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# Squares with one diagonal create 45–45–90 triangles

## Given the length of the legs, find the hypotenuse.

Given the hypotenuse find the length of each leg.
Make a generalization about relationship between leg and hypotenuse
Equilateral Triangles with an altitude drawn create 30–60–90 triangles

Given length of hypotenuse, find length of short leg and long leg.
Make a generalization about relationship between hypotenuse and
short leg and long leg.
Short
Hypotenuse
Vertical
30 Long Leg
Horizontal
Leg

1
Short = ( Hypotenuse )
2
Long = 3 ( Short )
3
Long
= ( Hypotenuse )
2
Hypotenuse Vertical
Leg
60
Short
Horizontal
Leg
45
Equal
Hypotenuse
Length
Leg
45
Equal Length Leg

Hypotenuse = 2 ( Leg )
Hypotenuse 2
Leg = i
2 2
2
Leg = ( Hypotenuse )
2
Squares with one diagonal create 45–45–90 triangles

4 2 5 2
4 5
2
4 5 2
2

1 2
3 2 1
3 2
3 1 2
2
Given the length of the legs, find the hypotenuse.
Given the hypotenuse find the length of each leg.
Make a generalization about relationship between leg and hypotenuse
Equilateral Triangles with an altitude drawn create 30–60–90 triangles

8 3 3 3
4 3
2
3
4
2
1 1
1 1 3 1 3 2
2 2
1
2

Given length of hypotenuse, find length of short leg and long leg.
Make a generalization about relationship between hypotenuse and
short leg and long leg.
Unit Circle: Fold your paper within the interior of the circle to find all special angles.
Color code the angles with a large dot.
Ex. 0-90-180-270 are black, 30s – green, 45s – blue, 60s – red
Using a ruler connect origin to each dot using the appropriate color from your coding
Label angles in degree measure in the appropriate color. We’ll label radian measures later.
Label ( x, y ) coordinates in the appropriate color.
Save this unit circle for reference in your Trigonometry class. This will be your most useful paper unit circle ever!!
Unit Circle: Fold your paper within the interior of the circle to find all special angles.

## Each special point on the circle has the information

( x, y ) = ( cos θ ,sin θ ) ; tan θ

Save this unit circle for reference in your Trigonometry class. This will be your most useful paper unit circle ever!!

π
90
2π 2 π
120 60
3 ( 0,1) ; und . 3
3π ⎛ 1 3⎞ π
135 3 ⎛1 3⎞ 3 45
4 ⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ; 4
⎝ 2 2 ⎠ 1
⎝ 2 2 ⎠ 1
⎛ 2 2⎞ ⎛ 2 2⎞
⎜⎜ − , ⎟⎟ ; − 1 ⎜⎜ , ⎟⎟ ; 1
⎝ 2 2 ⎠ ⎝ 2 2 ⎠

5π π
150 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

180 π 0 0
( −1, 0 ) ; 0 (1, 0 ) ; 0

7π π
210 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , − ⎟⎟ ; ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

5π 7π
225 315
4 4π 5π 4
240 300
⎛ 2 2⎞ 3 3 ⎛ 2 2⎞
⎜⎜ − ,− ⎟; 1 ⎛ 1 3⎞ ⎛1 3⎞ ⎜⎜ ,− ⎟ ; −1
⎝ 2 2 ⎟⎠ 3 3π 3
⎝ 2 2 ⎟⎠
⎜⎜ − , − ⎟⎟ ; 270 ⎜⎜ , − ⎟⎟ ; −
⎝ 2 2 ⎠ 1 2 ⎝2 2 ⎠ 1
( 0, −1) ; und .
( x, y ) = ( cos θ ,sin θ ) ; tan θ
π
90
2π 2 π
120 60
3 ( 0,1) ; und . 3
3π ⎛ 1 3⎞ π
135 3 ⎛1 3⎞ 3 45
4 ⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ; 4
⎝ 2 2 ⎠ 1
⎝2 2 ⎠ 1
⎛ 2 2⎞ ⎛ 2 2⎞
⎜⎜ − , ⎟⎟ ; − 1 ⎜⎜ , ⎟⎟ ; 1
⎝ 2 2 ⎠ ⎝ 2 2 ⎠

5π π
150 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

180 π 0 0
( −1, 0 ) ; 0 (1, 0 ) ; 0

7π π
210 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , − ⎟⎟ ; ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

5π 7π
225 315
4 4π 5π 4
240 300
⎛ 2 2⎞ 3 3 ⎛ 2 2⎞
⎜⎜ − ,− ⎟; 1 ⎛ 1 3⎞ ⎛1 3⎞ ⎜⎜ ,− ⎟⎟ ; − 1
⎝ 2 2 ⎟⎠ 3 3π 3 2 2
⎜⎜ − , − ⎟; ⎜⎜ , − ⎟⎟ ; − ⎝ ⎠
2 ⎟⎠
270
⎝ 2 1 2 ⎝2 2 ⎠ 1
( 0, −1) ; und .
The text passage contains a series of mathematical problems and generalizations about the relationships between sides in 45-45-90 and 30-60-90 triangles, as well as instructions for creating a unit circle. It also includes specific examples and calculations for these relationships.
multiple_questions

explanation:
- the passage includes multiple questions and prompts for calculations, such as "given the length of the legs, find the hypotenuse," "given the hypotenuse find the length of each leg," and "given length of hypotenuse, find length of short leg and long leg."
- no answers to these specific questions are provided in the text, only the general formulas and relationships.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
You are on page 1of 9

# Squares with one diagonal create 45–45–90 triangles

## Given the length of the legs, find the hypotenuse.

Given the hypotenuse find the length of each leg.
Make a generalization about relationship between leg and hypotenuse
Equilateral Triangles with an altitude drawn create 30–60–90 triangles

