Transcript Unit 3: Right Triangle Trigonometry

UNIT 3: RIGHT TRIANGLE
TRIGONOMETRY
Vocabulary Builder
Similar Triangles
leg
Pythagorean
Theorem
Square Root
Simplest radical
form
45-45-90 Triangles
Hypotenuse
Variable
30-60-90 Triangles
UNIT 3: RIGHT TRIANGLE
TRIGONOMETRY
3.1 Trigonometric Ratios
Square root
Perfect Square
Radical expression
radicand
Perfect Square
5
8
14
30
18
𝟔 𝟐
2 𝟔
3 𝟏𝟏
10 𝟑
3 𝟏𝟎
𝟓 𝟑
23
right triangle
equals
hypotenuse
Side lengths
42
+
82
=
𝑥2
16 + 64 = 𝑥
80 =
2
𝑥2
62
+
32
=
𝑥2
𝑥 2 + 3.92 = 112
2
𝑥 2 + 15.21 = 121
36 + 9 = 𝑥
45 =
𝑥2
4 5=𝑥
3 5=𝑥
8.9 = 𝑥
6.7 = 𝑥
𝑥 2 = 95.79
𝑥 = 9.75
32
+ 102
=
𝑥2
9 + 100 =
𝑥2
109 =
𝑥2
10.44 = 𝑥
72
+
22
=
𝑥2
𝑥 2 + 3.62 = 7.42
2
𝑥 2 + 12.96 = 54.76
49 + 4 = 𝑥
𝑥2
𝑥 2 = 41.8
7.3 = 𝑥
𝑥 = 6.47
53 =
hypotenuse
adjacent
opposite
𝑌𝑍
𝑋𝑌
𝑋𝑍
𝐻𝑈
𝑈𝐼
𝐻𝐼
trigonometric ratio
SOH CAH TOA
right
length
acute
SOH CAH TOA
4
𝑂𝑃𝑃 20
=
=
3
𝐴𝐷𝐽
15
𝑂𝑃𝑃 16
8
=
=
𝐴𝐷𝐽
30 15
7
𝑂𝑃𝑃 14
=
=
𝐴𝐷𝐽
48 24
𝑂𝑃𝑃 15
3
=
=
𝐻𝑌𝑃 25
5
𝑂𝑃𝑃 30 15
=
=
𝐻𝑌𝑃 34 17
𝐴𝐷𝐽
48 24
=
=
𝐻𝑌𝑃 50 25
=
11
29
25
=
11
=
11
29
18
=
7
=
25
29
=1
UNIT 3: RIGHT TRIANGLE
TRIGONOMETRY
3.2 Finding Sides and Angles
of Right Triangles
function
angle
output
input
x
x
x
sin 𝑅 =
𝑂𝑃𝑃
𝐻𝑌𝑃
𝑥
sin 19° =
11
11 ∙ sin 19° = 𝑥
3.58 = 𝑥
cos 𝐶 =
𝐴𝐷𝐽
𝐻𝑌𝑃
19.2
cos 46° =
𝑥
19.2
𝑥=
cos 46°
𝑥 = 27.64
sin 𝐻 =
𝑂𝑃𝑃
𝐻𝑌𝑃
11
sin 34° =
𝑥
11
𝑥=
sin 34°
𝑥 = 19.67
x
tan 26° =
𝑂𝑃𝑃
𝐴𝐷𝐽
14.63 = 𝑥
𝐴𝐷𝐽
𝐻𝑌𝑃
𝑂𝑃𝑃
tan 20° =
𝐴𝐷𝐽
𝑥 = 15.49
𝑥 = 6.92
tan 50° =
sin-1 cos-1 tan-1
5
5
tan−1 𝑥 = 1
tan−1 𝑥 =
tan−1 1 = 𝑥
45° = 𝑥
tan−1 𝑥 =
tan−1
5
12
5
=𝑥
12
22.6° = 𝑥
cos −1 𝑥 =
cos −1
40
42
20
=𝑥
21
17.8° = 𝑥
cos −1
15
=𝑥
28
57.6° = 𝑥
sin−1
19
=𝑥
23
55.7° = 𝑥
cos −1
35
=𝑥
43
35.5° = 𝑥
UNIT 3: RIGHT TRIANGLE
TRIGONOMETRY
3.3 Sine and Cosine of
Complementary Angles
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
180˚
90˚
complementary
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
Opposite angle
cosine
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
What am I going to do today?
