Transcript Right Triangle

Unit 9 -Right Triangle
Trigonometry
•This unit finishes the analysis of triangles with
Triangle Similarity (AA, SAS, SSS).
•This unit also addressed Geometric Means, and
triangle angle bisectors, and the side-splitter
theorem. (Different set of slides)
•This unit also contains the complete set of
instructions addressing Right Triangle
Trigonometry (SOHCAHTOA).
Standards
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SPI’s taught in Unit 9:
SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space.
SPI 3108.4.7 Compute the area and/or perimeter of triangles, quadrilaterals and other polygons when one or more
additional steps are required (e.g. find missing dimensions given area or perimeter of the figure, using trigonometry).
SPI 3108.4.9 Use right triangle trigonometry and cross-sections to solve problems involving surface areas and/or
volumes of solids.
SPI 3108.4.15 Determine and use the appropriate trigonometric ratio for a right triangle to solve a contextual problem.
CLE (Course Level Expectations) found in Unit 9:
CLE 3108.1.4 Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial,
graphical, tabular, and symbolic), to solve problems, to model mathematical ideas, and to communicate solution
strategies.
CLE 3108.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis
on formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of
solution strategies.
CLE 3108.1.7 Use technologies appropriately to develop understanding of abstract mathematical ideas, to facilitate
problem solving, and to produce accurate and reliable models.
CLE3108.2.3 Establish an ability to estimate, select appropriate units, evaluate accuracy of calculations and
approximate error in measurement in geometric settings.
CLE 3108.4.8 Establish processes for determining congruence and similarity of figures, especially as related to scale
factor, contextual applications, and transformations.
CLE 3108.4.10 Develop the tools of right triangle trigonometry in the contextual applications, including the Pythagorean
Theorem, Law of Sines and Law of Cosines
Standards
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CFU (Checks for Understanding) applied to Unit 9:
3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of
geometry, including specialized vocabulary (e.g. graphing calculators, interactive geometry software such as
Geometer’s Sketchpad and Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs, mirrors, spinners,
geoboards, conic section models, volume demonstration kits, Polyhedrons, measurement tools, compasses,
PentaBlocks, pentominoes, cubes, tangrams).
3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems.
.. 3108.1.8 Understand how the similarity of right triangles allows the trigonometric functions sine, cosine, and tangent
to be defined as ratio of sides.
3108.4.11 Use the triangle inequality theorems (e.g., Exterior Angle Inequality Theorem, Hinge Theorem, SSS
Inequality Theorem, Triangle Inequality Theorem) to solve problems.
3108.4.27 Use right triangle trigonometry to find the area and perimeter of quadrilaterals (e.g. square, rectangle,
rhombus, parallelogram, trapezoid, and kite).
3108.4.36 Use several methods, including AA, SSS, and SAS, to prove that two triangles are similar.
3108.4.37 Identify similar figures and use ratios and proportions to solve mathematical and real-world problems (e.g.,
Golden Ratio).
3108.4.42 Use geometric mean to solve problems involving relationships that exist when the altitude is drawn to the
hypotenuse of a right triangle.
3108.4.47 Find the sine, cosine and tangent ratios of an acute angle of a right triangle given the side lengths.
3108.4.48 Define, illustrate, and apply angles of elevation and angles of depression in real-world situations.
3108.4.49 Use the Law of Sines (excluding the ambiguous case) and the Law of Cosines to find missing side lengths
and/or angle measures in non-right triangles.
Unit 9 Bellringer: 10 points
• MT. Rainier is found in
Washington State, and is
both an active volcano, and
has active glaciers on the
side.
• From the center base of the
mountain to the outside
edge (along the ground), it is
22882.12 feet
• From the top of the
mountain down the slope to
the edge, it is 26422 feet
• How tall is the mountain?
Tallest US Mtns:
McKinley (AK)
Ebert (CO)
Massive (CO)
Harvard (CO)
Rainer (WA)
1. Draw the triangle the mountain
creates (3 points)
2. Write the equation (3 points)
3. Calculate the height (3 points)
4. Write your name somewhere
on it (1 point)
From here to the end of the
building
• It is 15 feet from the podium to the wall
• It is about 4 or 5 degrees deflection
measured from the podium and from the
wall
• Tan(85) = x/15
• 15Tan85) = X
• It is 42 steps to the corner of the building
• I take 65 steps to walk 100 meters
• 42/65 (100) = 64.61 meters
• = 212 feet
Building
x
15
A Look at Triangle Relationships
• What can you conclude about these
three partial Right triangles?
