Right Triangles and Trigonometry

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Right Triangles and Trigonometry

• Unit 7

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Chapter 7 Right Trianlges

• Goal 7.2 The Pythagorean Theorem Its Converse
• Goal 7.3 Special Right Triangles
• Goal 7.4 Trigonometry
• Goal 7.5 Angle of Elevation Depression
• Goal 7.6 The Law of Sines
• Goal 7.7 The Law of Cosines

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Goal 7.2 We will be able to

• Use and apply the Pythagorean theorem and its
  converse.

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Partner Activity

• Cut out the triangle according to the dimensions
  given.
• Cut a square that matches the dimension of each
  side. Example 1 Side is 3, then construct 3 x 3
  square.

• Find the area of each square.
• Answer the following question
• What is the relationship of the three sides of
  the right triangle?

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Right Triangle
Hypotenuse
Leg
Leg
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Pythagorean Theorem

• In a right triangle, the sum of the squares of
  the measures of the legs equals the square of the
  measure of the hypotenuse.
• a2 b2 c2

B
c
a
A
C
b
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Longitude and Latitude

• Carson City, Nevada, is located at about 120
  degrees longitude and 39 degrees latitude. NASA
  Ames I 122 degrees longitude and 27 degrees
  latitude.

• Find the degree distance to the nearest tenth
  between NASA Ames to Carson City, Nevada.

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More Right Triangle Practice

• Find d.

3 cm
d
6 cm
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Converse of the Pythagorean Theorem

• If the sum of the squares of the measures of 2
  sides of a triangle equals the square of the
  measure of the longest side, then the triangle is
  a right triangle.
• If a2 b2 c2, then ?ABC is a right triangle

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Verify that ?ABC is a Right Triangle

• The coordinates of the triangle are as follows
• A(-9, -3)
• B(1, -1)
• C(-3, -7)

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Pythagorean Triple

• Pythagorean Triple is 3 whole numbers that
  satisfy the equation
• a2 b2 c2.

• Pythagorean Triple?
• 9, 12, 15
• 21, 42, 54
• 4v3, 4, 8

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Practice 7.2

• Practice 7.2 WS

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Goal 7.3 We will be able to

• Recognize and use properties of
• 45-45-90 and 30-60-90 triangles

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Special Triangles

• 45-45-90

• 30-60-90

30
45
45
a
60
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45-45-90

• The length of the hypotenuse is v2 times the
  length of the leg.

45
45
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30-60-90

• The hypotenuse is 2 times the size of the
  smallest leg.
• The largest leg is v3 times the size of the
  smallest leg.

30
60
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Lets Practice

• Find x and y.

• Find x and y.

x
y
9.6
11
y
x
60
60
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Practice 7.3

• Practice 7.3 WS

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Goal 7.4 We will be able to

• Find trigonometric ratios using right triangles.
• Solve problems using trig. ratios.

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Trigonometry

• Greek word Trigon means triangle.

• Greek word Metron meaning measure.

A ratio of the lengths of the sides of a right
triangle are called the trigonometric ratios.
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SOH-CAH-TOA

• Sine A Opposite
• Hypotenuse
• Cosine A Adjacent
• Hypotenuse
• Tangent A Opposite
• Adjacent

B
C
A
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Sine

• Sine A

• Sine B

B
5
3
A
4
C
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Cosine

• Cosine A

• Cosine B

B
5
3
A
4
C
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Cosine

• Tangent A

• Tangent B

B
5
3
A
4
C
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Practice 7.4

• Practice 7.4 WS

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Goal 7.5 We will be able to

• Solve problems involving angles of elevation and
  depression.

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Angle of Elevation

• Angle of elevation is the angle between the line
  of sight and the horizontal when an observer
  looks upward.

Angle of Elevation
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Aviation

• A plane takes off from the end of the runway in
  the direction of the mountain at an angle that is
  kept constant until the peak has been cleared.
  If the pilot wants to clear the mountain by 50
  meters, what should the angle of elevation be for
  the takeoff to the nearest tenth of a degree?

400 m
2025 m
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Angle of Depression

• Angle of depression is the angle between the line
  of sight when an observer looks downward, and the
  horizontal.

Angle of Depression
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Salvage

• A salvage ship uses sonar to determine that the
  angle of depression to a wreck on the ocean
  floor. The depth chart shows that the ocean
  floor is 40 meters below the surface. How far
  must a diver lowered from the salvage ship walk
  along the ocean floor to reach the wreck?

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One Last Practice Problem

• From the top of a 150 ft high tower, an air
  traffic controller observes an airplane on the
  runway. To the nearest foot, how far from the
  base of the tower is the airplane?

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Practice 7.5

• Practice 7.5 WS