Apply the Tangent Ratio

Chapter 7.5

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.

Sides of a right triangle

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

Sides of a right triangle

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

Sides of a right triangle

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

Which side does the 22 represent? The hypotenuse, adjacent, or opposite?

Which side is which?

Which side is which?

Which side is which?

Tangent

• The ratio that we’ll focus on today is the tangent.

• The tangent is the opposite side over the adjacent side.

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

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

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

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

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.

Example

adjacent

opposite55tan27

x

2727

55tan27 x

Multiply both sides by the denominator!

x 55tan27

x56.38

Example

adjacent

opposite36tan18

x

1818

36tan18 x

Multiply both sides by the denominator!

x 36tan18

x08.13

Example

adjacent

opposite24tan44

x

4444

24tan44 x

Multiply both sides by the denominator!

x 24tan44

x59.19

Example

adjacent

opposite52tanx

67

xx

x 67

52tan

Multiply both sides by the denominator!

6752tan x

35.52x

Divide both sides by the tangent!

Example

adjacent

opposite22tanx

14

xx

x 14

22tan

Multiply both sides by the denominator!

1422tan x

65.34x

Divide both sides by the tangent!

Example

adjacent

opposite47tanx

55

xx

x 55

47tan

Multiply both sides by the denominator!

5547tan x

29.51x

Divide both sides by the tangent!

Example

adjacent

opposite49tanx

6

xx

x 6

49tan

Multiply both sides by the denominator!

649tan x

22.5x

Divide both sides by the tangent!

Find the length of x for each problem.

1. 2.

3. 4.

X = 8.66 X = 21.98

X = 42.84X = 25

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

What length must x be?

What must x be?

What must x be?

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

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?