Download Chapter 8 Notes Packet Notes #1—Section 8.2 Triangle Theorem In

Chapter 8 Notes Packet
Notes #1—Section 8.2
45° − 45° − 90° Triangle Theorem
In Words:
Symbols/Picture:
In a 45° − 45° − 90° triangle,
both ________ are congruent
and the length of the
_____________ is 2 times the
length of a leg.
Example:
Find the lengths of the missing
sides of the triangle.
1.)
18
Hypothesis =
2 ⋅ leg
45 °
2.)
45 °
4 2
30° − 60° − 90° Triangle Theorem
In Words:
In a 30° − 60° − 90° triangle, the
length of the _____________ is
twice the length of the
________________ leg. The
length of the longer leg is 3
times the length of the
___________ leg.
Symbols/Picture:
Example:
Find the lengths of the missing
sides of the triangle.
3.)
12
Hypothesis = 2 ⋅ shorter leg
Longer leg =
30 °
3 ⋅ shorter leg
4.)
60 °
18
Examples: Find the value of each variable. If your answer is not an integer, leave it in simplest radial
form.
5.)
6.)
45 °
y
15 2
x
x
45 °
y
2
7.)
8.)
y
16 3
30 °
9 3
x
y
60 °
x
9.)
10.)
b
4 3
d
a
14
a
60 °
c
30 °
45 °
60 °
b
c
d
11.)
12.)
6
b
60 °
12 2
2 3
c
a
45 °
b
d
13.)
14.)
a
2 2
45°
45°
b
m
5 3
60
30
y
15.)
16.)
6
10
x
8 2
30
60
x
30
y
y
Algebra Review: Solving Quadratic Equations using the Quadratic Formula:
Step 1: Set the equation equal to 0 & write it in standard form ( ax 2 + bx + c = 0 )
Step 2: Identify a, b, and c.
Step 3: Substitute the values of a, b, and c into the quadratic formula:
x=
−b ± b 2 − 4ac
2a
Step 4: Simplify the radical as much as possible.
Step 5: Simplify the numerator as much as possible.
Step 6: If possible, divide both terms on the numerator by the number in the denominator.
Examples: Solve each equation by the quadratic formula.
17.) x 2 + 3x = 4
18.) 2 x 2 = 4 x + 3
19.) 9 x 2 + 12 x + 4 = 0
20.) x 2 + 7 = 6 x
Notes #2—Section 8.3 & 8.4 (Finding Trig Values)
Trigonometry can be used to find
for non-special right triangles & to find
when you are given two side lengths of a right triangle.
B
Three Trig Definitions:
Given right + ABC where (C is a right angle:
A
sin A =
C
cos A =
tan A =
Examples: Write the sine, cosine, & tangent ratio for (A and (B
1.)
2.)
29
A
B
B
13
21
20
2
C
A
3
C
Using a Trig Table to Find the Trig Value:
• If you know the angle, find the angle in the left-most column and read to the right to find its
sine, cosine, and/or tangent
Use your trig table. Round your answers to the nearest hundredth:
3.)
a) sin32◦ = _____
b) tan19◦ = _____
c) cos75◦ = _____
d) cos48◦ = _____
e) sin80◦ = _____
f) tan59◦ = _____
Examples: Use the trig table to find the missing value.
x
30
4.) sin 42° =
5.) tan 67° =
x
20
6.) cos12° =
58
x
To Find a Missing Side on a Right Triangle:
Step 1: Pick an acute angle & label the sides as O (opposite), A (adjacent), and H (hypotenuse)
from the point of view of your angle.
Step 2: Choose a trig function (sine, cosine, or tangent) and write an equation.
Step 3: Solve for the variable—wait to use your calculator until the last step!
Examples: Find the value of the variable to the nearest tenth.
9.)
8.)
7.)
10
x
54 °
17 °
w
x
6.5
29 °
48
11.)
10.)
12.)
1.0
w
w
33 °
2.5
28 °
57 °
25
32 °
w
Algebra Review: Adding & Subtracting Radicals
A. Simplify all radicals first.
B. Add or subtract the coefficients of the radicals for the terms with identical radicals. The radical will
NOT change
13.)