O
Xo
O
Xo
A
A
O
Xo
A
• 1) There is only one hypotenuse that will fit each one,
based on how long the Opposite (O) side, and Adjacent
(A) Side are
• 2) There is only one angle that will fit each triangle, based
on how long the Opposite and Adjacent sides are
Labeling the Parts
• We will use the same approach to all
triangles during Right Triangle
Trigonometry
1. We do not apply the rules of R.T. Trig
to the right angle (I.E. solving for
tangent etc.)
2. If possible, we try to set the problem up
to use the bottom angle
3. We always label the side farthest from
o
X
the angle as “Opposite”
4. We always label the side that touches
the angle we are using as “Adjacent”
5. The Hypotenuse is the diagonal that
touches our angle
H
O
A
Tangent Ratios
• Big Idea: In Right Triangle ABC, the
ratio of the length of the leg opposite
(O) angle A to the length of the leg
adjacent (A) to angle A is constant, no
matter what lengths are chosen for one
side or the other of the triangle. This
trigonometric ratio is called the Tangent
Ratio.
Tangent Ratios
• Tangent of A =
Length of leg opposite A
Length of leg adjacent to A
• You can abbreviate this
A
As Tan A = Opposite
Adjacent
Leg adjacent to A
B
Leg
opposite
A
C
Writing Tangent Ratios
• Tan T = Opposite/Adjacent
– Or UV/TV = 3/4
• Tan U = Opposite/Adjacent
U
5
3
– TV/UV = 4/3
• What is the Tan for K?
• What is the Tan for J?
Tan K = 3/7
Tan J = 7/3
What relationship is there
between them?
They are reciprocals
V
T
4
J
3
L
K
7
So you’re a skier
• Imagine you want to know how far it is to a mountain top from
where you are.
• Aim your compass at the mountain top, and get a reading. Turn
left or right, and walk 90 degrees from your first reading. -So if
you read 200 degrees, and turned left, it would be 200 - 90, or
110, and if you turned right, it would be 200 + 90, or 290.
• Walk 50 feet in the new direction.
• Stop, and take a new compass reading to the mountain top.
• Suppose it is now 86 degrees to the mountain top
• Using the Tan ratio, you can now calculate how far it is to the
mountain top
How Far?
M
Your new angle to the MTN Top
50’ -how far
860 You walk
Here’s How
• You have created a right triangle, with one leg of 50 feet, and an
angle of 86 degrees. The other leg is unknown, or X.
• So, Tan 86o = x/50 (Remember, opposite / adjacent)
• NOTE: “Tan 86o” is just a number –remember, it is just the ratio
of the opposite to the adjacent. It’s just a fraction, which we can
write as a decimal
• Therefore, x = 50(Tan 860) (multiply both sides by 50)
• Type into your calculator 50 TAN 86 ENTER, and you get
715.03331
• Knowing you measured your first leg in feet, it is 715 feet to the
mountain top.
M
X (Opposite)
50 (Adjacent)
860
Set your Calculator
• This is the part where people try to solve a
problem and get the wrong answer, and they ask
me why 
• The problem is the default setting for graphing
calculators is in radians, not degrees
• To check, click on the “MODE” button on your
calculator. See if “RADIANS” is highlighted
instead of “DEGREES”
• Scroll down, and highlight DEGREES and hit
“ENTER”
• Click on “2ND” and then “QUIT” (MODE Button) to
get out of this setup
Find the value of W
Remember Tan(xo) = O/A
330
280
W
10
540
Tan 54 = W/10
W = 10 (tan 54)
W = 13.76
W
W
1.0
Tan 28 = 1.0/ W
W (Tan 28) = 1.0
W = 1.0 / (Tan 28)
W = 1.88
2.5
570
Tan 57 = W/2.5
W = 2.5 (Tan 57)
W = 3.84
OR….
Tan 33 = 2.5 / W
W (Tan 33) = 2.5
W = 2.5 / (Tan 33)
W = 3.84
Inverse of Tangent
• If you know the leg lengths for a right triangle, you
can find the tangent ratio for each acute angle.
