1. Find the values of the variables.1. 2.
ANSWER
x = 2 6, y =
ANSWER
15 2

2. Use trigonometric ratios to solve right triangles.
Target
Use trigonometric ratios to solve right triangles.
You will…
Use tangent ratio to find indirect measurement.

3. Vocabulary
trigonometric ratio –
the ratio between the lengths of two sides of a right triangle
sine, cosine, tangent, cosecant, secant, and cotangent
Opposite angle C Hypotenuse
Opposite angle A Adjacent angle B
Opposite angle B Adjacent angle A
tangent ratio –
tan(reference angle) =
opposite side adjacent side

4. 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 R
adj. to R =
opp S
adj. to S
tan S
tan R =
STRT =
RTST
tan S
tan R =
80 18 =
18 80
tan S
tan R
tan S
4.4444
tan R
0.2250 =

5. GUIDED PRACTICE
for Example 1
Find tan J and tan K. Round to four decimal places.
ANSWER
tan J = , tan K =
ANSWER
tan J = , tan K =

6. 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

7. 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.

8. Use a special right triangle to find a tangent
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.
long leg short leg 3 1 =
30o- 60o- 90o Triangle Theorem 3 1 = x
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
19.3
ANSWER
12.2
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 =