Download Geometry Chapter 5 Lesson 8 – Applying Special Right Triangles

Geometry Chapter 5 Lesson 8 – Applying Special Right Triangles
Learning Targets
Success Criteria
•
•
LT5-8: Solve problems involving special right
triangles.
Find side lengths in 45º-45º-90º triangles.
Find side lengths in 30º-60º-90º triangles.
Ex#1: Find missing sides in 45º-45º-90º Triangles.
Find the value of x. Give your answer in simplest radical form
a.
b.
d.
e.
c.
f.
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Ex#2: Find missing sides in 30º- 60º - 90º Triangles.
Find the values of x and y. Give your answers in simplest radical form.
a.
b.
c.
d.
e.
f.
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Chapter 8 Trigonometric Ratios
Lesson 8.1 Similarity in Right Triangles
Learning Targets
Success Criteria
•
LT8-1: Apply similarity relationships, including
the geometric mean, in right triangles to
solve problems.
Find the geometric mean of two numbers.
Geometric Mean:
1. Write a similarity statement comparing the three
triangles.
2. Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
A. 4 and 25
B. 5 and 30
C. 8 and 9
3. Find x, y, and z.
4. To estimate the height of a Douglas fir, Jan positions
herself so that her line of sight to the top and bottom of the
tree form a 90°angle. Her eyes are about 1.6m above the
ground, and she is standing
7.8m from the tree. What
is the height of the tree to
the nearest meter?
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Lesson 8.2 Trigonometric Ratios
Learning Targets
Success Criteria
LT8-2: Use trigonometric ratios of right triangles
to find unknown lengths.
•
•
Find trigonometric ratios.
Approximate trigonometric ratios using a
calculator.
•
Use trigonometric ratios to find lengths.
•
Find exact values for sin, cos, and tan of
45º, 30º, and 60º angles.
Find exact values for trig ratios of special
triangles.
•
•
trigonometric ratio:
By the AA Similarity Theorem, a right triangle with a given acute angle is similar to every other right
triangle with that same acute angle measure.
Ex#1: Find Trigonometric Ratios.
Write each trigonometric ratio as a fraction and a decimal rounded to the nearest hundredth.
A. sin J
D. cos A
B. cos J
E. tan B
C. tan K
F. sin B
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Ex#2: Approximate trigonometric ratios using a calculator.
Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
A. sin 52°
B. cos 19°
C. tan 65°
Ex#3: Use trigonometric ratios to find lengths.
Find each length. Round to the nearest hundredth.
A. BC
B. QR
C. FD
D. The Pilatusbahn in Switzerland is the world's steepest cog railway. Its steepest section makes an angle of about 25.6°
with the horizontal and rises about 0.9 km. To the nearest hundredth of a kilometer, how long is this section of the railway
track?
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Ex#4: Find exact values for sin, cos, and tan of 30º, 45º, and 60º angles. Answers must be exact
values in simplified radical form.
30º
45º
60º
sin
cos
tan
Ex#5: Find Exact Values of Trig Ratios of Special Right Triangles.
A. Use a special right triangle to write sin 45° as a fraction. B. Use a special right triangle to write sin 60° as a fraction.
Lesson 8.3 – Solving Right Triangles
Learning Targets
Success Criteria
LT8-3: Use inverse trigonometric ratios of right
triangles to find unkown angles.
•
•
•
•
Solving a triangle –
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Identify angles from trigonometric ratios.
Calculate angle measures from
trigonometric ratios.
Solve triangles.
Use inverse trigonometric ratios to find
angles.
Ex:#1 Identify angles from trigonometric ratios.
Use the trigonometric ratio cosA =
24
25
to determine which angle of the triangle is ∠A.
Ex#2: Calculate angle measures from trigonometric ratios.
Use your calculator to find each angle measure to the nearest degree.
A. cos-1 (0.87)
B. sin-1 (0.85)
C. tan-1 (0.71)
Ex#3: Solve each triangle.
A. Round lengths to the nearest hundredth and angle measures to the nearest degree.
B. The coordinates of the vertices of ΔPQR are P(-3, 3), Q(2, 3), and R(-3, -4). Find the side lengths to the nearest
hundredth and the angle measures to the nearest degree.
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Ex#4: Use inverse trigonometric ratios to find angles.
