1. Bell Ringer

2. 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.

3. Example 1
Find tan S and tan R as fractions in simplified form and as decimals rounded to four decimal places.
Find Tangent Ratio
SOLUTION
leg opposite S
tan S =
leg adjacent to S = ≈ 4 3
tan R =
leg opposite R
leg adjacent to R = ≈ 4 3 1 3

4. Approximate tan 74° to four decimal places.
Example 2
Use a Calculator for Tangent
Approximate tan 74° to four decimal places.
SOLUTION
Calculator keystrokes 74 or
Display
Rounded value
3.4874 4

5. 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.
ANSWER
tan S = 3 4
= 0.75;
tan R = ≈ 2.
ANSWER
tan S = 5 12
≈ ;
tan R =
= 2.4

6. Checkpoint Now you Try 
Find Tangent Ratio
Use a calculator to approximate the value to four decimal places. 3.
tan 35°
ANSWER
0.7002 4.
tan 85°
ANSWER 5.
tan 10°
ANSWER
0.1763

7. Example 3
Use a tangent ratio to find the value of x. Round your answer to the nearest tenth.
Find Leg Length
SOLUTION
tan 22° =
opposite leg
adjacent leg
Write the tangent ratio.
tan 22° = 3 x
Substitute.
x · tan 22° = 3
Multiply each side by x.
x = 3
tan 22°
Divide each side by tan 22°.
x ≈ 3
0.4040
Use a calculator or table to approximate tan 22°.
x ≈ 7.4
Simplify. 7

8. First, find the measure of the other acute angle: 90° – 35° = 55°.
Example 4
Use two different tangent ratios to find the value of x to the nearest tenth.
Find Leg Length
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° = 4 x
tan 55° = x 4
x · tan 35° = 4
4 tan 55° = x 8

9. The two methods yield the same answer: x ≈ 5.7.
Example 4
Find Leg Length
x = 4
tan 35°
4(1.4281) ≈ x
x ≈ 4
0.7002
x ≈ 5.7
x ≈ 5.7
ANSWER
The two methods yield the same answer: x ≈ 5.7. 9

10. Example 5
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.
Estimate Height
SOLUTION
tan 59° =
opposite leg
adjacent leg
Write ratio.
tan 59° = h 45
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. 10

11. The tree is about 75 feet tall.
Example 5
Estimate Height
ANSWER
The tree is about 75 feet tall. 11

12. Checkpoint Now you Try 
Find Side Length
Write two equations you can use to find the value of x. 6.
ANSWER
tan 44° = 8 x
and
tan 46° =
tan 37° = 4 x
and
tan 53° =
ANSWER 7. 8.
tan 59° = 5 x
and
tan 31° =
ANSWER

13. Checkpoint Now you Try 
Find Side Length
Find the value of x. Round your answer to the nearest tenth. 9.
ANSWER
10.4
10.
ANSWER
12.6
11.
ANSWER
34.6

14. Page 560

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