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Lesson 14.3
Solving Right Triangles
pp. 594-598
Objectives:
1. To find the trigonometric ratios for
angles using a calculator or tables.
2. To find missing sides or angles of
right triangles.
Solving a right triangle
means finding all the angle
measures and all the side
lengths of the triangle from
the information given.
Find the following using a
calculator. Your answers
should be accurate to four
decimal places.
1. cos 72° = 0.3090
2. tan 9° = 0.1584
3. sin 59° = 0.8572
4. tan 61° = 1.8040
Find mA to the nearest
degree given the following trig
ratios.
5. tan A = 1.3250 mA = 53°
6. cos A = 0.9455 mA = 19°
7. sin A = 0.9130 mA = 66°
There are two types of right
triangles to solve.
1. The right triangle given a side and
an acute angle.
2. The right triangle given two sides.
Steps to solve a right triangle
given a side and an acute
angle.
1. Find the other acute angle by
subtracting the one given from 90°.
2. Set up a trig equation involving the
acute angle and side given, and
one of the unknown remaining
sides.
Steps to solve a right triangle
given a side and an acute
angle.
3. Use the Pythagorean theorem and
the two known sides to find the
third.
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
A
A=
B = 72°
C = 90°
b
C
a =8
b=
c=
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
A
b
A = 90° - B
= 90° - 72°
= 18°
C
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
A
A = 18°
B = 72°
C = 90°
b
C
a =8
b=
c=
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
18°
A
b
C
b
tan 72 =
8
b=
(tan
24
.6 72)
8
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
18°
A
A = 18°
B = 72°
C = 90°
b
C
a =8
b = 24.6
c=
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
18°
A
24.6
82 + 24.62 = c2
64 + 605.16 = c2
C
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
18°
A
24.6
669.16 = c2
c  25.9
C
EXAMPLE 1 Given right
ABC, find the measure of each
side and angle.
B
c
72°
8
A
A = 18°
B = 72°
C = 90°
b
C
a =8
b = 24.6
c = 25.9
EXAMPLE 2 Solve right ABC
if C is the right angle, mA =
38°, and c = 26 units.
B A = 38°
B = 52°
26
C = 90°
a=
38°
b=
A
C c = 26
EXAMPLE 2 Solve right ABC
if C is the right angle, mA =
38°, and c = 26 units.
a
B sin 38° = 26
a = sin 38°(26)
a  16.0
26
38°
A
C
EXAMPLE 2 Solve right ABC
if C is the right angle, mA =
38°, and c = 26 units.
B A = 38°
B = 52°
26
C = 90°
a = 16.0
38°
b=
A
C c = 26
EXAMPLE 2 Solve right ABC
if C is the right angle, mA =
38°, and c = 26 units.
b
B cos 38° = 26
b = cos 38°(26)
b  20.5
26
38°
A
C
EXAMPLE 2 Solve right ABC
if C is the right angle, mA =
38°, and c = 26 units.
B A = 38°
B = 52°
26
C = 90°
a = 16.0
38°
b = 20.5
A
C c = 26
Steps to solve a right triangle
given two sides.
1. Find the third side using the given
sides and the Pythagorean
theorem.
2. Use the two given sides to set up a
trig equation to find one of the
acute angles.
Steps to solve a right triangle
given two sides.
3. Subtract the acute angle from 90°
to find the final angle.
EXAMPLE 3 Solve right ABC.
A=
A
B=
C
=
90°
11
a = 17
b = 11
C
17
B c=
172 + 112 = c2
c2 = 410
EXAMPLE 3 Solve right ABC.
A=
A
B=
C
=
90°
11
a = 17
b = 11
C
17
B c = 20.2
172 + 112 = c2
c  20.2
EXAMPLE 3 Solve right ABC.
A = 57°
A
B=
C
=
90°
11
a = 17
b = 11
C
17
B c = 20.2
tan A = 17/11
tan-1 (17/11) = A  57°
EXAMPLE 3 Solve right ABC.
A = 57°
A
B = 33°
C
=
90°
11
a = 17
b = 11
C
17
B c = 20.2
B = 90° - 57°
B = 33°
EXAMPLE 3 Solve right ABC.
A = 57°
A
B = 33°
C
=
90°
11
a = 17
b = 11
C
17
B c = 20.2
Homework
pp. 597-598
►A. Exercises
Find the indicated trigonometric ratios. See
the table on p. 616.
1. sin 41°
►A. Exercises
Find the indicated trigonometric ratios. See
the table on p. 616.
3. tan 82°
►A. Exercises
Find mA, given the following trigonometric
ratios. See the table on page 616. Find the
angles to the nearest degree.
7. cos A = 0.8746
►B. Exercises
Use the triangles shown. Name the ratio or
theorem that you would use to find the
indicated measurement and then calculate it.
E
11. DF
20
17°
F
D
►B. Exercises
Solve each right triangle. Round your
answers to the nearest tenth or to the nearest
X
degree.
17.
15
Z
12
Y
►B. Exercises
Solve each right triangle. Round your
answers to the nearest tenth or to the nearest
degree.
M
19.
L
26°
5
N
►B. Exercises
Solve right ∆ABC if C is the right angle.
Round your answers to the nearest tenth or
to the nearest degree.
23. mA = 47°, b = 18 units
■ Cumulative Review
Which are congruent, similar, or neither?
Why?
D
27.
B
E
C
A
AB || CD
■ Cumulative Review
Which are congruent, similar, or neither?
Why?
28.
P
R
Q
S
■ Cumulative Review
Which are congruent, similar, or neither?
Why?
29.
F
G
H
E
■ Cumulative Review
Which are congruent, similar, or neither?
Why?
M
30.
I
J
K
L
N
■ Cumulative Review
Which are congruent, similar, or neither?
Why?
31.
Z
W
X
Y
V
Analytic Geometry
Measurement
►Exercises
Use the figure for exercises 1-2.
1. Find the perimeter of the triangle.
A (3, 5)
B (-2, 1)
C (1, -1)
►Exercises
Use the figure for exercises 1-2.
2. Find the area of the triangle.
A (3, 5)
B (-2, 1)
C (1, -1)