Unit 9 -Right Triangle Trigonometry

1
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).

2
Standards

• SPIs 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

3
Standards

• 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 Geometers
  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.

4
Unit 9 Bellringer 10 points
Tallest US Mtns McKinley (AK) Ebert (CO) Massive
(CO) Harvard (CO) Rainer (WA)

• 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?

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)

5
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
6
A Look at Triangle Relationships

• What can you conclude about these three partial
  Right triangles?

Xo
Xo
Xo

• 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

7
Labeling the Parts

• We will use the same approach to all triangles
  during Right Triangle Trigonometry
• We do not apply the rules of R.T. Trig to the
  right angle (I.E. solving for tangent etc.)
• If possible, we try to set the problem up to use
  the bottom angle
• We always label the side farthest from the angle
  as Opposite
• We always label the side that touches the angle
  we are using as Adjacent
• The Hypotenuse is the diagonal that touches our
  angle

H
O
Xo
A
8
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.

9
Tangent Ratios

• Tangent of ?A
• Length of leg opposite ?A
• Length of leg adjacent to ?A
• You can abbreviate this
• As Tan A Opposite
• Adjacent

B
Leg opposite ?A
C
A
Leg adjacent to ?A
10
Writing Tangent Ratios

• Tan T Opposite/Adjacent
• Or UV/TV 3/4
• Tan U Opposite/Adjacent
• TV/UV 4/3
• What is the Tan for ?K?
• What is the Tan for ?J?

U
5
3
V
T
4
J
Tan K 3/7
3
Tan J 7/3
What relationship is there between them?
L
K
7
They are reciprocals
11
So youre 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
50 -how far You walk
860
Your new angle to the MTN Top
12
Heres 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.
  Its 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.

X (Opposite)
M
50 (Adjacent)
860
13
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

14
Find the value of W
Remember Tan(xo) O/A
330
280
W
W
W
1.0
570
10
2.5
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
Tan 28 1.0/ W W (Tan 28) 1.0 W 1.0 / (Tan
28) W 1.88
540
Tan 54 W/10 W 10 (tan 54) W 13.76
15
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

16
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
• So m ?X Tan -1 (.75)
• TAN -1 (.75) ENTER 36.86
• You can also type TAN -1 (6/8)
• So, m ?X 37 degrees

H
10
6
X
B
8
17
Example of Inverse

• Find the m of ?Y to the nearest degree

We need the tangent ratio so 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
T
100
P
41
Y
18
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 --gt rise/run

• The question is, can you use the tangent
  function to determine the measure of angle A?

• Tangent is a ratio of
  Opposite/Adjacent
• In this case, Opposite is the rise, and Adjacent
  is the run

Op Adjacent
A

• So Tan(A) is the slope, or 3/4
• Therefore, we use Tan-1(3/4)
• The measure of angle A is 370

19
Example

• Find the measure of the acute angle that the
  given line makes with the x-axis
• Y1/2x-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

20
Assignment

• Calculate Tangent Ratio Worksheet
• Visualize Tangent Worksheet
• Worksheet 9-1

21
Sine and Cosine Ratios

• We now understand the concepts were using to
  determine ratios, so we wont 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

22
SIN and COS

• There are two ways (among others) to remember
  these
• SOHCAHTOA
• This means
• SINOpposite/Hypotenuse
• COSAdjacent/Hypotenuse
• TANOpposite/Adjacent
• Oscar Has A Heap Of Apples (This uses the same
  order SIN, COS, TAN

23
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)?
R
15
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
24
Example
What is the Sin and Cos for angle X and Angle Z?
X
Sin(x) 64/80 Cos(x) 48/80 Sin(z)
48/80 Cos(z) 64/80
80
48
Z
Y
64

• 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 vice-versa

25
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

26
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)

