Download Warm up - cloudfront.net

Ch 8: Right Triangles and Trigonometry
8-1 The Pythagorean Theorem and Its Converse
8-2 Special Right Triangles
8-3 The Tangent Ratio
8-1: The Pythagorean Theorem and Its Converse
Focused Learning Target: I will be able to
 Use the Pythagorean Theorem
 Use the Converse of the Pythagorean
Theorem.
Vocabulary:
 Pythagorean Triple
8-4 Sine and Cosine Ratios
8-5 Angles of Elevation and Depression
CA Standard(s):
Geo 15.0 Students use the Pythagorean theorem to
determine distance and find missing lengths of sides of
right triangles.
Although the sides of a right triangle will always satisfy the Pythagorean Theorem, Pythagorean triples are a
special set of positive whole numbers that satisfy the equation a 2  b 2  c 2 .
Examples of common Pythagorean Triples:
3, 4, 5
8, 15, 17
5, 12, 13
7, 24, 25
Not Pythagorean Triples:
7, 10, 12.5
1 3 13
, ,
2 4 4
-3, -4, 5
1, 2, 3
Finding the length of a side using the Pythagorean Theorem and determining if it forms a Pythagorean triple:
I’ll do one:
Find the missing side, and then determine if it forms a Pythagorean triple with the other two sides.
We’ll do one together:
Find the missing side(s), and then determine if all three sides form a Pythagorean triple.
1
You Try:
Find the missing side(s) and determine if it forms a Pythagorean triple with the other two sides.
Simplifying Radicals:
Using Simplest Radical Form:
When solving for a variable by taking the square root, the value is often
not a perfect square. In those cases it is often best to leave your
answer as a radical in simplest form. Here is one method:
1) Divide the number by 2, 3, 4, 5, 6, etc
2) When the result is a perfect square you can simplify by taking the
square of the perfect square. Leave the other factor inside the radical.
3) If you divide by a perfect square and the result is a prime number (or
a number that you know cannot be simplified), then simplify by leaving
the prime number inside the radical and taking the square root of the
perfect square.
The first 20 Perfect Squares chart
1
12 =
112 = 121
122 = 144
22 = 4
9
132 = 169
32 =
142 = 196
4 2 = 16
25
152 = 225
52 =
16 2 = 256
6 2 = 36
17 2 = 289
7 2 = 49
64
182 = 324
82 =
81
19 2 = 361
92 =
10 2 = 100
20 2 = 400
4) If it does not factor in this way, then it is already in simplest radical form.
I’ll do one:
Simplify:
75
We’ll do one:
Simplify:
726
You try one:
Simplify:
980
Examples: Using the Pythagorean Theorem and Simplifying Radicals
I’ll do one:
If a = 2, b = 4, find c. Leave your answer in simplest radical form.
2
We’ll do one together:
Find x. Leave your answer in simplest radical form.
You Try:
If a = 6, c = 12, find b. Leave your answer in simplest radical form.
Examples: Applying the Pythagorean Theorem in real-world situations
I’ll do one:
The Parks Department rents Paddle boats at docks near each entrance to the park. To the nearest meter, how
far is it to paddle from one dock to the other?
We’ll do one together:
A highway detour affects a company’s delivery route. The plan showing the old route and the detour is below.
How many extra miles will the trucks travel once the detour is established? Round your answer to the nearest
tenth of a mile.
3
You Try:
A bike messenger has just been asked to make an additional stop. Instead of biking straight from the law office
to the court, she is going to stop at City Hall in between. Using the picture below, determine how many
additional miles she will need to travel. Leave your answer in simplest radical form.
Using the converse to determine if it is a right triangle:
I’ll do one:
Is the triangle below a right triangle? Explain.
We’ll do one together:
Is the triangle below a right triangle? Explain.
You Try:
Is the triangle below a right triangle? Explain.
4
8-2 Special Right Triangles
Focused Learning Target: I will be able to
 Use the properties of
45  45  90 triangles.
 Use the properties of
30  60  90 triangles.
CA Standard(s):
Geo 15.0: Students use the Pythagorean theorem to determine
distance and find missing lengths of sides of right triangles.
Geo 20.0: Students know and are able to use angle and side
relationships in problems with special right triangles, such as 30,
60, 90 triangles and 45, 45, and 90 triangles.
There are two special triangles that are often encountered. In this section, we will explore the characteristics
that are unique to these triangles.
The 45  45  90 triangle is an isosceles right triangle. It has all of the properties of an isosceles triangle and
all of the properties of a right triangle. The theorem below is often used as a quicker way to find missing
lengths of these special right triangles compared to the Pythagorean Theorem.
Finding the hypotenuse
I’ll do one:
We’ll do one together:
You try:
Find the value of the variable:
Find the value of the variable:
Find the value of the variable:
Finding the length of a leg:
I’ll do one:
Find the length of a leg of a 45  45  90 Triangle with a hypotenuse of length 10.
5
We’ll do one together:
Find the value of the variables:
You try:
Find the value of the variable:
Another type of special triangle is the 30  60  90 triangle for which we use the theorem below:
I’ll do one:
Find the lengths of the missing sides.
We’ll do one together:
Find the missing variables:
6
You Try:
Find the missing variables:
Some Real-World Situations
I’ll do one:
The moose warning sign below is an equilateral triangle. It is one meter high. Find the lengths of each of the
sides.
We’ll do one:
After heavy winds damaged a farmhouse, workers placed a 6-m brace against its side at a 45 angle. Then, at
the same spot on the ground, they placed a second brace at 30 angle.
a) How long is the longer brace? Round your answer to the nearest tenth of a meter.
b) About how much higher on the house does the longer brace reach than the shorter brace?
7
You try:
Jefferson Park sits on one square city block 300 ft on each side. Sidewalks connect the opposite corners.
a) Draw the diagram
b) About how long is each diagonal sidewalk?
8-3 The Tangent Ratio
Focused Learning Target: I will be able to:
 Use the tangent ratios to
determine side lengths in triangles
CA Standard(s):
Geo 18.0: students know the definitions of the basic
trigonometric functions defined by the angles of a right triangle.
Geo 19.0: Students use trigonometric functions to solve for an
unknown length of a side of a right triangle, given an angle and a
length of a side.
Vocabulary:
Tangent: the tangent of acute A in a right triangle is the ratio of the length of the leg opposite of A to the
length of the leg adjacent to A .
Example 1: Writing tangent ratios:
I’ll do one:
We’ll do one:
Write the tangent ratios for
T and U .
You try:
Write the tangent ratios for
E and F .
8
Example 2: Real-World-Connection
I’ll do one:
We’ll do one:
Find the distance from the boathouse on shore to the cabin on the island.
You try:
Find x to the nearest whole number.
 opp 
 , to find the missing angle
Example 3: Using the Inverse of Tangent, tan 1 
adj


