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Bell Ringer. Tangent Ratios. A trigonometric ratios is a ratio of the lengths of two sides of a right triangle. For any acute angle of a right triangle, there is a leg opposite the angle and a leg adjacent to the angle. The ratio of these legs is the tangent of the angle. Example 1.

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Bell Ringer

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## Bell Ringer

### Tangent Ratios

• A trigonometric ratios is a ratio of the lengths of two sides of a right triangle.

• For any acute angle of a right triangle, there is a leg opposite the angle and a leg adjacent to the angle. The ratio of these legs is the tangent of the angle.

Example 1

SOLUTION

leg opposite S

4

tan S =

=

=

≈ 1.7321

4

leg opposite R

1

4

3

3

3

3

=

=

tan R =

≈ 0.5774

4

Find Tangent Ratio

Find tan S and tan R as fractions in simplified form and as decimals rounded to four decimal places.

Example 2

SOLUTION

Rounded value

Calculator keystrokes

Display

3.487414444

3.4874

Use a Calculator for Tangent

Approximate tan74° to four decimal places.

or

74

74

Now you Try 

Find Tangent Ratio

Find tan Sand tan R as fractions in simplified form and as decimals. Round to four decimal places if necessary.

1.

tan S =

tan R =

≈1.3333

=0.75;

12

3

4

5

=2.4

tan S =

tan R =

≈0.4167;

12

3

4

5

2.

Checkpoint

Now you Try 

Find Tangent Ratio

Use a calculator to approximate the value to four decimal places.

3.

tan 35°

4.

tan 85°

11.4301

0.7002

0.1763

5.

tan 10°

Example 3

Find Leg Length

Use a tangent ratio to find the value of x. Round your answer to the nearest tenth.

SOLUTION

tan 22° =

Write the tangent ratio.

Substitute.

3

tan 22° =

x

opposite leg

x· tan 22° = 3

Multiply each side by x.

Divide each side by tan22°.

x =

3

Use a calculator or table to approximate tan22°.

tan 22°

x ≈

3

x≈ 7.4

Simplify.

0.4040

Example 4

Method 1

Method 2

Find Leg Length

Use two different tangent ratios to find the value of x to the nearest tenth.

SOLUTION

First, find the measure of the other acute angle: 90° – 35° = 55°.

tan 55° =

tan 35° =

x

4

tan 55° =

tan 35° =

x

4

opposite leg

opposite leg

x· tan 35° = 4

4 tan 55° = x

Example 4

Find Leg Length

4(1.4281)≈x

x≈ 5.7

x≈ 5.7

The two methods yield the same answer: x≈5.7.

x =

4

tan 35°

x ≈

4

0.7002

Example 5

Estimate Height

You stand 45 feet from the base of a tree and look up at the top of the tree as shown in the diagram. Use a tangent ratio to estimate the height of the tree to the nearest foot.

SOLUTION

tan 59° =

Write ratio.

h

tan 59° =

45

opposite leg

Substitute.

45 tan 59° = h

Multiply each side by 45.

45(1.6643)≈h

Use a calculator or table to approximate tan 59°.

74.9≈h

Simplify.

Example 5

Estimate Height

The tree is about 75 feet tall.

Checkpoint

8

tan 44° =

and

tan 46° =

x

Now you Try 

Find Side Length

Write two equations you can use to find the value of x.

6.

x

4

5

x

x

7.

x

x

8

4

5

tan 37° =

and

tan 53° =

8.

tan 59° =

and

tan 31° =

Checkpoint

Now you Try 

Find Side Length

Find the value of x. Round your answer to the nearest tenth.

9.

10.

12.6

10.4

34.6