Geometry Day 62

1
Geometry Day 62

• Trigonometry III
• The Sine of the Times

2
Todays Objective

• The Law of Sines
• The Law of Cosines

3
Review

• The trigonometric functions sine, cosine, and
  tangent give the ratios of certain sides given
  an acute angle of a right triangle.
• Remember, you take the sin, cos, and tan of
  angles, and you will get a ratio.

4
Review

• The inverse trig functions will take a ratio and
  provide the angle that forms that ratio.
• In other words, it allows you to solve for an
  angle given two sides of a right triangle.

5
Trig and non-right triangles

• Sine, cosine, and tangent only work on right
  triangles, but there is a way we can expand this
  idea to other types of triangles.
• Take any triangle.
• Lets say we know two angles anda side opposite
  one of the knownangles
• and we want to know the sideacross from the
  other angle.

B
15
x
57?
21?
A
C
6
Trig and non-right triangles

• If we are to use trigonometry, we need right
  triangles. Can we create right triangles in this
  diagram?
• Any triangle can be divided into tworight
  triangles by drawing its altitude.
• Can you use the information in the diagram to
  solve for x?
• Hint solve for h first.

B
15
x
h
57?
21?
A
C
7
Lets generalize the process

• Can you use sine to find a relationship between
  the angles of a triangle and their opposite sides?

B
a
c
h
A
C
8
The Law of Sines

• This idea becomes what is known as the Law of
  Sines
• You can use the Law of Sines if you know ASA,
  AAS, or SSA. (In other words, if you can
  compare an angle to the side across from it.)
• SSA can be tricky, in that there may be one,
  none, or two triangles that exist for a given set
  of values. All of the problems in this class
  will only have one solution youll learn more in
  Pre-Calculus.

B
a
c
A
C
b
9
Practice

• Solve for the variables

x?
10
The Law of Cosines

• If we know SAS or SSS, then we cannot use the Law
  of Sines, since we cant form a ratio between an
  angle and its opposite side.
• Lets examine
• Well create right triangles by drawing the
  altitude.
• Well give CD a length of y.
• What is AD?
• If we know what y is, we could use the
  Pythagorean Theorem to find BD, then use it again
  to find x.
• We can use cosine to find y.

B
7
x
y
62?
4 y
A
C
4
D
11
Lets generalize the process

• Lets label the other segments.
• We can use the Pythagorean Theorem twice

B
a
c
h
b x
x
A
C
b
12
Lets generalize the process

• We can use cosine to write an expression for x

B
a
c
h
b x
x
A
C
b
13
The Law of Cosines

• The Law of Cosines states
• Remember, the Law of Cosines can be used if you
  know SAS or SSS.

B
a
c
A
C
b
14
Practice

• Solve for the variables

a

15
Summary

• Solving a triangle means to find the measures of
  all sides and angles.
• For a right triangle
• If you know two sides, you can find the third by
  the Pythagorean Theorem. You can find the acute
  angles by using inverse trig functions.
• If you know a side and acute angle, you can find
  the other sides with trig functions. You can
  find the other acute angle with the Triangle
  Interior Angle Sum Theorem.
• For non-right triangles
• If you know ASA or AAS, you can use the Interior
  Angle Theorem to find the third angle, and the
  Law of Sines to find the other sides.
• If you know SSA, you can use the Law of Sines to
  find one of the angles. Then youll have either
  ASA or AAS, and see above.
• If you know SAS, you can use the Law of Cosines
  to find the third side. Then youll have SSA
  (see above) and SSS (see below).
• If you know SSS, you can use the Law of Cosines
  to solve for an angle. Then youll have SSA (see
  above).

16
Homework 37

• Workbook, p. 107