Definition of Sine and Cosine Revisited

1
Definition of Sine and Cosine Revisited

• Suppose P (x,y) is the point on the unit circle
  specified by the angle . We define the
  functions, cosine of , or ,
  and the sine of , or , by either
  pair of formulas
  

Hypotenuse
(-1,0)
Side Opposite
Side Adjacent
2
The Tangent Function Revisited

• Suppose P (x, y) is the point on the unit
  circle specified by the angle . We define the
  tangent of , or tan , by tan y/x, or by
  

Side Opposite
Side Adjacent
3
Solving Right Triangles

• One of the angles of a right triangle is 90.
  Thus, one part is already known and the only
  additional information necessary is either two
  sides or an acute angle and a side. With this
  additional information, it is possible to solve
  for the remaining parts.
• Making a carefully labeled sketch of the triangle
  is important in solving these problems.
• Problem 23, page 305. The top of a 200-foot
  tower is to be anchored by cables that make an
  angle of 30 with the ground. How long must the
  cables be? How far from the base of the tower
  should the anchors be placed?
  sin 30 200/h gt h 200/sin 30 400
  ft tan 30 200/x gt x
  200/tan 30 346.4 ft

h
200 ft
30
x
4
Solving Non-right Triangles

• Suppose the triangle to be solved does not
  contain a right angle. If we are given three
  parts of the triangle, not all angles, then we
  can solve for the remaining parts. The solution
  exists and is unique except in one
  case--discussed in a later slide.
• Usually we label the angles of the triangle as A,
  B, C and the sides opposite these angles as a, b,
  and c, respectively.
• Law of Cosines
• Law of Sines

C
b
a
A
B
c
5
An example using the Law of Cosines

• A person leaves her home and walks 5 miles due
  east and then 3 miles northeast. How far away
  from home (as the crow flies) is she? As seen
  from her home, what is the angle ? between the
  easterly direction and the direction to her
  destination?
• By applying the Law of Cosines, we can solve for
  x
• A second application of the Law of Cosines
  gives

Destination
N
x
3
45
135
Home
5
6
An example using the Law of Sines

• Problem 29, page 312. Two fire stations are
  located 56.7 miles apart, at points A and B.
  There is a forest fire at point C. If ?CAB 54
  and ?CBA 58, which fire station is closer?
  How much closer?
• The fire station at point B is closer to the fire
  by 51.861 49.474 2.387
  miles.

C
68
b
a
54
58
A
B
c 56.7 miles
7
The ambiguous case for solving a triangle

• Suppose we are given two sides of a triangle and
  the angle opposite one of them, and we are asked
  to find the remaining parts of the triangle.
  Depending on the data given, there may be two
  possible triangles, one possible triangle, or no
  possible triangle.
• Example. ?CAB 45, b 1, and various values of
  a

C
a3/10
b1
No triangle exists.
45
A
C
C
One triangle exists.
b1
b1
a11/10
a?2/2
45
45
B
A
A
B
C
b1
Two triangles exist.
a8/10
45
A
8
Summary for Trigonometry

• Sine, cosine, and tangent were redefined in terms
  of right triangles.
• Right triangles in which either two sides or an
  acute angle and a side are given can be solved.
• We may consider solving non-right triangles in
  which three parts, not all angles, are given.
  The solution exists and is unique except in the
  ambiguous case when two sides and the angle
  opposite one of them are given.
• In the ambiguous case, there may be two
  possible triangles, one possible triangle, or no
  possible triangle.
• The law of cosines is a generalization of the
  Pythagorean theorem that works for any triangle.
• The law of sines is useful when we know a side
  and the angle opposite it.