Triangles

Let at arbitrary triangle have three angles fA;B;Cg and side lengths fa; b; cg where the angle is opposite the corresponding side. Use right triangle trig, law of sines, and law of cosines to compute the following triangles and nd the area of the triangle:

1. a = 5; b = 4;A = 60 3. a = 7; b = 1;A = 108

2. B = 90; b = 17; a = 3 4. A = 134; a = 38; b = 24

5. a = 21:2; b = 24:60c = 12:0 6. C = 58:4; b = 7:23; c = 22:0

7. A small plane is

ying from Chicago to St. Louis, a distance of 258 miles. After ying at 130 miles per hour for 45 minutes, the pilot nds that she is 12 o course. If she corrects the course and maintains speed and direction for the remainder of the ight, how much longer will the ight take?

8. A goose ying due west at 20 mph is initially 55.4 SE of a pond. After 1.5 hours,the pond is 66.8 NE of the goose. How far is the goose from the pond the second time?

How close does the goose pass to the pond?

9. The angle of depression from a tower on top of a building to the ground is 58o36: If a cable worker is standing 15 feet from the base of the building, how tall is the building to the nearest foot?

10. Find the perimeter of a regular pentagon inscribed in a circle with radius 5 cm.

11. Find the exact values (without using a calculator) of cos(tan??1(23)) ^ sin(cos??1(??14))

Part 4: Trigonometric Models

Graph the following trig functions:

1. m(x) = 2 sin(x+)??1 2. n(x) = 12 cos(x??)+ 12

3. A normal seated adult breathes in and exhales aout 0.80 L of air every 4.00 seconds.
The volume of air V(t) in the lungs is modeled by V (t) = 0:45 ?? 0:40 cos(t2 ).

a. What is the maximum amount of air in the lungs and what is the minimum?
b. What is the period of breathing?
c. How many breaths are taken per minute?
d. Graph the function for 8 seconds of breathing.
4. The average temperature in Fargo, ND can be approximated by the functionT(t) = 12 sin(6 (x ?? 3:9)) + 72 where t = 1 represents January, t = 2 represents February,etc.

a. What is the average temperature in April?
b. What is the average temperature?
c. What is the lowest average temperature and the highest average temperature and when do they occur within the year?

5. The horizontal distance that a projectile will travel in the air (ignoring air resistance) is given by

R() =
v2
0 sin(2)
g

a. If you can throw a lacrosse ball with initial speed 100 meters per second, at what angle of elevation  should you launch the ball if you wish it to travel 500 meters?

b. How far can you launch the ball?

6. In the study of heat transfer, the equation x + tan(x) = 0 arises. Verify that it indeed has a solution and nd how many solutions there are.

7. What is Snell's law?

Part 3: True/False

Treat this as a small research project. Stated below is a two mathematical deductions derived from Euler's Formula: (1) and (2). One of them is true and one is false. Determine which one is true and which one is false and show why. The additional information will be necessary in showing why.

Statements
cos (i) = cosh () (1)
sin (i) = sinh () (2)
Additional Information
sin (x) =
eix ?? e??ix
2i
cos (x) =
eix + e??ix
2
sinh (x) =
ex ?? e??x
2
cosh (x) =
ex + e??x
2