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 Find Tangent Ratio

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

SOLUTION

leg opposite Stan S =

leg adjacent to S = = ≈ 1.732144 3

3

tan R =leg opposite R

leg adjacent to R = = ≈ 0.57744

4 313

Example 2 Use a Calculator for Tangent

Approximate tan 74° to four decimal places.

SOLUTION

Calculator keystrokes

74 or 74

Display

3.487414444

Rounded value

3.4874

Now you Try Find Tangent Ratio

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

1.

2.

ANSWER tan S = 34 = 0.75;

tan R = 43 ≈ 1.3333

ANSWER tan S = 512 ≈ 0.4167;

tan R = 125 = 2.4

Checkpoint Find Tangent Ratio

ANSWER 0.7002

ANSWER 11.4301

ANSWER 0.1763

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

3. tan 35°

4. tan 85°

5. tan 10°

Now you Try

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° =opposite leg

adjacent leg Write the tangent ratio.

tan 22° = 3x Substitute.

x · tan 22° = 3 Multiply each side by x.

x = 3tan 22°

Divide each side by tan 22°.

x ≈ 30.4040

Use a calculator or table to approximate tan 22°.

x ≈ 7.4 Simplify.

Example 4 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°.

Method 1

tan 35° =opposite leg

adjacent leg

Method 2

tan 55° =opposite leg

adjacent leg

tan 35° = 4x tan 55° = x4

x · tan 35° = 4 4 tan 55° = x

Example 4 Find Leg Length

x ≈ 5.7

x = 4tan 35° 4(1.4281) ≈ x

x ≈ 40.7002 x ≈ 5.7

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

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° =opposite leg

adjacent leg Write ratio.

tan 59° = h45 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

ANSWER The tree is about 75 feet tall.

Checkpoint Find Side Length

Write two equations you can use to find the value of x.6.

7.

8.

ANSWER

tan 44° = 8x and tan 46° = x8

tan 37° = 4x and tan 53° = x4

ANSWER

tan 59° = 5x and tan 31° = x5

ANSWER

Now you Try

Checkpoint Find Side Length

ANSWER 10.4

ANSWER 12.6

ANSWER 34.6

Find the value of x. Round your answer to the nearest tenth.9.

10.

11.

Now you Try

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