1. Find the values of the variables.1.
ANSWER
x = 2 6, y =

2. Find the values of the variables.2.
ANSWER
15 2

3. EXAMPLE 1
Find tangent ratios
Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places.
SOLUTION =
opp S
adj. to S =
RTST =
80 18 =
40 9
tan S
4.4444 =
opp R
adj. to R =
STRT =
18 80 =
9 40
tan R
0.2250 =

4. GUIDED PRACTICE
for Example 1
Find tan J and tan K. Round to four decimal places.
ANSWER
0.7500,
ANSWER
0.5333,

5. Use the tangent of an acute angle to find a leg length.
EXAMPLE 2
Find a leg length
ALGEBRA
Find the value of x.
SOLUTION
Use the tangent of an acute angle to find a leg length.
tan 32o =
opp. adj.
Write ratio for tangent of 32o.
tan 32o 11 = x
Substitute.
x tan 32o
= 11
Multiply each side by x.
x = 11
tan 32o
Divide each side by tan 32o x 11
0.6249
Use a calculator to find tan 32o x
17.6
Simplify

6. Estimate height using tangent
EXAMPLE 3
Estimate height using tangent
LAMPPOST
Find the height h of the lamppost to the nearest inch.
tan 70o =
opp. adj.
Write ratio for tangent of 70o.
tan 70o h = 40
Substitute.
Multiply each side by 40.
40 tan 70o
= h
Use a calculator to simplify.
109.9 h
ANSWER
The lamppost is about 110 inches tall.

7. EXAMPLE 4
Use a special right triangle to find a tangent
Use a special right triangle to find the tangent of a 60o angle.
STEP 1
Because all 30o-60o-90o triangles are similar, you can simplify your calculations by choosing 1 as the length of the shorter leg. Use the 30o-60o-90o Triangle Theorem to find the length of the longer leg.

8. Use a special right triangle to find a tangent
EXAMPLE 4
Use a special right triangle to find a tangent
longer leg = shorter leg 3
30o- 60o- 90o Triangle Theorem x
= 1 3
Substitute. x = 3
Simplify.

9. Use a special right triangle to find a tangent
EXAMPLE 4
Use a special right triangle to find a tangent
STEP 2
Find tan 60o
tan 60o =
opp. adj.
Write ratio for tangent of 60o.
tan 60o = 3 1
Substitute.
tan 60o = 3
Simplify.
ANSWER
The tangent of any 60o angle is 3
1.7321

10. Find the value of x. Round to the nearest tenth.
GUIDED PRACTICE
for Examples 2, 3, and 4
Find the value of x. Round to the nearest tenth.
ANSWER
12.2
ANSWER
19.3
What If? In Example 4, suppose the side length of the shorter leg is 5 instead of 1. Show that the tangent of 60° is still equal to 5. 3
ANSWER
shorter leg = 5, longer leg = , 3 5
tan 60 =

11. Daily Homework Quiz
Use this diagram for Exercises 1- 4. 1.
If a = 18, b = 80 and c = 82, find tan B and tan A. Write each answer as a fraction and a decimal rounded to 4 places.
ANSWER
tan B = 40 9
= ;
tan A = =

12. Use this diagram for Exercises 1- 4.
Daily Homework Quiz
Use this diagram for Exercises 1- 4. 2.
If a = 17 and m A = 31 , find b to the nearest tenth. o
ANSWER
28.3

13. Use this diagram for Exercises 1- 4.
Daily Homework Quiz
Use this diagram for Exercises 1- 4. 3.
If a = 9 and m B = 74 , find b to the nearest tenth. o
ANSWER
31.4

14. Use this diagram for Exercises 1- 4.
Daily Homework Quiz
Use this diagram for Exercises 1- 4. 4.
If a = 5 and m B = 79 , find the number of square units in the area of ABC to the nearest tenth. o
ANSWER
64.3