precalculus 4/10/13 obj: midterm review
TRANSCRIPT
PreCalculus 4/10/13
Obj: Midterm Review
Agenda
1. Bell Ringer: None
2. #35, 72 Parking lot 37, 39, 41
3. Homework Requests: Few minutes on Worksheet
4. Exit Ticket: In Class Exam Review
Homework:
Study for Midterm Exam
Announcements:
30th Week Exam 4/11
Exit Ticket:
Pg 368 #6, 10 also find the measure of the
angle.
Pg 368 #31, 33, 35,39, 49, 51
Pg 358 #4, 6, 14, 22, 26, 30,
Pg 346 #27, 32,33, 63, 75
Pg 331 #18, 37
Trigonometry is… • A branch of geometry used first by the
Egyptians and Babylonians (Iraq)
• Used extensively is astronomy and building
• Based on ratios between angles in RIGHT
Triangles
The Trigonometric (trig) ratios: FUNCTION
INVERSE FUNCTION
Also true are…
Sample keystrokes Careful about Deg or
Rad Setting Exit Ticket pg 369 #30-40
evens Sample keystroke
sequences
Sample calculator display Rounded
Approximation
74
74
0.961262695 0.9613
0.275637355 0.2756
3.487414444 3.4874
sin
sin
ENTER
74
74
COS
COS
ENTER
74
74
TAN
TAN
ENTER
A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is
150 feet high. What is the number of feet from the swimmer to the shore?
18º
150
Tan 18 =
x
150
x
0.3249 =
150
x
0.3249x = 150
0.3249 0.3249
X = 461.7 ft 1
A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly
below the dragon. At what angle does the archer need to aim his arrow to slay the dragon?
x
60
120
Tan x =
60
120Tan x = 0.5
Tan-1(0.5) = 26.6º
A person is 200 yards from a river. Rather than
walk directly to the river, the person walks along a
straight path to the river’s edge at a 60° angle.
How far must the person walk to reach the river’s
edge?
200
x
Ex. 5
60°
cos 60°
x (cos 60°) = 200
x
X = 400 yards
Exit Ticket WS 2, 4, 10, 12, 18, 20 For 18 find values of all trig functions
An explorer is standing 14.3 miles from the base of
Mount Everest below its highest peak. His angle of
elevation to the peak is 21º. What is the number of feet
from the base of Mount Everest to its peak?
21º 14.3
x
Tan 21 =
x
14.30.3839 =
x
14.3
x = 5.49 miles
= 29,000 feet
1
The Trigonometric Functions
SINE
COSINE
TANGENT
Pronounced “theta”
Greek Letter q
Represents an unknown angle
Pronounced “alpha”
Greek Letter α
Represents an unknown angle
Pronounced “Beta”
Greek Letter β
Represents an unknown angle
Finding Trig Ratios
• A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.
Trigonometric Ratios • Let ∆ABC be a right
triangle. The sine,
the cosine, and the
tangent of the acute
angle A are defined
as follows.
ac
bside adjacent to angle A
Side
opposite
angle A
hypotenuse
A
B
C
sin A = Side opposite A
hypotenuse
= a
c
cos A = Side adjacent to A
hypotenuse
= b
c
tan A = Side opposite A
Side adjacent to A
= a
b
q
opposite hypotenuse
SinOpp
Hyp
adjacent
CosAdj
Hyp
TanOpp
Adj
hypotenuse opposite
adjacent
C B
A
We could ask for the trig functions of the angle by using the definitions.
a
b
c
You MUST get them memorized. Here is a
mnemonic to help you.
The sacred Jedi word:
SOHCAHTOA
c
b
hypotenuse
oppositesin
adjacentcos
hypotenuse
a
c opposite
tanadjacent
b
a
adjacent
SOHCAHTOA
It is important to note WHICH angle you are talking
about when you find the value of the trig function.
a
b
c
Let's try finding some trig functions
with some numbers. Remember that
sides of a right triangle follow the
Pythagorean Theorem so
222 cba
Let's choose: 222 5 43 3
4
5
sin = Use a mnemonic and
figure out which sides
of the triangle you
need for sine.
h
o
5
3
tan =
a
o
3
4
adjacent
Use a mnemonic and
figure out which sides
of the triangle you
need for tangent.
