Download - 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.
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Sides of a right triangle
• Adjacent – the side next to the angle you are looking at.
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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.
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Which side does the 22 represent? The hypotenuse, adjacent, or opposite?
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Which side is which?
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Which side is which?
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Which side is which?
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Tangent
• The ratio that we’ll 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
adjacent
oppositeR tan
80
18
adjacent
oppositeS tan
18
80
225.0
44.4
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.
68.12)225.0(tan 1
32.77)444.4(tan 1
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Find the tangent of ŸJ and ŸK
adjacent
oppositeJ tan
32
24
adjacent
oppositeK tan
24
32
4
3
3.13
4
87.36)75.0(tan 1
13.53)333.1(tan 1
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Find the tangent of ŸJ and ŸK
adjacent
oppositeJ tan
15
8
adjacent
oppositeK tan
8
15 875.1
06.28)533.0(tan 1
93.61)875.1(tan 1
533.0
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Find the Tangent of ŸA and ŸB, then the angle measures.
Tan A = 0.75Tan B = 1.333móA = 36.87ômóB = 53.12ô
Tan A = 1.05Tan B = 0.95móA = 46.4ômóB = 43.5ô
Tan A = 0.4166Tan B = 2.4móA = 22.62ômóB = 67.38ô
Tan A = 3.43Tan B = 0.29móA = 73.75ômóB = 16.17ô
Tan A = 1.61Tan B = 0.622móA = 58.15ômóB = 31.88ô
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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
adjacent
opposite55tan27
x
2727
55tan27 x
Multiply both sides by the denominator!
x 55tan27
x56.38
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Example
adjacent
opposite36tan18
x
1818
36tan18 x
Multiply both sides by the denominator!
x 36tan18
x08.13
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Example
adjacent
opposite24tan44
x
4444
24tan44 x
Multiply both sides by the denominator!
x 24tan44
x59.19
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Example
adjacent
opposite52tanx
67
xx
x 67
52tan
Multiply both sides by the denominator!
6752tan x
35.52x
Divide both sides by the tangent!
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Example
adjacent
opposite22tanx
14
xx
x 14
22tan
Multiply both sides by the denominator!
1422tan x
65.34x
Divide both sides by the tangent!
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Example
adjacent
opposite47tanx
55
xx
x 55
47tan
Multiply both sides by the denominator!
5547tan x
29.51x
Divide both sides by the tangent!
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Example
adjacent
opposite49tanx
6
xx
x 6
49tan
Multiply both sides by the denominator!
649tan x
22.5x
Divide both sides by the tangent!
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Find the length of x for each problem.
1. 2.
3. 4.
X = 8.66 X = 21.98
X = 42.84X = 25
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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
3
2 2
3
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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?
3
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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?