Given length of hypotenuse, find length of short leg and long leg.
Make a generalization about relationship between hypotenuse and
short leg and long leg.
Short
Hypotenuse
Vertical
30 Long Leg
Horizontal
Leg

1
Short = ( Hypotenuse )
2
Long = 3 ( Short )
3
Long
= ( Hypotenuse )
2
Hypotenuse Vertical
Leg
60
Short
Horizontal
Leg
45
Equal
Hypotenuse
Length
Leg
45
Equal Length Leg

Hypotenuse = 2 ( Leg )
Hypotenuse 2
Leg = i
2 2
2
Leg = ( Hypotenuse )
2
Squares with one diagonal create 45–45–90 triangles

4 2 5 2
4 5
2
4 5 2
2

1 2
3 2 1
3 2
3 1 2
2
Given the length of the legs, find the hypotenuse.
Given the hypotenuse find the length of each leg.
Make a generalization about relationship between leg and hypotenuse
Equilateral Triangles with an altitude drawn create 30–60–90 triangles

8 3 3 3
4 3
2
3
4
2
1 1
1 1 3 1 3 2
2 2
1
2

Given length of hypotenuse, find length of short leg and long leg.
Make a generalization about relationship between hypotenuse and
short leg and long leg.
Unit Circle: Fold your paper within the interior of the circle to find all special angles.
Color code the angles with a large dot.
Ex. 0-90-180-270 are black, 30s – green, 45s – blue, 60s – red
Using a ruler connect origin to each dot using the appropriate color from your coding
Label angles in degree measure in the appropriate color. We’ll label radian measures later.
Label ( x, y ) coordinates in the appropriate color.
Save this unit circle for reference in your Trigonometry class. This will be your most useful paper unit circle ever!!
Unit Circle: Fold your paper within the interior of the circle to find all special angles.

## Each special point on the circle has the information

( x, y ) = ( cos θ ,sin θ ) ; tan θ

Save this unit circle for reference in your Trigonometry class. This will be your most useful paper unit circle ever!!

π
90
2π 2 π
120 60
3 ( 0,1) ; und . 3
3π ⎛ 1 3⎞ π
135 3 ⎛1 3⎞ 3 45
4 ⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ; 4
⎝ 2 2 ⎠ 1
⎝ 2 2 ⎠ 1
⎛ 2 2⎞ ⎛ 2 2⎞
⎜⎜ − , ⎟⎟ ; − 1 ⎜⎜ , ⎟⎟ ; 1
⎝ 2 2 ⎠ ⎝ 2 2 ⎠

5π π
150 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

180 π 0 0
( −1, 0 ) ; 0 (1, 0 ) ; 0

7π π
210 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , − ⎟⎟ ; ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

5π 7π
225 315
4 4π 5π 4
240 300
⎛ 2 2⎞ 3 3 ⎛ 2 2⎞
⎜⎜ − ,− ⎟; 1 ⎛ 1 3⎞ ⎛1 3⎞ ⎜⎜ ,− ⎟ ; −1
⎝ 2 2 ⎟⎠ 3 3π 3
⎝ 2 2 ⎟⎠
⎜⎜ − , − ⎟⎟ ; 270 ⎜⎜ , − ⎟⎟ ; −
⎝ 2 2 ⎠ 1 2 ⎝2 2 ⎠ 1
( 0, −1) ; und .
( x, y ) = ( cos θ ,sin θ ) ; tan θ
π
90
2π 2 π
120 60
3 ( 0,1) ; und . 3
3π ⎛ 1 3⎞ π
135 3 ⎛1 3⎞ 3 45
4 ⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ; 4
⎝ 2 2 ⎠ 1
⎝2 2 ⎠ 1
⎛ 2 2⎞ ⎛ 2 2⎞
⎜⎜ − , ⎟⎟ ; − 1 ⎜⎜ , ⎟⎟ ; 1
⎝ 2 2 ⎠ ⎝ 2 2 ⎠

5π π
150 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , ⎟⎟ ; − ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

180 π 0 0
( −1, 0 ) ; 0 (1, 0 ) ; 0

7π π
210 30
6 6
⎛ 3 1⎞ 1 ⎛ 3 1⎞ 1
⎜⎜ − , − ⎟⎟ ; ⎜⎜ , ⎟⎟ ;
⎝ 2 2⎠ 3 ⎝ 2 2⎠ 3

5π 7π
225 315
4 4π 5π 4
240 300
⎛ 2 2⎞ 3 3 ⎛ 2 2⎞
⎜⎜ − ,− ⎟; 1 ⎛ 1 3⎞ ⎛1 3⎞ ⎜⎜ ,− ⎟⎟ ; − 1
⎝ 2 2 ⎟⎠ 3 3π 3 2 2
⎜⎜ − , − ⎟; ⎜⎜ , − ⎟⎟ ; − ⎝ ⎠
2 ⎟⎠
270
⎝ 2 1 2 ⎝2 2 ⎠ 1
( 0, −1) ; und .
user
The text passage contains a series of mathematical problems and generalizations about the relationships between sides in 45-45-90 and 30-60-90 triangles, as well as instructions for creating a unit circle. It also includes specific examples and calculations for these relationships.

TYPE: MULTIPLE_QUESTIONS

Explanation:
- The passage includes multiple questions and prompts for calculations, such as "Given the length of the legs, find the hypotenuse," "Given the hypotenuse find the length of each leg," and "Given length of hypotenuse, find length of short leg and long leg."
- No answers to these specific questions are provided in the text, only the general formulas and relationships.