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The Tangent Ratio

CHAPTER 7

RIGHT TRIANGLE TRIGONOMETRY

You will need a protractor for this activity

1. Each person draw a right triangle (∆ABC) where ے A has a measure of 30º.

2. Each person in the group should draw the triangle with different side lengths, then measure the legs using inches.

3. Compute the ratio leg opposite A ے

4. Compare the ratio with the others in the group. Make a conjecture.

Trigonometry and the Tangent Ratio

Objectives:• Use tangent ratios to determine side lengths in

triangles

Now, we will use Trigonometry (triangle measure).We will investigate 3 of the 6 trigonometric functions:

• tangent• sine• cosine

Previously, to find measures in a right triangle, we used:

• Pythagorean Theorem• Distance Formula• 30-60-90 or 45-45-90 special right triangles theorems

Trigonometry and the Tangent Ratio

Tangent Ratio:In a right triangle, the ratio of the length of the leg opposite ے P to the length of the leg adjacent to ے P In a right triangle, the ratio of the length of the leg opposite ے P to the length of the leg adjacent to ے P . This is called the tangent ratio.

Tangent of ے P = opposite leg adjacent leg

Write a Tangent Ratio

Write the tangent ratios for ےT and ے U.

tan T = opp = UV = 3 adj TV 4

tan U = opp = TV = 4 adj UV 3

what is a tangent ratio?

•We use tangent ratios to determine side lengths and angles in right triangles.

•In a right triangle, the ratio of the length of the leg opposite to an angle to the length of the leg adjacent to the same angle.

•This ratio is always constant

The Tangent Ratio Tangent of <A:

Length of the leg opposite <A

Length of the leg adjacent to <A

Tan =

A

B C

Writing Tangent Ratios Write the tangent ratios for <K and <J:

K L

J

7

3

tan<K =

tan<J =

7/3

3/7

Find the Tangent Ratios Find the tangent ratios for <A and <B:

A

BC

1

2

tan <A =

tan <B =

2/1 or 2

1/2

Find the Tangent Ratios Find the tangent ratios for <A and <B:

A

BC

3

6

tan <A =

tan <B =

6/3 or 2

3/6 or 1/2

Solve for the missing side

Find the value of w to the nearest tenth:

10

w

Start at 54°. We have sides opposite and adjacent of that angle. We can use tangent to solve for the missing side.

Set up the tangent ratio:

tan 54 = w 10

Cross multiply

w = (tan 54)(10)

w = 13.8

Solve for the missing side

Find the value of w to the nearest tenth:

57° 2.5

w

Start at 57°. We have sides opposite and adjacent of that angle. We will use tangent to solve for the missing side.

Set up the tangent ratio:

tan 57 = w 2.5

Cross multiply

w = (tan 57)(2.5)

w = 3.8

Solve for the missing side

Find the value of w to the nearest tenth:

28°

1

w

Start at 28°. We have sides opposite and adjacent of that angle. We will use tangent to solve for the missing side.

Set up the tangent ratio:

tan 28 = 1 w

Cross multiply

(tan 28)(w) = 1 Divide by tan 28

w = 1 tan 28 w = 1.9

The Tangent Ratio

CHAPTER 7

RIGHT TRIANGLE TRIGONOMETRY

Solving for Angle Measures

Using the Inverse of Tangent The Inverse Tangent Button on your calculator looks like this:

tan-1

(You must press the “2nd” or “SHIFT” or “INV” Button and then press “tan”)

*We use inverse tangent when solving for a missing angle measure

Using Inverse Tangent Find the m < X to the nearest degree:

H

B X

5

8

We will use tangent to find the measure of <X

Set up the tangent ratio:

tan X = 5 8Use inverse tangent:

X = tan-1 5 8

X = 32°

If no calculator: divide 5/8, =0.6250Look under the Tangent column of the trig table for the number closest to 0.6250, then move your finger to the left to find the degrees (32) under the Angle column.

Using Inverse Tangent Find the m < Y to the nearest degree:

Y

T P

25

23

We will use tangent to find the measure of <Y

Set up the tangent ratio:

tan Y = 23 25

Use inverse tangent:

Y = tan-1 23 25

Y = 43°