You need to pay attention to which angle you want the trig function
of so you know which side is opposite that angle and which side is
adjacent to it. The hypotenuse will always be the longest side and
will always be opposite the right angle.
This method only applies if you have
a right triangle and is only for the
acute angles (angles less than 90°)
in the triangle.
3
4
5
Oh,
I'm
acute!
So
am I!
Ex. 1: Finding Trig Ratios Large Small
15
817
A
B
C
7.5
48.5
A
B
C
sin A = opposite
hypotenuse
cosA = adjacent
hypotenuse
tanA = opposite
adjacent
8
17 ≈ 0.4706
15
17 ≈ 0.8824
8
15 ≈ 0.5333
4
8.5 ≈ 0.4706
7.5
8.5 ≈ 0.8824
4
7.5 ≈ 0.5333
Trig ratios are often expressed as decimal approximations.
Ex. 1: Finding Trig Ratios Large Small
15
817
A
B
C
7.5
48.5
A
B
C
sin A = opposite
hypotenuse
cosA = adjacent
hypotenuse
tanA = opposite
adjacent
8
17 ≈ 0.4706
15
17 ≈ 0.8824
8
15 ≈ 0.5333
4
8.5 ≈ 0.4706
7.5
8.5 ≈ 0.8824
4
7.5 ≈ 0.5333
Trig ratios are often expressed as decimal approximations.
Ex. 2: Finding Trig Ratios
S
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
tanS = opposite
adjacent
5
13 ≈ 0.3846
12
13 ≈ 0.9231
5
12 ≈ 0.4167
adjacent
opposite
12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios
S
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
tanS = opposite
adjacent
5
13 ≈ 0.3846
12
13 ≈ 0.9231
5
12 ≈ 0.4167
adjacent
opposite
12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of the indicated angle.
R
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
tanS = opposite
adjacent
12
13 ≈ 0.9231
5
13 ≈ 0.3846
12
5 ≈ 2.4
adjacent
opposite12
13 hypotenuse5
R
T S
Ex. 2: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of the indicated angle.
R
sin S = opposite
hypotenuse
cosS = adjacent
hypotenuse
tanS = opposite
adjacent
12
13 ≈ 0.9231
5
13 ≈ 0.3846
12
5 ≈ 2.4
adjacent
opposite12
13 hypotenuse5
R
T S
Ex: 5 Using a Calculator
• You can use a calculator to approximate the
sine, cosine, and the tangent of 74. Make
sure that your calculator is in degree mode.
The table shows some sample keystroke
sequences accepted by most calculators.
3.2 cm
7.2 cm 24º
3.2
7.2
0.45 Tangent 24º
0.45
Tangent A =
opposite
adjacent
3.2 cm 7.9 cm
24º
9.7
2.3
0.41 Sin 24º
0.41
Sin α =
hypotenuse
opposite
5.5 cm
7.9 cm
46º
9.7
5.5
0.70 Cos 46º
0.70
Cosine β =
hypotenuse
adjacent
A plane takes off from an airport an an angle of 18º and
a speed of 240 mph. Continuing at this speed and angle,
what is the altitude of the plane after 1 minute?
18º
x
After 60 sec., at 240 mph, the plane
has traveled 4 miles
4
18º
x 4
opposite
hypotenuse
SohCahToa
Sine A =
opposite
hypotenuseSine 18 =
x
4
0.3090 =
x
4
x = 1.236 miles
or
6,526 feet
1
Soh
Solving a Problem with
the Tangent Ratio
60º
53 ft
h = ?
We know the angle and the
side adjacent to 60º. We want to
know the opposite side. Use the
tangent ratio:
ft 92353
531
3
5360tan
h
h
h
adj
opp
1
2 3
Why?
A surveyor is standing 50 feet from the base of
a large tree. The surveyor measures the
angle of elevation to the top of the tree as
71.5°. How tall is the tree?
50
71.5
°
?
tan
71.5°
tan
71.5° 50
y
y = 50 (tan 71.5°)
y = 50 (2.98868)
149.4y ft
Ex.