• Conversely, if you know the tangent ratio for an
angle, you can use the inverse of tangent or Tan -1
to find the measure of an angle
• Bottom Line:
– We use the Tangent if we know the angle, and need a
length of a leg -these are ones we just did
– We use the Tangent Inverse if we know the lengths of
the legs, and need the angle
Example of Inverse
• You have triangle HBX with lengths of
the sides as given:
• Find the measure of X to the nearest
degree
•
•
•
•
•
We know that Tan X = 6/8, or .75
H
So m X = Tan -1 (.75)
TAN -1 (.75) ENTER = 36.86
6
You can also type TAN -1 (6/8)
So, m X = 37 degrees
B
10
8
X
Example of Inverse
• Find the m of Y to the nearest degree
We need the tangent ratio so P
that we can plug it in to the
calculator and solve for Tan-1
Tan Y = O/A
Tan Y = 100/41, or 2.439
M Y = Tan -1 (2.439) (or use
100/41)
M Y = 67.70
Or, m Y = 68 degrees
100
T
41
Y
Tangents on Graphs
• Graph the line y - 3/4x = 2
• Rewrite the equation as y = 3/4x + 2
• What is the slope?
• The slope is 3/4, or rise over run --> rise/run
• The question is, can you use
the tangent function to determine
the measure of angle A?
• Tangent is a ratio of
Op
Opposite/Adjacent
A
Adjacent
•In this case, Opposite is the
rise, and Adjacent is the run
•So Tan(A) is the slope, or 3/4
•Therefore, we use Tan-1(3/4)
•The measure of angle A is 370
Example
• “Find the measure of the acute angle that the
given line makes with the x-axis”
• Y=1/2x-2
• Do we need to graph this? No. all we need is
the slope
• The slope is 1/2. Therefore Tan(x)= 1/2
• We need the measure of the angle, therefore
use Tan-1(1/2)
• Tan-1(1/2)= 26.56, or 27 degrees
Assignment
• Calculate Tangent Ratio Worksheet
• Visualize Tangent Worksheet
• Worksheet 9-1
Sine and Cosine Ratios
• We now understand the concepts we’re using to
determine ratios, so we won’t have to re-explain
those.
• Tangent (of angle)= Opposite/Adjacent
• Sine (of angle) = Opposite/Hypotenuse
• Cosine (of angle) = Adjacent /Hypotenuse
• These are abbreviated
• SIN(A)
• COS(A
SIN and COS
•
There are two ways (among others) to
remember these
1. SOHCAHTOA
•
This means
1. SIN:Opposite/Hypotenuse
2. COS:Adjacent/Hypotenuse
3. TAN:Opposite/Adjacent
2. Oscar Has A Heap Of Apples (This
uses the same order: SIN, COS, TAN
Examples
G
1. What is the ratio for Sin(T)?
17
2. What is the ratio for Sin(G)?
8
3. What is the ratio for Cos(T)?
4. What is the ratio for Cos(G)?
1. Sin(T) = 8/17
3. Cos(T) = 15/17
2. Sin(G) = 15/17
4. Cos(G) = 8/17
T
15
R
Example
What is the Sin and
Cos for angle X and
Angle Z?
Sin(x) = 64/80
Cos(x) = 48/80
Sin(z) = 48/80
Cos(z) = 64/80
X
80
48
Y
64
Z
•What conclusions can I draw when I look at these ratios?
•If the two angles are complimentary (and they are in a right
triangle) then the Sin(1st angle) = Cos(2nd angle) and viceversa
Sine and Cosine
• There is a relationship between Sine and
Cosine:
• Sin(X0) = Cos(90-X)0 for values of x between
0 and 90. -Remember they are equal to each
other when the two acute angles (not the 90
degree angle) are complimentary, which is
always in a right triangle
• This equation is called an Identity, because it
is true for all allowed values of X
Real World
• Trig functions have been known for centuries
• Copernicus developed a proof to determine
the size of orbits of planets closer to the sun
than the Earth using Trig
• The key was determining when the planets
were in position, and then measuring the
angle (here angle a)
Mercury's mean distance from the
sun is 36 million miles. Mercury runs
around the sun in a tight little elliptical
path. At it's closest to the Sun,
Mercury is 28.6 million miles , at it's
farthest it is 43.4 million miles.
