Download Trigonometry - TangHua2012-2013

TRIGONOMETRY
Lesson 4: Solving Problems Involving 2 Right
Triangles and Direct or Indirect Measure
Todays Objectives
• Students will be able to develop and apply the
primary trigonometric ratios (sine, cosine,
tangent) to solve problems that involve right
triangles, including:
• Solve a problem that involves one or more right
triangles by applying the primary trigonometric ratios or
the Pythagorean theorem
• Solve a problem that involves indirect and direct
measurement, using the trigonometric ratios, the
Pythagorean theorem and measurement instruments
such as a clinometer or meter stick
Solving Problems Involving 2 Right Triangles
• Many problems involving right triangles require
more than a single calculation in order to find the
desired information. In some questions the order
of these calculations is unimportant, while in
others, the order is crucial to determine the
unknown values.
Example
• Determine the length of side x to the nearest tenth of a
centimeter.
y
x
15 cm
30°
20°
z
• Solution:
x=y+z
For this question, it does not matter
which length is calculated first
• Since the value of x is the sum of the lengths of two legs
of two right triangles, assign variables to those two legs
and find their values separately.
Example
• Apply the tangent ratio.
• tan30º =
• tan20º =
𝑦
,𝑦
15
𝑧
,𝑧
15
= 15tan30º
= 15tan20º
• 𝑥 = 𝑦 + 𝑧, 15tan30º + 15tan20º = 𝑥
• 𝑥 = 14.1 𝑐𝑚
• The length of side x is approximately 14.1 cm.
• *Do not calculate the values of 15tan30º and 15tan20º
until the final step to avoid error that may affect your final
answer!
Solving Problems Involving 2 Right Triangles
• In some problems, the required unknown value cannot be
calculated directly because it is part of a triangle with
insufficient information (side lengths and angle
measures).
• These problems require determining a side length or
angle measure of a triangle that shares a common side
with the one containing the required unknown.
Example
• Determine the length of side x to the nearest tenth of a
meter.
y
43° 47°
x
z
47°
43°
22 m
• Solution:
• Begin by filling in any of the measures that you can
determine easily. In this case, we can determine the
measures of the acute angles in the diagram because we
know the sum of angles in a triangle = 180º.
Example
• Since the lengths of all the sides for the right triangle that
x is part of are unknown, the first step is to determine the
length of one other side of the triangle containing side x.
The sides labeled y and z in the previous diagram are
possibilities.
y
x
z
• Knowing the value of either y or z allow you to calculate
side length x. However, the value of x is probably more
easily obtained by first determining the value of z, as
follows:
Example
• Apply the tangent ratio.
• tan47º =
𝑧
, 22tan47º
22
= 𝑧 (expression for z)
• 𝑧 = 23.59 (approximate value for z)
• Now x can be determined using the approximate value of z, or
the expression for z.
• *Use 22tan47º instead of 23.59 to reduce error!
• Apply the sine ratio.
𝑧
, 𝑥𝑠𝑖𝑛43º = 𝑧
𝑥
𝑧
22tan47
,𝑥 =
,𝑥
sin43°
sin43
• sin43º =
• 𝑥 =
= 34.6
• The length of side x is approximately 34.6 meters.
Example (You do)
• Determine the length of side x to the nearest tenth of a centimeter.
6 cm
4 cm
y
50°
• Solution:
• Apply the sine ratio.
• sin50º =
6
,𝑦
𝑦
=
x
6
,=
sin50
7.8
• Apply the Pythagorean theorem to find x.
6
)2, =
sin50
• 𝑥2 = 42 + (
16 + 61.3, = 77.3
• Take the square root of both sides.
• 𝑥 = 8.8
• The length of side x is approximately 8.8 centimeters.
Example
• From a position in a tower 120 meters above the ground,
a forest ranger named Laughing observes a fire that is
directly west at an angle of depression of 18º. The ranger
also observes a herd of deer directly east of the tower at
an angle of depression of 35º. How far is the fire from the
herd of deer, to the nearest meter?
• Solution: First, sketch and label a diagram
Tower
18°
35°
120 m
35°
18°
x
y
Example
• Notice that the angles of depression from the tower to the fires and to the
deer are equal to the angles of elevation from the fire and from the deer.
The distance between the deer and the fire is x + y.
• Apply the tangent ratio to solve for x.
• tan18º =
• 𝑥 =
120
, 𝑥𝑡𝑎𝑛18º
𝑥
= 120
120
tan18
• Apply the tangent ratio to solve for y.
• tan35º =
• 𝑦 =
120
, 𝑦𝑡𝑎𝑛35º
𝑦
= 120
120
tan35
• Distance = x + y
•
120
tan18
120
+ tan18 = 541 𝑚𝑒𝑡𝑒𝑟𝑠.
• The fire is approximately 541 meters from the herd of deer.
Example (You do)
• Jackie Chan is standing on the street 60 m from the base
of a tall office building that has a flagpole on the top edge
that he is facing. He uses a clinometer to measure the
angles of elevation from his position on the street to the
top of the building and to the top of the flagpole as 48º
and 54º, respectively. How high is the flagpole, to the
D
nearest tenth of a meter?
• Solution: Draw a diagram, then calculate
C
y
A
48°
54°
60 m
B
x
Example
• Apply the tangent ratio in triangle ABC to solve for y.
• tan48º =
𝑦
,𝑦
60
= 60tan48º
• Similarly, apply the tangent ratio in triangle ABD to solve
for x.
• tan54º =
𝑥
,𝑥
60
= 60tan54º
• The required length is the difference between x and y.
• 𝑥 – 𝑦 = 60tan54º − 60tan48º = 60(tan54º − tan48º)
• =15.9
• The flagpole is approximately 15.9 meters high.