viewed from tower B, the fire has compass bearing North 25 degrees West. Use a calculator to compute the distance of the fire from each of the fire towers. Show your work.
Make an accurate GSP sketch and check your answer from the sketch.
For an explanation of compass bearings, see the website Discovering Lewis and Clark.
2. The perimeter of a protected wetland is surveyed. The perimeter has five sides. The lengths of the sides and the angles between them are measured as follows:
Side a length 39.78 meters, angle from side a to side b 87.7 degrees
Side b length 40.33 meters, angle from side b to side c 128.2 degrees
Side c length 33.65 meters, angle from side c to side d 80.2 degrees
Side d length 38.10 meters, angle from side d to side e 141.1 degrees
Side e length 26.53 meters, angle from side e to side a 102.7 degrees
Use a calculator to compute the area of the wetland (in square meters). Show your work.
3. Prove that for a triangle ABC labelled as in homework 14 above, if D is the midpoint of side AB, then the length m of the median CD satisfies the equation
m2 = 1/2 a2 + 1/2 b2 - 1/4 c2.
4. Find formulas in terms of trig functions for the edge length and the area of a regular n-sided polygon that is inscribed in a circle of radius R.
5. Suppose that the quadrilateral ABCD is inscribed in a circle, and that the diagonal AC is a diameter of the circle. Prove that
(AC)(DB) = (AB)(CD) + (BC)(AD)
Hint: Use the addition law for the sine function.
Email your homework to Dr. McCrory by 9 am Monday.
Homework 16, due Friday, November 7
Prove the following theorems. You may use any theorems we have proved earlier in the course, and you may also use the trig formulas we have discussed in class or in the homework.
1. If a triangle has side lengths a, b, c, and circumradius R, then its area is abc/4R.
2. Let ABC be a triangle, and let D be a point on the side AB. As usual, let a = BC, b = AC, and c = AB. Also let d = CD, and let p = AD/AB, q = DB/AB. Prove that
d2 = pa2 + qb2 - pqc2 .
(This is a generalization of problem 3 on the last homework.)
3. Prove Heron's formula: A triangle with sides a, b, c has area √[s(s - a)(s - b)(s - c)], where s = 1/2(a + b + c) is the "semiperimeter."
Use the following formulas:
(1) Area = 1/2 ab sin C (square both sides)
(2) (sin C)2 + (cos C)2 = 1 (solve for (sin C)2)
(3) c2 = a2 + b2 - 2ab cos C (solve for cos C)
After a lot of high school algebra, you should get
(Area)2 = 1/16(a + b + c)(a + b - c)(a - b + c)(- a + b + c),
which simplifies to Heron's formula.
4. Extra credit problem: Let ABC be a triangle with circumradius R. Let p and q be the radii of two circles through A, tangent to BC at B and C, respectively. Then pq = R2. (Warning: Trig may not be the best way to do this.)
Email your homework to Dr. McCrory by 9 am Friday.
Homework 17, due Wednesday, November 12
Download the GSP file: Half Plane model of Hyperbolic Geometry. Use the tools in this file to do the following hyperbolic constructions.
1. Construct a hyperbolic equilateral triangle. Measure its angles and its area. What is the relation between the angles and the area?
2. Construct a hyperbolic regular hexagon. (Hint: Use problem 1.)
3. Construct a hyperbolic regular quadrilateral. (Hint: Use the perpendicular bisector theorem. Note that the angles of the quadrilateral will not be 90 degrees.)
Email your homework to Dr. McCrory by 9 am Wednesday.
Homework 18, due Friday, November 14
In class Wednesday we discussed which of our axioms hold for hyperbolic geometry. The only axioms that fail are the parallel axiom and the area axiom. (The area axiom fails only because there are no rectangles in hyperbolic geometry.)
1. Study our list of basic theorems, and decide which of them holds for hyperbolic geometry. For each basic theorem, just list the axioms that are used in its proof. If the parallel and area axioms are not used, then the theorem holds in hyperbolic geometry. If the parallel or area axioms are used in the proof, then the theorem may not hold in hyperbolic geometry, and you may need to experiment with hyperbolic GSP to determine whether it holds.
2. The similarity theorems (SAS, AA, SSS) do NOT hold in hyperbolic geometry. For each of these theorems, use hyperbolic GSP (as in homework 17) to construct a counterexample.
Comments on problem 2:
For a counterexample to SAS similarity, just make a counterexample to the midpoint theorem: If ABC is a triangle, D is the midpoint of AB, and E is the midpoint of AC, then ABC is similar to ADE.
For a counterexample to AA similarity, you need to copy angles. To copy an angle, use that the angle between two hyperbolic lines is the same as the euclidean angle between the tangent lines to the two circles that represent the hyperbolic lines.
For a counterexample to SSS similarity, use problem 1 of the last homework.
Note: A user's guide to the hyperbolic GSP package can be found at the web site of the author, Judit Abardia Bochaca.
Email your homework to Dr. McCrory by 9 am Friday.
Homework 19, due Friday, November 21
1. Study our list of axioms and basic theorems, and decide which of them hold for spherical geometry. Explain your reasoning for each axiom. Be careful with axioms 1-4. For each basic theorem, list the axioms that are used in its proof. If an axiom or theorem does not hold, explain how it fails, and explain how the axiom or theorem can be modified to hold in spherical geometry. This is a discussion question. Use what you know about the geometry of a sphere. You may use drawings to illustrate your conclusions.
2. Compute the angle excess and the area for each of the following two spherical polygons, and check the formula:
(sum of angles) - (number of sides - 2) π = area.
(Angles are measured in radians, and the sphere has radius 1.)
(a) a face of a spherical dodecahedron (projection of a dodecahedron to its circumsphere)
(b) a face of a spherical icosahedron (projection of a icosahedron to its circumsphere)
You may turn in your homework by email before class or on paper at the start of class.