Download Using Trigonometry to Find Side Le

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Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle
Objective: I will be able find the missing side length of an acute triangle given two side lengths and the measure
of the included angle.
Opening Exercise
a.
Find the lengths of and .
b.
Find the lengths of and . How is this different from part (a)?
LAW OF SINES: For an acute triangle △ with angles ∠, ∠, and ∠ and the sides opposite them , , and
, the law of sines states:
sin ∠ sin ∠ sin ∠
=
=
Example 1
A surveyor needs to determine the distance between two points and that lie on opposite banks of a river. A
point is chosen 160 meters from point , on the same side of the river as . The measures of angles ∠ and
∠ are 41˚ and 55˚, respectively. Approximate the distance from to to the nearest meter.
Exercises 1–2
1.
In △ , ∠ = 30, = 12, and = 10. Find sin∠. Include a diagram in your answer.
2.
A car is moving towards a tunnel carved out of the base of a hill. As the accompanying diagram shows, the
top of the hill, , is sighted from two locations, and . The distance between and is 250ft. What is
the height, ℎ, of the hill to the nearest foot?
LAW OF COSINES: For an acute triangle △ with angles ∠, ∠, and ∠ and the sides opposite them , ,
and , the law of cosines states
= + − 2
cos ∠.
Example 2
Our friend the surveyor from Example 1 is doing some further work. He has already found the distance between
points and (from Example 1). Now he wants to locate a point $ that is equidistant from both and and on
the same side of the river as . He has his assistant mark the point $ so that the angles ∠$ and ∠$ both
measure 75˚. What is the distance between $ and to the nearest
meter?
Exercise 3
3.
Parallelogram $ has sides of lengths 44mm and 26mm, and one of the angles has a measure of 100˚.
Approximate the length of diagonal to the nearest mm.
Problem Set
1.
Given △ , = 14, ∠ = 57.2°, and ∠ = 78.4°, calculate the measure of
angle to the nearest tenth of a degree, and use the Law of Sines to find the lengths
of and to the nearest tenth.
Calculate the area of △ to the nearest square unit.
2.
Given △ $*+, ∠+ = 39°, and *+ = 13, calculate the measure of ∠*, and use
---- and $*
---- to the nearest hundredth.
the Law of Sines to find the lengths of $+
3.
Does the law of sines apply to a right triangle? Based on △ , the following ratios were set up according
to the law of sines.
sin ∠ sin ∠ sin 90
=
=
Fill in the partially completed work below.
sin ∠ sin 90
=
sin ∠ sin 90
=
sin ∠ =
sin ∠ =
sin ∠ =
sin ∠ =
What conclusions can we draw?
4.
Given quadrilateral ./0, ∠ = 50°, ∠/. = 80°, ∠/.0 = 50°, ∠0
is a right angle and . = 9in., use the law of sines to find the length of
--- and --./, and then find the lengths of .0
0/ to the nearest tenth of an
inch.
5.
Given triangle 123, 12 = 10, 13 = 15, and ∠1 = 38°, use
the law of cosines to find the length of ----23 to the nearest tenth.
6.
Given triangle , = 6, = 8, and ∠ = 78°. Draw a diagram of triangle , and use the law of
cosines to find the length of --- .
Calculate the area of triangle .
Problem Set Sample Solutions
1.
Given △ 456, 45 = 78, ∠4 9:. ;°, and ∠6 :<. 8°, calculate the measure of angle 5 to the nearest tenth of a
degree, and use the law of sines to find the lengths of 46 and 56 to the nearest tenth.
By the angle sum of a triangle, ∠5 88. 8°.
=>? 4 =>? 5 =>? 6
A
B
@
=>? 9:. ; =>? 88. 8 =>? :<. 8
@
A
78
@
78 =>? 9:. ;
C 7;. D
=>? :<. 8
A
78 =>? 88. 8
C 7D. D
=>? :<. 8
Calculate the area of 456 to the nearest square unit.
7
AB =>? 4
;
7
4EF@ G7DHG78H =>? 9:. ;
;
4EF@ 4EF@ :D =>? 9:. ; C 9I
2.
Given JKL, ∠L MI°, and KL 7M, calculate the measure of ∠K, and use the Law of Sines to find the lengths of ---JL
---- to the nearest hundredth.
and JK
By the angle sum of a triangle, N∠K 99°.
=>? J =>? K =>? L
O
F
P
=>? <Q =>? 99
F
7M
7M =>? 99
F
C 7D. Q:
=>? <Q
=>? <Q =>? MI
7M
P
P
7M =>? MI
C <. ;D
=>? <Q
3.
Does the law of sines apply to a right triangle? Based on 456, the following ratios were set up according to the law
of sines.
=>? ∠4 =>? ∠5 =>? ID
A
B
@
Fill in the partially completed work below:
=>? ∠4 =>? ID
@
B
=>? ∠4 7
B
@
=>? ∠4 R
B
=>? ∠5 =>? ID
A
B
=>? ∠5 7
B
A
=>? ∠5 S
B
What conclusions can we draw?
The law of sines does apply to a right triangle. We get the formulas that are equivalent to =>? ∠4 =>? ∠5 4.
TUU
, where 4 and 5 are the measures of the acute angles of the right triangle.
VWU
TUU
and
VWU
Given quadrilateral XYZ[, ∠Y 9D°, ∠YZX <D°, ∠ZX[ 9D°, ∠[ is a right angle and XY I>?., use the Law of
--- and ---[Z to the nearest tenth of an inch.
Sines to find the length of XZ, and then find the lengths of X[
By the angle sum of a triangle, ∠YXZ 9D°; therefore,
XYZ is an isosceles triangle since its base ∠'s have equal
measure.
=>? 9D =>? <D
I
V
I =>? 9D
V
C :. D
=>? <D
\ : BT] 9D C 8. 9
^ : ]_` 9D C 9. 8
5.
----- to the nearest
Given triangle abc, ab 7D, ac 79, and ∠a M<°, use the Law of Cosines to find the length of bc
tenth.
d; 7D;
;
d 7DD
79; ! ;G7DHG79H ef= M<
;;9 ! MDD ef= M<
;
d M;9 ! MDD ef= M<
d √M;9 ! MDD ef= M<
d C I. 8
bc I. 8
6.
Given triangle 456, 46 Q, 45 <, and ∠4 :<°. Draw a diagram of triangle 456, and use the law of cosines to
find the length of ---56.
@ ; Q;
;
@ MQ
<; ! ;GQHG<HGef= :<H
Q8 ! IQGef= :<H
;
@ 7DD ! IQ ef= :<
@ √7DD ! IQ ef= :<
@ C <. I
---- is approximately <. I.
The length of 56
Calculate the area of triangle 456.
7
ABG=>? 4H
;
7
4EF@ GQHG<HG=>? :<H
;
4EF@ 4EF@ ;M. 9G=>? :<H
4EF@ C ;M. 9
The area of triangle 456 is approximately ;M. 9 square units.