288 + 50
15.) 3 72 − 8 50
14.)
32 − 5 8
16.) 6 20 + 4 45
Notes #3—Section 8.3 & 8.4 (Finding Angle Measures)
To find the angle measure for a given trig value, you use the
,
, and
.
indicated as
, which are
Using Your Trig Table to Find an Angle:
• If you know the decimal value of its sine, cosine, and/or tangent, look down the sin/cos/tan
column until you find the closest match. The read to the left to find the angle.
Use your trig table. Round your answers to the nearest degree:
b) tan____◦ = 0.21
1.)
a) sin____◦ = 0.9903
d) cos____◦ = 0.454
e) sin____◦ = 0.7
c) cos____◦ = 0.79
f) tan____◦ = 2.5
Examples: Use the trig table to find the missing value to the nearest degree.
3.) tan x = 1.6429
4.) sin x = 0.6120
2.) cos x = 0.2419
To Find an Angle on a Right Triangle:
Step 1: From the unknown angle, label the sides of the triangle as O (opposite), A (adjacent),
and H (hypotenuse)
Step 2: Choose a trig function (sine, cosine, or tangent) and write an equation.
Step 3: Solve for the variable—wait to use your calculator until the last step!
Examples: Find the value of x to the nearest degree.
7.)
6.)
5.)
x°
42
x°
20
92
42
82
26
w°
8.)
10.)
9.)
52
55
x°
x°
17
65
29
x°
12
Algebra Review: Multiplying & Dividing Radicals
To Multiply Radicals:
Step 1: Simplify all radicals as much as possible
Step 2: Multiply the coefficients together & multiply the numbers under the radicals (the radicands)
together.
Step 3: Simplify the radical
To Divide Radicals:
Step 1: Simplify the radicals as much as possible
Step 2: Multiply the numerator & denominator by the radical that is on the denominator
Step 3: Simplify the remaining radical (if necessary) and reduce the fraction (if necessary)
Examples: Simplify each expression as much as possible.
11.) 8 ⋅ 48
12.)
13.)
12
6
14.)
24 ⋅ 120
3 18
2 27
Notes #4—Section 8.5
Angle of Elevation is the angle from a
________________ line looking _________ to an
object
Picture:
Angle of Depression is the angle from a
_________________ line looking __________ to
an object.
Picture:
Examples: Describe each angle as it relates to the situation in the diagram.
1.) (1
2.) ( 2
3.) (3
4.) ( 4
Examples: Solve each word problem involving angles of elevation & depression.
5.) A surveyor stands 200 ft from a building to the measure its height with a 5-ft tall tool. The angle of
elevation from the top of the tool to the top of the building is 35° . How tall is the building? Round to
the nearest tenth of a foot.
6.) An airplane flying 3500 ft above ground begins a 2° angle of descent (depression) to land at an
airport. What is the plane’s horizontal distance to the airport when it starts its descent? Round your
answer to the nearest tenth of a foot.
7.) A 6 ft man (from his feet to his eyes) stands 12 ft from the base of a tree. The angle of elevation
from his eyes to the top of the tree is 76 ° . How tall is the tree? Round to the nearest tenth of a foot.
8.) Students in a hang gliding class stand on the top of a cliff 70 meters high. They watch a hang glider
land on the beach below. The angle of depression from the top of the cliff to the hang glider on the
beach below is 72 ° . How far is the hang glider form the base of the cliff? Round to the nearest tenth of
a meter.
9.) A pedestrian sights the top of a building at an angle of elevation of 75 ° . She is standing 50 feet
from the base of the building. How high above her eye level is the top of the building to the nearest
foot?
10.) Linda is flying a kite. She lets out 45 yards of string and anchors it to the ground. She determines
that the angle of elevation from the ground (where the string is anchored) to the top of the kite is 58 ° .
How high off the ground is the kite? Round to the nearest tenth of a yard.
11.) A plane flying at an altitude of 10,000 feet spots a hot air balloon in the distance. The balloon is
9000 ft above ground. The angle of depression from the plane to the balloon is 30 ° . Find the
horizontal distance from the plane to the balloon.