I’ll do one:
We’ll do one:
You Try:
Find the measure of angle X to the Find the value of x to the nearest
Find the value of x to the nearest
nearest degree.
degree.
degree.
9
8-4 Sine and Cosine Ratios
Focused Learning Target:
 Use sine and cosine to determine side lengths in triangles
CA Standard(s):
Geo 18.0: Students know the definitions of the basic trigonometric functions defined by the angles of a right
triangle. They are able to use elementary relationships between them.
Geo 19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle,
given an angle and a length of a side.
Vocabulary:
Example 1: Writing the Sine and Cosine Ratios
I’ll do one:
We’ll do one:
Write the cosine and sine ratios for X and Y .
You Try:
Write the ratios for sin P and cos Q.
10
How can we remember the ratios of Sine, Cosine, and Tangent? SOH, CAH, TOA!!!!
O
Opp
 S
 SOH
Sin (angle ) 
H
Hyp
A
Adj
 C
 CAH
Cos(angle ) 
H
Hyp
O
Opp
 T
 TOA
Tan(angle ) 
A
Adj
Example 2: Real-World Connection
I’ll do one:
We’ll do one:
Find the value of x. Round answer to the nearest tenth.
You Try:
Find the value of x. Round answer to the nearest tenth.
11
 adj 
1  opp 
 and Cosine, cos 1 

Example 3: Using the Inverse of Sine, sin 
 hyp 
 hyp 
I’ll do one:
Find the value of x. Round your answer to the nearest degree.
We’ll do one:
Find the value of x. Round your answer to the nearest degree.
You try:
Find the value of x. Round your answer to the nearest degree.
12
8-5 Angles of Elevation and Depression
Focused Learning Target:
 Use angles of elevation and
depression to solve problems.
Vocabulary:
 Angle of elevation
 Angle of depression
CA Standard(s):
Geo 18.0: Students know the definitions of the basic
trigonometric functions defined by the angles of a right triangle.
Geo 19.0: Students use trigonometric functions to solve for an
unknown length of a side of a right triangle, given an angle and
a length of a side.
The angle of elevation is the angle formed by taking the
horizontal line (as if you’re looking straight ahead) and then
looking upwards.
The angle of depression is the angle formed by taking the
horizontal line and then looking downwards.
In the picture to the right, notice that the angle of
depression from the person in the hot air balloon is
congruent to the angle of elevation from the person on the
ground. Why are those angles congruent?
Example 1: Identifying angles of elevation and depression
I’ll do one:
If you were at the peak of the mountain at
1 , what type of angle would that be; an
angle of elevation or depression?
We’ll try some together:
What about at 2 ? 3 ?
You try:
What about at 4 ?
13
Example 2: Finding lengths in real-world scenarios using trigonometry
I’ll do one:
You are hiking and you spot a rock climber on a nearby cliff at a 32 o angle of elevation. The horizontal ground
distance to the base of the cliff is 1000 ft. Find the height of the rock climber. (hint: always try to draw the
situation if none is given)
We’ll do one together:
To approach runway 17 of the Ponca City Municipal Airport in Oklahoma, the pilot must begin a 3o descent
starting from an altitude of 2714 ft. The airport’s altitude is 1007 ft above sea level. How many feet from the
runway is the airplane at the start of this approach? Then convert your answer into miles.
You try:
A coast guard helicopter pilot sights a life raft at a 26 o angle of depression. The helicopter’s altitude is 3 km.
What is the helicopter’s horizontal distance from the raft?
14