.379 x 93 million =
35.25 million miles
Real World
If A0 = 22.3 degrees for
Mercury, how far is
Mercury from the sun in
AU? (about 93 million
miles)
Sin(22.3) = X/1
X = Sin(22.3)
X = .379 (AU)
Venus’ distance from the sun
varies from 67.7 million miles to
about 66.8 million miles. The
average distance is about 67.2
million miles from the sun.
x
a0
.72 x 93 million =
66.96 million miles
Sun
If A0 = 46 for Venus,
how far from the sun
is Venus in AU?
Sin(46) = X/1
X = .72 (AU)
Inverse Sine and Cosine
• Again, the inverse function on the
calculator finds the degree, not the ratio
• Find the measure of angle L to the
nearest degree
L
Cos(L) = 2.5/4.0
Cos-1(2.5/4.0) = 51.37,
or 51 degrees
Or, Sin(L) = 3.1/4.0
Sin-1(3.1/4.0) = 50.8
or 51 degrees
4.0
2.5
F
3.1
O
Assignment
•
•
•
•
Page 510-511 7-27
Page 511 33-36 (honors)
Visualizing Sine Cosine Worksheet
Worksheet 9-2
Unit 9 Quiz 1
1. If X0 = 34, and O = 5, what is
the measure of A?
H
0
2. If X = 62, and A = 4.7 what is
the measure of O?
3. If O = 5.5, and A = 3, what is Xo
A
the measure of X0?
4. If A = 4.7, and O = 2.1, what is
the measure of X0?
5. If X0= 45, and O = 7, what is
the measure of A?
O
Unit 9 Quiz 2
1. If X0 = 54, and O = 5, what is
the measure of A?
H
0
2. If X = 22, and A = 4.7 what is
the measure of O?
3. If O = 3.5, and A = 3, what is Xo
A
the measure of X0?
4. If A = 7.7, and O = 2.1, what is
the measure of X0?
5. If X0= 45, and O = 3, what is
the measure of A?
O
Unit 9 Quiz 3
1. If X0 = 24, and O = 5, what is
the measure of H?
H
0
2. If X = 72, and A = 4.7 what is
the measure of h?
Xo
3. If H = 6.5, and A = 3, what is
A
the measure of X0?
4. If H = 4.7, and O = 3.1, what is
the measure of X0?
5. If X0= 15, and H = 7, what is
the measure of A?
O
Angles of Elevation and
Depression
• Suppose you were on the ground, and looked up to a balloon.
From the horizontal line, to the balloon the angle is 38 degrees.
This is the angle of elevation
• At the same time, someone looking down from the horizontal
would see you on the ground at an angle of 38 degrees. This is
the angle of depression.
• If you look, you see that these are opposite interior angles on a
transversal crossing parallel lines, thus they are the same
measure.
Horizontal Line
0
Angle of Depression 38
Parallel Lines
380
Angle of Elevation
Horizontal Line
Elevation and Depression
• Key Point: No matter what the angle
of depression is, USE THAT AS THE
ANGLE OF ELEVATION!!!
• The angle of depression is OUTSIDE
the triangle, so we move it INSIDE
and call it the angle of elevation
• Do NOT put it at the top of the triangle
Xo
Xo
Real World
• Surveyors use 2 instruments -the transit and
the theodolite- to measure angles of elevation
and depression.
• On both instruments, the surveyor sets the
horizon line perpendicular to the direction of
gravity.
• By using gravity to establish the horizontal
line (a bubble level), they avoid the problems
presented by sloping surfaces
Real World
• A surveyor wants to find the height of
the “Delicate Arch” in Arches National
Park in Utah.
• To do this, she sets the theodolite at the
bottom of the arch, and moves to a
point where she can measure the angle
to the top
• Then she measures how far she walked
out to measure the arch
Real World
• How high is the arch?
In this case it’s opposite over
adjacent, so we use Tan(48)
And get 39.98, or 40 ft
But we need to add
The 5 feet for
The tripod
So 45 ft.
480
X FT
Theodolite sits
on a tripod 5
feet off the
ground
Assignment
• Page 519 9-23
• Workbook 9-3
• Trig Word Problems Worksheet
Unit 9 Quiz 4
1. If X0 = 24, and O = 5, what is
the measure of A?