Opp
Hyp
Notes: • If you look back at Examples 1-5, you
will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.
Ex
A 100 degree sector cut from a circular disc has a
length of 7 cm. To the nearest cm., what is the
radius of the circle? What is the area of the sector?
S = r ɵ
Must be in radians
Just convert degrees to
radians
ɵ
Ans. 4 cm
Ans.13.95 sq. cm
What is
SohCahToa? Is it in a tree, is it in a car, is it in the sky
or is it from the deep blue sea ?
This is an example of a sentence
using the word SohCahToa.
I kicked a chair in the middle of
the night and my first thought was
I need to SohCahToa.
An example of an acronym for SohCahToa.
Seven
old
horses
Crawled
a
hill
To
our
attic..
Old Hippie
Some Old Hippie Came A Hoppin’ Through Our Apartment
SOHCAHTOA
Old Hippie
Sin Opp Hyp Cos Adj Hyp Tan Opp Adj
Other ways to remember SOH CAH TOA
1. Some Of Her Children Are Having Trouble
Over Algebra.
2. Some Out-Houses Can Actually Have
Totally Odorless Aromas.
3. She Offered Her Cat A Heaping Teaspoon
Of Acid.
4. Soaring Over Haiti, Courageous Amelia Hit
The Ocean And ...
5. Tom's Old Aunt Sat On Her Chair And
Hollered. -- (from Ann Azevedo)
Other ways to remember SOH CAH TOA
1. Stamp Out Homework Carefully, As Having
Teachers Omit Assignments.
2. Some Old Horse Caught Another Horse
Taking Oats Away.
3. Some Old Hippie Caught Another Hippie
Tripping On Apples.
4. School! Oh How Can Anyone Have Trouble
Over Academics.
A
Trigonometry Ratios
Tangent A =
opposite
adjacent
Sine A =
opposite
hypotenuse
Cosine A =
adjacent
hypotenuse
Soh Cah Toa
14º
24º
60.5º
46º 82º
1.9 cm
7.7 cm
14º
1.9
7.7
0.25 Tangent 14º
0.25
The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side.
opposite adjacent
hypotenuse
5.5 cm
5.3 cm
46º
5.5
5.3
1.04 Tangent 46º
1.04
Tangent A =
opposite
adjacent
6.7 cm
3.8 cm
60.5º
6.7
3.8
1.76
Tangent 60.5º
1.76
Tangent A =
opposite
adjacent
As an acute angle of a triangle
approaches 90º, its tangent
becomes infinitely large
Tan 89.9º = 573
Tan 89.99º = 5,730
Tangent A =
opposite
adjacent
etc.
very
large
very small
Since the sine and cosine functions always have the hypotenuse as the denominator,
and since the hypotenuse is the longest side, these two functions will always be less than 1.
Sine A =
opposite
hypotenuse
Cosine A =
adjacent
hypotenuse
A Sine 89º = .9998
Sine 89.9º = .999998
Ex. 3: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of 45
45
sin 45= opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
1
hypotenuse1
√2
cos 45=
tan 45=
1
√2 =
√2
2 ≈ 0.7071
1
√2 =
√2
2 ≈ 0.7071
1
1 = 1
Begin by sketching a 45-45-90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the hypotenuse is √2. 45
2
1
Ex. 4: Finding Trig Ratios—Find the sine, the
cosine, and the tangent of 30
30
sin 30= opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
√3
cos 30=
tan 30=
√3
2 ≈ 0.8660
1
2 = 0.5
√3
3 ≈ 0.5774
Begin by sketching a 30-60-90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. 30
√3
1 =
Trigonometric Functions on a
Rectangular Coordinate System
x
y
q
Pick a point on the
terminal ray and drop a
perpendicular to the x-axis.
r y
x
The adjacent side is x
The opposite side is y
The hypotenuse is labeled r
This is called a
REFERENCE TRIANGLE. y
x
x
y
x
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Ratios may be found by:
45 º
1
1
2Using ratios of special triangles
145tan
2
145cos
2
145sin
For angles other than 45º, 30º, 60º you will need to use a
calculator. (Set it in Degree Mode for now.)
We need a way to remember all of these ratios…