Download 8-4 Trigonometry 2-19

2/23/2016
Five-Minute Check (over Lesson 8–3)
CCSS
Then/Now
New Vocabulary
Key Concept: Trigonometric Ratios
Example 1: Find Sine, Cosine, and Tangent Ratios
Example 2: Use Special Right Triangles to Find Trigonometric
Ratios
Example 3: Real-World Example: Estimate Measures Using
Trigonometry
Key Concept: Inverse Trigonometric Ratios
Example 4: Find Angle Measures Using Inverse Trigonometric
Ratios
Example 5: Solve a Right Triangle
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Over Lesson 8–3
(1-2) Find x and y.
The length of the diagonal of a square is
centimeters. Find the perimeter of the
square.
The side of an equilateral triangle measures
21 inches. Find the length of an altitude of the
triangle.
Over Lesson 8–3
Find x and y.
A.
B.
C.
D.
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Over Lesson 8–3
Find x and y.
A. x = 5, y = 5
B. x = 5, y = 45
C.
D.
Over Lesson 8–3
The length of the diagonal of a square is
centimeters. Find the perimeter of the
square.
A. 15 cm
B. 30 cm
C. 45 cm
D. 60 cm
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Over Lesson 8–3
The side of an equilateral triangle measures
21 inches. Find the length of an altitude of the
triangle.
A.
in.
B. 12 in.
C. 14 in.
D.
in.
Over Lesson 8–3
∆MNP is a 45°-45°-90°triangle with right angle P.
Find the coordinates of M in Quadrant II for
P(2, 3) and N(2, 8).
A. (–1, 3)
B. (–3, 3)
C. (5, 3)
D. (6, 2)
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Over Lesson 8–3
The hypotenuse of a 30°-60°-90°triangle measures
inches. What is the length of the side
opposite the 30°angle?
A. 10 in.
B. 20 in.
C.
D.
Content Standards
G.SRT.6 Understand that by similarity, side ratios in
right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios
for acute angles.
G.SRT.7 Explain and use the relationship between
the sine and cosine of complementary angles.
Mathematical Practices
1 Make sense of problems and persevere in solving
them.
5 Use appropriate tools strategically.
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You used the Pythagorean Theorem to find
missing lengths in right triangles.
• Find trigonometric ratios using right
triangles.
• Use trigonometric ratios to find angle
measures in right triangles.
• trigonometry
• inverse tangent
• trigonometric ratio
• sine
• cosine
• tangent
• inverse sine
• inverse cosine
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Find Sine, Cosine, and Tangent Ratios
A. Express sin L as a
fraction and as a decimal to
the nearest hundredth.
sin 12
37
Answer:
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Find Sine, Cosine, and Tangent Ratios
B. Express cos L as a
fraction and as a decimal
to the nearest hundredth.
12
37
35
cos 37
sin Answer:
Find Sine, Cosine, and Tangent Ratios
C. Express tan L as a
fraction and as a decimal
to the nearest hundredth.
12
37
35
cos 37
12
tan 35
sin Answer:
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Find Sine, Cosine, and Tangent Ratios
D. Express sin N as a
fraction and as a decimal
to the nearest hundredth.
12
37
35
cos sin 37
12
tan 35
sin Answer:
Find Sine, Cosine, and Tangent Ratios
E. Express cos N as a
fraction and as a decimal to
the nearest hundredth.
12
cos 37
35
cos sin 37
12
tan 35
sin Answer:
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Find Sine, Cosine, and Tangent Ratios
F. Express tan N as a
fraction and as a decimal to
the nearest hundredth.
12
cos 37
35
sin cos 37
35
12
tan tan 12
35
sin Answer:
A. Find sin A.
A.
B.
C.
D.
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B. Find cos A.
A.
B.
C.
D.
C. Find tan A.
A.
B.
C.
D.
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D. Find sin B.
A.
B.
C.
D.
E. Find cos B.
A.
B.
C.
D.
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F. Find tan B.
A.
B.
C.
D.
Use Special Right Triangles to Find
Trigonometric Ratios
Use a special right triangle to express the cosine of
60°as a fraction and as a decimal to the nearest
hundredth.
Draw and label the side lengths of a
30°-60°-90° right triangle.
Answer: Cos 60°= 13
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Use a special right triangle to express the tangent
of 60° as a fraction and as a decimal to the nearest
hundredth.
A.
B.
C.
D.
Estimate Measures Using Trigonometry
EXERCISING A fitness trainer sets the incline on a
treadmill to 7°. The walking surface is 5 feet long.
Approximately how many inches did the trainer
raise the end of the treadmill from the floor?
Pay attention to units!
12
60
1
sin Answer: The treadmill is
!
about 7.3 inches
sin 7° high.
60
5 •
60 sin 7° ≈ 7.31216
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CONSTRUCTION The bottom of a handicap ramp
is 15 feet from the entrance of a building. If the
angle of the ramp is about 4.8°, about how high
does the ramp rise off the ground to the nearest
inch?
A. 1 in.
B. 11 in.
C. 16 in.
D. 15 in.
In more advanced math courses the word “inverse” is often referred
to as “arc”. For example “inverse sine” is essentially the same as
“arcsine”.
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Find Angle Measures Using Inverse
Trigonometric Ratios
Use a calculator to find the measure of ∠P to the
nearest tenth.
%&'%
cos !
cos ( 13
19
( *
13
≈ 46.82644889
19
KEYSTROKES: 2nd [COS] ( 13 ÷ 19 )
ENTER 46.82644889
Answer: So, m∠P is approximately 46.8°.
Use a calculator to find the measure of ∠D to the
nearest tenth.
A. 44.1°
B. 48.3°
C. 55.4°
D. 57.2°
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Solve a Right Triangle
Solve the right triangle. Round side measures to
the nearest hundredth and angle measures to the
nearest degree.
%&'%
4
tan - 7
4
- %* ≈ 29.74°
7
tan
/∠- ≈ 30°
tan . 7
4
. %*
7
≈ 60.26°
4
/∠. ≈ 60°
Solve a Right Triangle
Solve the right triangle. Round side measures to
the nearest hundredth and angle measures to the
nearest degree.
-1 + .1 -. -. -1 + .1 -. 7 + 4
-. 49 + 16
-. 65 ≈ 8.06
Answer: m∠A ≈ 30, m∠B ≈ 60, AB ≈ 8.06
/∠- ≈ 30°
/∠. ≈ 60°
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Solve the right triangle. Round side measures to
the nearest tenth and angle measures to the
nearest degree.
A. m∠
∠A = 36°, m∠
∠B = 54°,
AB = 13.6
B. m∠
∠A = 54°, m∠
∠B = 36°,
AB = 13.6
C. m∠
∠A = 36°, m∠
∠B = 54°,
AB = 16.3
D. m∠
∠A = 54°, m∠
∠B = 36°,
AB = 16.3
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