H
0
O
2. If X = 72, and A = 4.7 what is
the measure of O?
3. If O = 6.5, and A = 3, what is Xo
A
the measure of X0?
4. If A = 4.7, and O = 3.1, what is
the measure of X0?
5. If X0= 15, and O = 7, what is
Extra Credit: (From CPD Test)
the measure of A?
1. What is 8 percent of 42,000
2. What is 3/5 divided by 2/3
• (FYI: They weren’t allowed to
use a calculator
Unit 9 Quiz 5
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
What does SOHCAHTOA mean?
If you are given the lengths of Side O and Side A, and are
asked to find the measure of Angle X (in degrees), what
function do you use on the calculator?
If you are asked to find the length of Side A, and are given the
length of the Hypotenuse and the degree of the angle x, what
function do you use on the calculator?
What does A stand for?
What does O stand for?
What does H stand for?
If A = 12, and H = 13, what is the measure of X0?
If O = 7, and H =15 what is the measure of X0?
If X0 = 34, and O = 8, what is the measure of A?
H
If X0 = 62, and A = 4.7 what is the measure of H?
Xo
A
O
Unit 9 Quiz 6
1. If A is 5 and O is 7, what is the measure o
X
of X0?
2. If O is 5 and H is 9, what is the measure
of X0?
3. If A is 3 and H is 11, what is the measure
of X0?
4. If O is 7 and A is 9, what is the measure
of X0?
5. If A is 5 and H is 21, what is the measure
of X0?
H
O
A
Unit 9 Quiz 7
• Write a paragraph about what Veteran’s
day means to you.
• It must have more than three sentences
to be a paragraph.
• 10 minutes
• 10 points
How Tall is the Smokestack?
• To calculate how tall is the smoke stack, we need two
pieces of information:
• How far away is the smoke stack
• What is the angle of elevation to the smoke stack
• Then we can use the tangent ratio to calculate the
height:
Angle we calculate
Smokestack
There is only one problem….
Us
Distance (from Google Earth)
This is: 4371 meters (2.71 miles)
Height we
calculate
How Tall is the Smokestack?
Angle we calculate
Smokestack
Height we
calculate
Distance (from Google Earth)
This is: 4371 meters
Add 21.5
meters
• We are actually 20 meters higher in elevation than
the base of the smokestack
• So when we calculate the height, we need to add 20
meters
• We also need to add 5 feet, or 1.5 meters
• Therefore, overall we will add 21.5 meters to our final
calculation
Distance to Stack
• According to Google Earth
the distance from the
corner of the parking lot at
the front of the school to
the base of the
smokestack is 4371
meters
• We want to shoot an
azimuth to the top of the
smokestack
• And then measure the
angle from level ground, to
the top
• Now all we need is the
height of the tower, found
by calculating the tangent
ratio…
H
x0
And the Answer is…
• The actual
height of the
tallest
smokestack
is 305
meters…
Real World Application
Solution
1) Shot an angle from the
fire hydrant to the house
across the street (328
degrees)
X
3) Shot a new angle
to the house (318
degrees)
This means my
interior triangle
degree is 80 degrees
2) Turned left 90
degrees and walked
at that new angle for
100 meters (238
degrees)
800 100m
•To calculate the
distance to the
house across the
street, I created
a right triangle.
The distance is
the opposite side
–or X- the
adjacent side is
100 meters, and
the angle is 80
degrees.
•To solve, the
equation is
TAN(80) = X/100
•The solution is
567 meters
•According to
Google Earth, it
is 530 meters
•This is a
deviation of 37
meters, or I am
accurate to
within 90%…
4400 meters
2.73 miles
Extra Credit, worth 10 points
Draw picture
Write Equation
What is your answer (nearest foot)
• Tom wants to paint the Iwo Jima Memorial
• The Memorial is 60 feet to the top of the flag pole
“Among the men
• Tom measures the angle from where he is QuickTime™ and a
decompressor
who
fought on Iwo
standing, to the top of the flag pole, at 300 are needed to see this
picture.
Jima, uncommon
• Tom can’t see the statue very well, so he moves
back-he moves away from the statue
valor was a
• The angle to the top of the flag pole is now 200
common virtue.”
• Rounded to the nearest foot, how many feet
back did Tom move?
-Admiral Nimitz