Apply the Tangent Ratio

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Apply the Tangent Ratio

• Chapter 7.5

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Trigonometric Ratio

• A trigonometric ratio is a ratio of 2 sides of a
  right triangle.
• You can use these ratios to find sides lengths
  and angle measures.

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Sides of a right triangle

• Opposite the side opposite the angle you are
  looking at.

Opposite Side
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Sides of a right triangle

• Adjacent the side next to the angle you are
  looking at.

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Adjacent Side
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Sides of a right triangle

• Hypotenuse the side opposite the right angle.
  It is also the longest side on a triangle.

Hypotenuse
28
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Which side does the 22 represent? The hypotenuse,
adjacent, or opposite?
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Which side is which?
50
40
35
30
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Which side is which?
x
65
48
53
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Which side is which?
x
y
49
z
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Tangent

• The ratio that well focus on today is the
  tangent.
• The tangent is the opposite side over the
  adjacent side.

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Find the tangent of ŸR and ŸS
To find the measure of the angle R, find the
tangent. On a scientific calculator use the
inverse tangent button to calculate the angle
measure.
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Find the tangent of ŸJ and ŸK
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Find the tangent of ŸJ and ŸK
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Find the Tangent of ŸA and ŸB, then the angle
measures.
Tan A 0.4166 Tan B 2.4 móA 22.62ô móB
67.38ô
Tan A 0.75 Tan B 1.333 móA 36.87ô móB
53.12ô
Tan A 1.05 Tan B 0.95 móA 46.4ô móB 43.5ô
Tan A 1.61 Tan B 0.622 móA 58.15ô móB
31.88ô
Tan A 3.43 Tan B 0.29 móA 73.75ô móB
16.17ô
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Finding missing side lengths

• Some problems may require you to find a missing
  side length.
• In these problems you will be given a side length
  and a measure of an angle.
• You will then use the fact that the tangent of an
  angle is equal to the opposite side over the
  adjacent side to find the angle.

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Example
Multiply both sides by the denominator!
55
27
x
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Example
Multiply both sides by the denominator!
x
36
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Example
Multiply both sides by the denominator!
24
44
x
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Example
Multiply both sides by the denominator!
52
x
Divide both sides by the tangent!
67
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Example
Multiply both sides by the denominator!
22
x
Divide both sides by the tangent!
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Example
Multiply both sides by the denominator!
47
x
Divide both sides by the tangent!
55
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Example
Multiply both sides by the denominator!
49
x
Divide both sides by the tangent!
6
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Find the length of x for each problem.
X 21.98
1.
2.
X 8.66
3.
X 25
4.
X 42.84
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Tangents and Special Right Triangles

• Recall that for a 45-45-90 triangle the side
  lengths are
• leg x -or- leg 1
• leg x -or- leg 1
• Hypotenuse x -or- Hypotenuse
• Recall that for a 30-60-90 triangle the side
  lengths are
• Shorter leg x -or- Shorter leg 1
• Longer leg x -or- Longer leg
• Hypotenuse 2x -or- Hypotenuse 2

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What length must x be?
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What must x be?
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What must x be?
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A little more abstract

• If I tell you that a right triangle has a measure
  of 30 degrees, could you find the tangent of the
  angle?

1
30
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A little more abstract

• If I tell you that a right triangle has a measure
  of 45 degrees, could you find the tangent of the
  angle?

1
45
1