27
Real World
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.
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.
.379 x 93 million 35.25 million miles
If A0 22.3 degrees for Mercury, how far is
Mercury from the sun in AU? (about 93 million
miles)
x
Sun
Sin(22.3) X/1 X Sin(22.3) X .379 (AU)
1 AU (Astronomical Unit)
If A0 46 for Venus, how far from the sun is
Venus in AU?
a0
.72 x 93 million 66.96 million miles
Sin(46) X/1 X .72 (AU)
28
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
4.0
2.5
Or, Sin(L) 3.1/4.0 Sin-1(3.1/4.0) 50.8 or 51
degrees
F 3.1 O
29
Assignment

• Page 510-511 7-27
• Page 511 33-36 (honors)
• Visualizing Sine Cosine Worksheet
• Worksheet 9-2

30
Unit 9 Quiz 1

1. If X0 34, and O 5, what is the measure of A?
2. If X0 62, and A 4.7 what is the measure of O?
3. If O 5.5, and A 3, what is 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?

H
O
Xo
A
31
Unit 9 Quiz 2

1. If X0 54, and O 5, what is the measure of A?
2. If X0 22, and A 4.7 what is the measure of O?
3. If O 3.5, and A 3, what is 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?

H
O
Xo
A
32
Unit 9 Quiz 3

1. If X0 24, and O 5, what is the measure of H?
2. If X0 72, and A 4.7 what is the measure of h?
3. If H 6.5, and A 3, what is 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?

H
O
Xo
A
33
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
380
Angle of Depression
Parallel Lines
Angle of Elevation
380
Horizontal Line
34
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
35
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

36
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

37
Real World

• How high is the arch?

In this case its 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.
X FT
36 FT
480
Theodolite sits on a tripod 5 feet off the ground
38
Assignment

• Page 519 9-23
• Workbook 9-3
• Trig Word Problems Worksheet

39
Unit 9 Quiz 4

1. If X0 24, and O 5, what is the measure of A?
2. If X0 72, and A 4.7 what is the measure of O?
3. If O 6.5, and A 3, what is 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 the measure of A?

H
O
Xo
A

• Extra Credit (From CPD Test)
• What is 8 percent of 42,000
• What is 3/5 divided by 2/3
• (FYI They werent allowed to use a calculator

40
Unit 9 Quiz 5

1. What does SOHCAHTOA mean?
2. 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?
3. 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?
4. What does A stand for?
5. What does O stand for?
6. What does H stand for?
7. If A 12, and H 13, what is the measure of
   X0?
8. If O 7, and H 15 what is the measure of X0?
9. If X0 34, and O 8, what is the measure of A?
10. If X0 62, and A 4.7 what is the measure of H?

H
O
Xo
A
41
Unit 9 Quiz 6
H
O

1. If A is 5 and O is 7, what is the measure 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?

Xo
A
42
Unit 9 Quiz 7

• Write a paragraph about what Veterans day means
  to you.
• It must have more than three sentences to be a
  paragraph.
• 10 minutes
• 10 points

43
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.
Height we calculate
Us
Distance (from Google Earth) This is 4371 meters
(2.71 miles)
44
How Tall is the Smokestack?
Angle we calculate
Smokestack
Height we calculate
Add 21.5 meters
Distance (from Google Earth) This is 4371 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

45
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
Do you see the triangle?
x0
46
And the Answer is

• The actual height of the tallest smokestack is
  305 meters

47
Real World Application Solution

• 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

1) Shot an angle from the fire hydrant to the
house across the street (328 degrees)
2) Turned left 90 degrees and walked at that new
angle for 100 meters (238 degrees)
X
3) Shot a new angle to the house (318
degrees) This means my interior triangle degree
is 80 degrees
100m
800
48
4400 meters 2.73 miles
49
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
• Tom measures the angle from where he is
  standing, to the top of the flag pole, at 300
• Tom cant see the statue very well, so he
  moves back-he moves away from the statue
• The angle to the top of the flag pole is now
  200
• Rounded to the nearest foot, how many feet
  back did Tom move?

• Among the men who fought on Iwo Jima, uncommon
  valor was a common virtue.
• -Admiral Nimitz