Transcript Apply the Tangent Ratio

Apply the Tangent Ratio
5.2 (M2)
Vocabulary



Trigonometry: branch of mathematics that deals
with the relationships between the sides and angles
of triangles and the calculations based on these
relationships
Trigonometric ratio: lengths of 2 sides in a right
triangle
Tangent of the angle: ratio of the length of the leg
opposite an acute angle to the length of the leg
adjacent to the angle (constant)
B
legoppositeA BC
tan A 

legadjacent A AC
C
A
Complementary Angles: sum of their
measures is 90o
BC
tan A 

AC
B
5
3
C
A
4
AC
tan B 

BC
Can you do the tangent of C?
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
tan S = opp S = RT = 80
ST
18
adj. to S
= 40
9
4.4444
tan R = opp R = ST = 18
RT
80
adj. to R
= 9
40
= 0.2250
GUIDED PRACTICE
for Example 1
Find tan J and tan K. Round to four decimal places.
ANSWER
0.7500, 1.3333
ANSWER
0.5333, 1.8750
EXAMPLE 2
ALGEBRA
Find a leg length
Find the value of x.
SOLUTION
Use the tangent of an acute angle to
find a leg length.
opp.
tan 32o =
Write ratio for tangent of 32o.
adj.
Substitute.
tan 32o = 11
x
Multiply each side by x.
x tan 32o = 11
11
x =
Divide each side by tan 32o
o
tan 32
Use a calculator to find tan 32o
11
x
0.6249
x 17.6
Simplify
EXAMPLE 3
Estimate height using tangent
LAMPPOST
Find the height h of the lamppost to the nearest inch.
opp.
tan 70o =
adj.
h
40
tan 70o =
40 tan 70
o
109.9
Write ratio for
tangent of 70o.
Substitute.
= h
h
Multiply each
side by 40.
Use a calculator to
simplify.
ANSWER
The lamppost is about 110 inches tall.
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.
EXAMPLE 4 Use a special right triangle to find a tangent
longer leg = shorter leg
x =1
x = 3
3
3
Substitute.
Simplify.
30o- 60o- 90o Triangle Theorem
EXAMPLE 4 Use a special right triangle to find a tangent
STEP 2
tan 60o
Find tan 60o
opp.
=
adj.
Write ratio for tangent of 60o.
=
3
1
Substitute.
tan 60o =
3
Simplify.
o
tan 60
ANSWER
The tangent of any 60o angle is
3
1.7321
GUIDED PRACTICE
for Examples 2, 3, and 4
Find the value of x. Round to the nearest tenth.
ANSWER
12.2
ANSWER
19.3
5. 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 3 .
ANSWER
shorter leg = 5, longer leg = 5 3 , tan 60 = 5 3 =
5
3