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

- 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

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.

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?

- Draw the triangle the mountain creates (3 points)
- Write the equation (3 points)
- Calculate the height (3 points)
- 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?

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

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

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
- As Tan A Opposite
- Adjacent

B

Leg opposite ?A

C

A

Leg adjacent to ?A

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

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

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

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

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

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
- 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

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

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

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

Assignment

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

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

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

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)?

- Sin(T) 8/17

3. Cos(T) 15/17

2. Sin(G) 15/17

4. Cos(G) 8/17

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

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)

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)

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

Assignment

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

Unit 9 Quiz 1

- If X0 34, and O 5, what is the measure of A?
- If X0 62, and A 4.7 what is the measure of O?
- If O 5.5, and A 3, what is the measure of X0?
- If A 4.7, and O 2.1, what is the measure of

X0? - If X0 45, and O 7, what is the measure of A?

H

O

Xo

A

Unit 9 Quiz 2

- If X0 54, and O 5, what is the measure of A?
- If X0 22, and A 4.7 what is the measure of O?
- If O 3.5, and A 3, what is the measure of X0?
- If A 7.7, and O 2.1, what is the measure of

X0? - If X0 45, and O 3, what is the measure of A?

H

O

Xo

A

Unit 9 Quiz 3

- If X0 24, and O 5, what is the measure of H?
- If X0 72, and A 4.7 what is the measure of h?
- If H 6.5, and A 3, what is the measure of X0?
- If H 4.7, and O 3.1, what is the measure of

X0? - If X0 15, and H 7, what is the measure of A?

H

O

Xo

A

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

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 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

Assignment

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

Unit 9 Quiz 4

- If X0 24, and O 5, what is the measure of A?
- If X0 72, and A 4.7 what is the measure of O?
- If O 6.5, and A 3, what is the measure of X0?
- If A 4.7, and O 3.1, what is the measure of

X0? - 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

Unit 9 Quiz 5

- 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?
- If X0 62, and A 4.7 what is the measure of H?

H

O

Xo

A

Unit 9 Quiz 6

H

O

- If A is 5 and O is 7, what is the measure of X0?
- If O is 5 and H is 9, what is the measure of X0?
- If A is 3 and H is 11, what is the measure of X0?
- If O is 7 and A is 9, what is the measure of X0?
- If A is 5 and H is 21, what is the measure of X0?

Xo

A

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

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)

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

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

And the Answer is

- The actual height of the tallest smokestack is

305 meters

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

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 - 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