Transcript Document

Lesson 7-R
Chapter 7 Review
Objectives
• Review Chapter 7 Material in preparation for
the test
Vocabulary
• None new
Geometric and Arithmetic Means
Arithmetic Mean (AM) or average of 2 numbers: (a + b) / 2
Geometric Mean (GM) of 2 numbers: √ab
Altitude Length = GM of divided hypotenuse
= √ab
a
altitude
b
Special Right Triangles
c
a
Pythagorean Theorem:
a2 + b2 = c2
Pythagorean Triples: Whole numbers
that solve the theorem (example: 3,4,5)
b
Side opposite 30° angle is ½ the hypotenuse
Side opposite 45° angle is ½ the hypotenuse times √2
Side opposite 60° angle is ½ the hypotenuse times √3
45°
½ x√2
x
y
60°
½y
45°
½ x√2
30°
½ y√3
Trigonometric Functions
opposite
• Sin (angle) = Opposite / Hypotenuse
• Cos (angle) = Adjacent / Hypotenuse
• Tan (angle) = Opposite / Adjacent
angle
adjacent
• SOH – CAH – TOA (or others) to help remember the
definitions
• To find an angle use the inverse of the Trig Function
– Trig Fnc-1 (some side / some other side) = angle
• Remember the shortcut for the bottom of a fraction
8
--- = 0.781
x

8
--------- = x
0.781
just switch x and the = #
Trig Problems Steps to Solution
•
•
•
•
Step 1:
Step 2:
Step 3:
Step 4:
Label sides (A, H, O) based on angle
Identify trig function to use
Set up equation
Solve for variable (1 of these methods)
– if variable is in top of fraction, multiply both sides by
the bottom to get “x = …”
x
sin 23° = ------45
x = 45  sin 23°
– if variable is in bottom of fraction, x trades places
with what’s on the other side of the = sign to get “x =
…”
21
21
cos 41° = ------x
x = --------cos 41°
– if variable is the angle, use inverse trig function
notation to get “x = …”
23
tan x° = ------37
x = tan
-1
23
----37
Angles of Elevation or Depression
• To Solve:
–
–
–
–
–
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Draw the triangle below
Label sides (A, H, O) from problem information
Identify trig function to use
Set up equation
Solve for variable (use 1 of the 3 methods)
slant distance;
ski slope or road
angle always goes here
vertical distance
or height
Θ
horizontal distance
or length of shadow
Summary & Homework
• Summary:
– Arithmetic mean is the average – (a+b)/2
– Geometric mean
• Square root of the product -- √ab
• Length of the altitude (GM of divided hypotenuse)
– Pythagorean Theorem – a² + b² = c²
– Pythagorean Triples – whole numbers
– Special Case Right Triangles
• Side opposite 30° is ½ hypotenuse
• Side opposite 45° is ½ hypotenuse  √2
• Side opposite 60° is ½ hypotenuse  √3
– Trigonometric functions (SOH – CAH – TOA)
– Angle of Elevation or Depression
• Homework:
– study for the test
Problems
1. Find the Arithmetic Mean and the Geometric Mean of 3, 15
15
2. Find the altitude in the triangle to the right
a
10
3. Find the missing side in the triangle to the right
25
x
15
More Problems
4. Does 6, 8, 9 make a Right Triangle?
5. Does 1, 4/3, 5/3 make a Rt Triangle?
A Pythagorean Triple?
A Pythagorean Triple?
6. Solve for the variables in the triangle to the right
y°
26
x
30°
z
7. Solve for the variables in the triangle to the right
y°
z
x
45°
15
8. If a 20 ft ladder leans up against a barn at a 62° angle to the ground,
how high up the barn does it reach?
20
62°
x
Problems
1. Find the Arithmetic Mean and the Geometric Mean of 3, 15
AM = (3+15)/2 = 18/2 = 9
GM = √ (3•15) = √45 = 6.71
15
2. Find the altitude in the triangle to the right
a
10
GM = √ (10•15) = √150 = 12.25
3. Find the missing side in the triangle to the right
25
Pythagorean Theorem: hyp² = leg² + other leg²
or c² = a² + b²
(25)² = (15)² + x²
625 = 225 + x²
400 = x²
20 = x
x
15
6² + 8² ≠ 9²
More Problems
not a Rt ▲
4. Does 6, 8, 9 make a Right Triangle? NO! A Pythagorean Triple? NO!
5. Does 1, 4/3, 5/3 make a Rt Triangle? Yes! A Pythagorean Triple? NO!
(1)² + (4/3)² = (5/3)²
not all whole numbers
6. Solve for the variables in the triangle to the right
By Trig:
sin 30° = x / 26
0.5 = x / 26
13 = x
y°
26
cos 30° = z / 26
0.866 = z / 26
22.52 = z
y = 90 – 30 = 60°
x
30°
z
7. Solve for the variables in the triangle to the right
By Trig:
tan 45° = x / 15
1 = x / 15
15 = x
cos 45° = 15 / z
0.707 = 15 / z
0.707z = 15
z = 21.22
y = 90 – 45 = 45°
y°
z
x
45°
15
sin 62° = x / 20
0.883 = x / 20
17.66 = x
20
62°
opp
8. If a 20 ft ladder leans up against a barn at a 62° angle to the ground,
how high up the barn does it reach?
hyp
x