Download Indirect measurement Many problems involving right triangles

Indirect measurement
Many problems involving right triangles require more than a single
calculation in order to find the desired information. Thus, we need
indirect measurement.
Example
Determine the length of side x to the nearest tenth of a centimeter.
y
x
15 cm
z
30°
20
°
x=y+z
•
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.
•
For this question, it does not matter which length is calculated first,
while in others, the order is crucial to determine the unknown values.
Similar
triangles
• Similar triangles are triangles in which the corresponding angles
have the same measure. The corresponding sides in similar
triangles are proportional. One way of constructing similar right
triangles is shown in the given diagram below.
B3
26 cm
B2
29 cm
B1
25 cm
37 cm
14 cm
A
Compass and
Bearings
• Compass directions may be given as north (N), south (S), east (E), and
west (W); or, halfway between these, such as northwest (NW).
Directions can also be given as the number of degrees east or west,
north or south. This is called a heading.
• For headings, north or south is listed first.
For example, a heading of
N40ºE means 40º east of due north, as shown in this illustration.
40
W
°
N40°E
E
• A bearing is a three-digit number of degrees between 000º and 360º
that indicates a direction measured clockwise from north. For example, a
bearing of 127º is in a direction 127º measured clockwise from due north,
as shown below.
N
127°
Bearing
127°
of
Using Trigonometry to solve for a
side
Let’s review how to use trigonometry to solve for a side.
In order to find an unknown side measure in a right triangle, we
can use trig ratios. In addition, we must have two conditions:
length of one other side and the measure of one of the acute
angles.
Now, let’s see an example:
62
Hypotenuse=10m
Adjacent
Opposite=X
We should know three sides in a right triangle: hypotenuse side,
adjacent side and opposite side.
First, we identify the positions of the known and required side
lengths of the triangle relative to the acute angle, whose measure is
known. This is very important for using trigonometry to solve for
a side.
Since the length of the hypotenuse is known and the length of
the opposite side is required, we can use sine ratio.
Sine62 = opp / hyp = x / 10
X=8.8(m)
Do not forget the unit!
Using Trigonometry to Solve for
an Angle
In order to find the measure of one of the acute angles in a
right triangle when the measure of each acute angle is unknown,
we must know the lengths of two of the three sides.
6cm
15cm
θ
Here is an example:
We solve forθto the nearest tenth of a degree.
First, we label the given sides relative to the angle whose
measure you are trying to determine.
Since the lengths of the opposite side and the hypotenuse are
known, we use sine ratio.
sinθ= 6/15
θ= 23.6°
In addition, sum of the angles in a triangle is 180°.
Right Triangles
• A right triangle has a right (90º) angle. The other two
angles are acute (between 0º and 90º).
• The side opposite the right angle is the longest side, and is
called the hypotenuse.
• The two sides adjacent to the right angle are called legs
of the right triangle. The word adjacent means “beside”.
Pythagorean Theorem
In any right triangles, the square of the hypotenuse is equal to
the sum of the squares of the other two sides (legs).
You can see the following illustration.
A
b
C
c
a
B
Don’t forget: Vertices are often labeled with capital letters,
and the sides opposite the vertices are labeled with the
corresponding lower case letters. (You can see in the
illustration)
Sine, Cosine, and Tangent Ratios
The three primary trigonometric ratios: Sine, Cosine, and
Tangent Ratios describe the ratios of the different sides in a
right triangle.
• These ratios use one of the acute angles as a point of
reference. The 90º angle is never used. In the following
illustration, the ratios are described relative to angle θ.
• Notice that the abbreviations for sine, cosine, and tangent
are sin, cos, and tan.
Hypotenuse
Opposite side to
θ
Adjacent side to
SOHCAHTOA
Remenber!!
sinθ=opposite/Hypotenuse
cosθ= Adjacent/Hypotenuse
tanθ= Opposite/Adjacent
Angles of elevation and depression.
Question：
This leads out the knowledge of angles of elevation and
depression. An angle of elevation is measured between two rays
that have a common starting point called the vertex of the angle.
One ray is horizontal, while the other is above the horizontal.
An angle of depression is similar to an angle of elevation, except
that the second ray is below the horizontal ray.
•