A highway sign warns that a section of road ahead has a 7% grade. To the nearest degree, what angle does the road make
with a horizontal line?
Lesson 8-4 Angles of Elevation and Depression
Learning Targets
Success Criteria
•
LT8-4: Solve problems involving angles of
depression and angles of elevation.
•
•
Classify angles of elevation and
depression.
Find distance using angles of elevation.
Find distance using angles of depression.
Angle of elevation –
Angle of depression –
Ex#1: Classify Each Angle as an Angle of Elevation or an Angle of Depression.
Ex#2: Find Distance by Using Angle of Elevation.
The seattle Space Needle casts
a 67m. shadow. If the angle of elevation
from the tip of the shadow to the
top of the Space Needle is 70°,
how tall is the Space Needle?
Round to the nearest meter.
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Ex#3: Find Distance by Using Angle of Depression.
A. An ice climber stands at the edge of a crevasse that is 115ft. wide. The angle of depression from the edge where she
stands to the bottom of the opposite side is 52°. How deep is the crevasse at this point? Round to the nearest foot.
B. What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal
distance to this fire? Round to the nearest foot.
C. An observer in a lighthouse is 69ft. above the water. He sights two boats in the water directly in front of him. The
angle of depression to the nearest boat is 48°. The angle of depression to the other boat is 22°. What is the distance
between the two boats? Round to the nearest foot.
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Lesson 8.5A – Law of Sines
Learning Targets
Success Criteria
LT8-5A: Use the Law of Sines to solve problems.
•
•
Find trigonometric ratios for obtuse angles
using a calculator.
Use the Law of Sines to find side lengths
and angle measures of triangles.
Ex#1: Use a calculator to find each trigonometric ratio.
Round to the nearest hundredth.
A. tan 103°
B. cos 165°
C. sin 93°
Ex#2. Using the Law of Sines to find side lengths and angle measures of triangles.
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
A.
FG
B.
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m ∠Q
Lesson 8.5B Law of Cosines
Learning Targets
Success Criteria
•
LT8-5B: Use the Law of Cosines to solve
problems.
Use the Law of Cosines to find side
lengths and angle measures of triangles.
Ex#1: Use the Law of Cosines to find side lengths and angle measures of triangles.
Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
B. m ∠T
A. XZ
Ex#2: Use the Law of Sines or the Law of Cosines to find side lengths and angle measures.
A sailing club has planned a triangular racecourse, as shown in the diagram.
How long is the leg of the race along BC ? How many degrees must
competitors turn at point C? Round the length to the nearest tenth and the
angle measure to the nearest degree.
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Lesson 8.6 – Vectors
Learning Targets
Success Criteria
•
•
•
•
LT8-6: Use vectors to solve problems.
Identify equal and parallel vectors.
Write vectors in component form.
Find the magnitude of a vector.
Find the direction of a vector.
•
vector –
• component form –
•
magnitude-
•
direction –
•
equal vectors –
•
parallel vectors –
•
resultant vector –
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Ex#1: Identify equal vectors and parallel vectors.
Identify each of the following.
A Equal vectors.
B. Parallel vectors.
Ex#2: Write vectors in component form.
Write each vector in component form.
A.
%
HG
B.
Ex#3: Find the magnitude of a vector.
Draw the vector 〈−1, 5〉 on a coordinate plane.
Find its magnitude to the nearest tenth.
Ex#4: Finding the direction of a vector.
3, The force exerted by a skier is given by the vector 〈1, 4〉 .
Draw the vector on a coordinate plane. Find the direction
of the vector to the nearest degree.
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%
MN
with M(-8, 1) and N(2, -7).
Lesson
Problems
5.8 p. 360
#1-8, 17, 18 & Special Triangles Worksheet
8.1 p. 521
#15-39, 42, 47-49, 54, 57-59
8.2 p. 529
#22-50, 53, 54, 62, 68, 70, 84-86
8.3 p. 538
#21-29, 33, 34, 36, 39-44, 48, 52, 65-67
8.4 p. 547
#10-14, 17-22, 23
8.5 p. 555
Law of Sines
#23-28, 30, 39, 47, 49, 60, 61, 68-74
8.5 p. 555
Law of Cosines
#32-35, 38, 40, 44-46, 51-54, 59
8.6 p. 564
#18-35, 38, 39, 46, 47, 53, 54, 60, 63
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