8.1: The Pythagorean Theorem and Its Converse
Simplifying Radicals
1. Write radicand in prime factored form.
2. Circle and cancel pairs of numbers.
3. Bring circled numbers out and leave untouched number under
Examples: Simplify the following radicals.
1) √24
2) √48
3) √36
Notice: The legs are always “a” and “b” and the shortest lengths given to you.
Examples: Find the missing value. Leave answers in simplest radical form.
1.
2.
3.
Is it Right, Acute or Obtuse???
Pythagorean’s Thm. Will Tell Us
Remember that the shortest lengths are always “a” and “b”
In other words:
If
=
+
then the triangle is RIGHT
2
2
2
If c < a + b , then the triangle is ACUTE
If c2 > a2 + b2, then the triangle is OBTUSE
c2
a2
b2,
.
.
.
Examples: The lengths of the sides of a triangle are given. Classify each triangle as acute,
right, or obtuse.
4. 3, 8, 10
5. 12, 15, 19
6. 10, 24, 26
7. A square has side length 10 yd. What is the length of a diagonal of the square? Express in
simplest radical form.
8. A repairman leans the top of an 8-ft ladder against the top of a stone wall. The base
of the ladder is 5.5 ft from the wall. About how tall is the wall? Round to the nearest
tenth of a foot.
Pythagorean Triples: set of integer lengths that satisfy the Pythagorean triple
Decimals are not allowed and neither are radicals.
Examples: Are the following lengths a Pythagorean Triple?
9. 3, 4, 5
10. √24, 10, 26
11. 2, 5, 8
8.2: Special Right Triangles
We can use special right triangles and proportions to find missing sides.
MEMORIZE the two special triangles. (45-45 and 30-60)
Use these and a proportion to solve for missing sides.
Leave answers in radical form!
REMEMBER: No radical is allowed in the denominator. Rationalize to get rid of it!
In a 45-45-90 triangle, the length of the hypotenuse is
leg.
times as long as a
Examples: Use the given triangle and values to find the indicated value in the
following 45-45-90 right triangles.
DDD
F
45
D
45
E
1. DE = 16 2 , DF = _______.
2. DE = 10, EF = __________.
3. FE = 18, DE = _______.
4. DE = 5 2 , FE = ____________.
Examples: Use the given triangle and values to find the indicated value in the
following 30-60-90 right triangles.
5. Find the value of x .
6. Find the value of x.
12
30°
x
7. Find the value of x
14
30°
x
x
30°
9 3
8. Find the value of x
x
30°
20
Example: 9. An artist makes pendants in the shape of equilateral triangles. Suppose the sides are
18mm long. What is the height of the pendant?
8.3: Trigonometry
Trigonometric Ratios: ratios for that compare sides in a RIGHT triangle given a certain
trigonometric function
Trigonometric Functions
Trig Function
Abbreviation
Ratio
Sine
Cosine
Tangent
Trig ratios must always be related from the angle given.
The opposite leg is always opposite from the given angle.
The adjacent leg is always adjacent to the given angle.
The hypotenuse is the same no matter what. Always opposite the right angle.
When in doubt: Remember Chief SOH-CAH-TOA!!!
C
T
S
O
A
O
H
H
A
Examples: Write the sin X, cos X and tan X in fraction form.
1.
2.
3.
USING YOUR CALCULATOR AND TRIG FUNCTIONS
Different calculators work different ways. Type the first one in to try to get the value.
Examples: Use your calculator to find the following values to the nearest thousandth.
4. sin 37
5. Cos 54
6. Tan 26
≈.609
(Use you calc to get this.)
Using Trig Ratios to Find Missing Sides
1) Use a given triangle to relate the given sides to one of the acute angles.
2) Decide on the correct trig ratio to use.
3) Set up an equation and solve. YOU MUST USE A CALCULATOR!
Examples: Find the value of x to the nearest tenth.
7.
8.
9.
USING YOUR CALCULATOR TO DO INVERSE FUNCTIONS.
Different calculators work different ways. You will need to find the sin-1, cos-1 and tan-1.
Probably will need to use the [2nd] or [Shift] button to get these!
Examples: Use your calculator to find the following values to the nearest integer.
10. Sin  X = 0.7193
11. Cos X = 0.9205
12. Tan X = 0.5467
X ≈ 46
Using Trig Ratios to Solve for Missing Angles
Known as using inverses
You must still set up the trig equation relating sides to an angle
Your equation will have an “x” next to the trig function instead of an angle measure.
Examples: Find the value of x. Round to the nearest degree.
10.
11.
12.
8.4: Angles or Elevation and Depression
SOH-CAH-TOA Review: http://www.youtube.com/watch?v=8xwA4DHWLk0
Angle of Elevation:
Angle of Depression:
Examples: Identify the 4 angles as either angles of elevation or depression.
Trig functions can be used to solve these problems.
Example: You sight a rock climber on a cliff at a 32 degree angle of elevation. Your eye
level is 6 feet above the ground and you are 1000 ft from the base of the cliff. How high
is the climber? Use the diagram.
Example: A pilot sights a life raft at a 26 degree angle of depression. He is 3 km in the
air. What is his horizontal distance “d” from the raft?
Examples: Describe each angle as it relates to the situation in the diagram.
1. 1
5. 5
2. 2
6. 6
3. 3
7. 7
4. 4
8. 8
Examples: Find the value of x. Round to the nearest 10th.
9.
10.
11. A person is standing 40 ft from a flagpole and can see the top of the pole at a 35º angle of
elevation. The person’s eye level is 4 ft from the ground. What is the height of the flagpole to
the nearest foot?