= 𝐜𝐨𝐬 𝟐𝟔°
= 𝐜𝐨𝐬 𝟔𝟔°
= 𝐬𝐢𝐧 𝟔°
= 𝐬𝐢𝐧 𝟓𝟐°
= 𝐬𝐢𝐧 𝟏𝟖°
= 𝐜𝐨𝐬 𝟒𝟓°
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
= 𝐜𝐨𝐬 𝟔𝟖°
= 𝐬𝐢𝐧 𝟐𝟓°
= 𝐬𝐢𝐧 𝟒𝟔°
= 𝐜𝐨𝐬 𝟓𝟖°
= 𝐬𝐢𝐧 𝟔𝟓°
= 𝐜𝐨𝐬 𝟕𝟓°
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
𝟒
𝟓
𝟑
𝟓
𝟑
𝟓
𝟒
𝟓
Sin A and Cos B are the same which means The complement of sine and cosine are equal.
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
Then mQ = 60˚
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
𝟎. 𝟓
𝟎. 𝟖𝟔𝟔
𝟎. 𝟖𝟔𝟔
𝟎. 𝟓
The complement of sine and cosine are equal.
What am I learning today?
102 + 242 = 𝑥 2
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
Then mT = 45˚
100 + 576 = 𝑥 2
676 = 𝑥 2
26 = 𝑥
26
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
𝟐𝟒 𝟏𝟐
=
𝟐𝟔 𝟏𝟑
𝟏𝟎 𝟓
=
𝟐𝟔 𝟏𝟑
𝟎. 𝟕𝟎𝟕
𝟎. 𝟕𝟎𝟕
𝟏𝟎 𝟓
=
𝟐𝟔 𝟏𝟑
𝟐𝟒 𝟏𝟐
=
𝟐𝟔 𝟏𝟑
𝟎. 𝟕𝟎𝟕
𝟎. 𝟕𝟎𝟕
The complement of sine and cosine are equal.
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
𝐜𝐨𝐬 𝟓𝟐° ≈ 𝟎. 𝟔𝟏𝟔
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
𝐬𝐢𝐧 𝟑𝟎° ≈ 𝟎. 𝟓
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
What am I going to do today?
•Find the complement of
sine and cosine
𝟓
𝟏𝟑
𝟓
𝟏𝟑
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
A
13
52 + 122 = 𝑥 2
5
25 + 144 = 𝑥 2
169 = 𝑥 2
13 = 𝑥
C
12
B
A
37
𝟏𝟐
𝟑𝟕
𝟏𝟐
𝟑𝟕
13
122 + 352 = 𝑥 2
144 + 1225 = 𝑥
35
2
1369 = 𝑥 2
37 = 𝑥
H
37
12
T
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
What am I going to do today?
•Find the complement of
sine and cosine
10
𝟒
𝟓
𝟒
𝟓
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
X
82 + 62 = 𝑥 2
64 + 36 = 𝑥 2
100 = 𝑥 2
10 = 𝑥
10
𝟒
𝟓
𝟒
𝟓
10
8
Y
Z
6
X
32 + 𝑥 2 = 52
9 + 𝑥 2 = 25
5
3
𝑥 2 = 16
𝑥=4
I
4
M
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
What am I learning today?
•G.SRT.7 Explain and use
the relationship between
the sine and cosine of
complementary angles
What am I going to do today?
•Find the complement of
sine and cosine
How will I show you that I’ve
learned it?
•5 Question Daily
Assessment
•Daily Ticket out the Door
UNIT 3: RIGHT TRIANGLE
TRIGONOMETRY
3.4 Solve Right Triangles
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
solve
Side lengths
angle
The length of two sides
The length of 1 side and 1 acute angle
The length of 1 side and 1 trigonometric ratio
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
9.4
9.4
19.2˚
70.8˚
3.12 + 8.92 = 𝑥 2
9.61 + 79.21 = 𝑥 2
88.82 = 𝑥 2
9.4 = 𝑥
tan−1
3.1
= ∠𝑃
8.9
19.2° = ∠𝑃
R = 90 – 19.2 = 70.8˚
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
3.9
6.3
3.9
6.3˚
58˚
𝐵𝐶
sin 32° =
7.4
3.58 = 𝑥
𝑥 2 + 3.92 = 7.42
𝑥 2 + 15.21 = 54.76
𝑥 2 = 39.55
𝑥 = 6.3
C = 90 – 32 = 58˚
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
Angle of Elevation
Angle of Depression
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
618.75 ft2
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the Pythagorean
Theorem to solve triangles
in applied problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios to
Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
991.86 ft2
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the
Pythagorean Theorem to
solve triangles in applied
problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios
to Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door
What am I learning today?
•G.SRT.8 Use trigonometric
ratios and the
Pythagorean Theorem to
solve triangles in applied
problems
What am I going to do today?
•Use the Pythagorean
Theorem to Solve Right
Triangles
•Use Trigonometric Ratios
to Solve Right Triangles
How will I show you that I’ve
learned it?
• 5 Question Daily Assessment
•Daily Ticket out the Door