Simplify square roots. expression, or radical. The number or

DATE
Simplify square roots.
An expression written with a radical symbol is called a radical
expression, or radical. The number or expression inside the radical
symbol is the radicand.
Find the square root of 95. Round your answer to the nearest tenth.
Check that your answer is reasonable.
Use your calculator and the following keystrokes.
Rounded value
Display
Calculator Keystrokes
9.74679
V~ ~ 9.7
95
95
or
This is reasonable because 95 is between the perfect squares 81 and 100. So, ~ should
be between V~- = 9 and ~ = 10. The answer 9.7 is between 9 and 10. Furthermore,
(9.7)a = 94.09, which is close to the original value of 95.
Use the Pythagorean Theorem to find the length of the
hypotenuse to the nearest tenth.
a2 + b2 = c2
31 + 15 = c2
46 = ca
V~=c
6.8 ~ c
Chapter 10 Resource Book
Write the Pythagorean Theorem.
Substitute ~ for a and ~ for b.
Simplify.
Add.
Take the square root of each side.
Use a calculator.
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Find the missin9 side ~ength of the ri~h,t triangle. Round
answer to the nearest tenth.
7.
8.
x
9.
x
Multiply the radicals. Then simplify if possible.
Exercises for Example 3
b. (10~/-~)~. - 10V~"OV~
1
= 10. 10. V~. ~/~
= 100.2
= 200
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For use with pa0es 5o,2-5o,7
Find the side
A right triangle with angle measures of 45°, 45°, and 90° is called a
45°-45°= 9~)° triangle°
Theorem 1&1 dt5°-45°-9~)° Triangle Theorem
In a 45°-45°-90° triangle, the length of the hypotenuse is the length of a
leg times ~/~.
Find the length x of the hypotenuse in the 45°-45°-90°
triangle shown at the right.
By the 450-450-90° Triangle Theorem, the length of the
hypotenuse is the length of a leg times ~.
hypotenuse = leg ¯ ~
x= 10~/~ ¯ ~v/-~
x= 10"V~
x= 10"2
x = 20
45°-45°-90° Triangle Theorem
Substitute x for hypotenuse and 10~x/~ for leg.
Product Property of Radicals
Evaluate the square root.
Simplify.
Answer: The length of the .hypotenuse is 20.
Find the va~ue of x.
25
25
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LI~SSOi~
~
~
~
’CONT~NU~3)
DATE
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Far use with pages 5~2-5~7
e
Fin
shown, at the right.
By the 450-450-90° Triangle Theorem, the length of the hypotenuse
is the length of a leg times
45°-45°-90° Triangle Theorem
hypotenuse = leg"
Substitute 9~/~ for hypotenuse and x for leg.
Divide each side by V~.
9=x
.Simplify.
Find the va~ue of xo
Show that the triangle is a 45°-45°-90° triangle. Then find the
value of x. Round your answer to the nearest tenth.
17
The triangle is an isosceles right triangle. By the Base Angles Theorem,
its acute angles are congruent. The acute angles of a right triangle are
complementary. Because the two acute angles are congruent, the measure
of each must be 45°. Therefore the triangle is a 45°-45°-90° triangle. You
can use the 45°-45°-90° Triangle Theorem to find the value of x.
hypotenuse = leg ¯ V~
45°-45°-90° Triangle Theorem
17 = x~v/~
Substitute 17 for hypotenuse and x for leg.
17
Divide each side by
Use a calculator to approximate.
12.0 ~ x
g,~er~,i~e~ for Example 3
Show that the triangle is a 45°-45°=90° triangle. Then find the
va~ue of x. Round your answer to the nearest tenth.
7.x~/42
Chapter 10 Resource Book
8.
x~
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Find the side ~engths of 30°=60°=90° trian~j~eso
A right triangle with angle measures of 30°, 60°, and 90° is called a
30°-60°-90° triangle.
Theorem l&2 3~)°-6~)°-90° Triangle Theorem
In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter
leg, and the longer leg is the length of the shorter leg times ~/~.
In the 300-600-90° triangle at the right, the length of the shorter
leg is given. Find the length of the hypotenuse x.
48
The hypotenuse of a 30°-60°-90° triangle is twice as long as the
shorter leg.
30°-60°-90° Triangle Theorem
hypotenuse = 2. shorter leg
x=2.48
Substitute x for hypotenuse and 48 for shorter leg.
x = 96
Simplify.
Answer: The length of the hypotenuse is 96.
............................................................................................
Find the ~en~th x of the hvpotenuse of the triangle.
2.
15
30°
3.
60°
~
2
Find Leg ~en~th
a. In the 30o-60o-90° triangle at the right, the length of the
longer leg is given. Find the length of the shorter leg x.
Round your answer to the nearest tenth.
b. In the 30°-60°-’90° triangle at the right, the length of the
shorter leg is given. Find the length of the longer leg y.
Round your answer to the nearest tenth.
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a. The length of the longer leg of a 300-600-90° triangle is the length of
the shorter leg times V~.
30°-60°-90° Triangle Theorem
longer leg = shorter leg ¯
Substitute 9 for longer leg and x for shorter leg.
9 = x~f~
9
Divide each side by ~/~.
Use a calculator.
5.2 ~ x
Answer: The length of the shorter leg is about 5.2.
b. The length of the longer leg of a 30°-60°-90° triangle is the 16ngth of
the shorter leg times ~/~..
longer leg = shorter leg ¯ ~ 300-600-90° Triangle Theorem
y = 8 ¯ ~/~
Substitute y for longer leg and 8 for shorter leg.
y ~ 13.9
Use a calculator.
Answer: The length of the longer leg is about 13.9.
Find the length x of the shorter leg and the length
y of the longer leg. Write your answer in radical form.
Use the 30°-60°-90° Triangle Theorem to find the length of the,
shorter leg. Then use that value to find the length of the longer leg.
Shorter leg
Longer leg
longer
leg = shorter leg ¯ V~
hypotenuse = 2 ¯ shorter leg
y=17"~f~
34=2ox
y = 17~v/~
17 = x
7.
Y
2
~
y
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For use wi~h pages 55~;-5~2
Find the tangent of an acute angle.
A trigonometric ratio is a ratio of the lengths of two sides of a right
triangle. For any acute angle of a right triangle, there is a leg oppasite
the angle and a leg adjacent to the angle. The ratio of the leg opposite
the angle to the leg adjacent to the angle is the tangent of the angle.
leg opposite Z A
Tangent Rafi~: tan A = leg adjacent to ZA
Find tan D and tan E as fractions in simplified form and as
decimals rounded to four decimal places.
34
16
E
30
F
tan D = leg opposite/D = 30 = 15 = 1.875
leg adjacent to/D 16 8
8
tan E = leg opposite Z E .... 16
~ 0.5333
leg adjacent to /E
30 15
tan # and tan ~ as fractions in simplified form and as
deCimals. Round to four de¢i~a~ #~aCes if neCessaW.
1. D
E
2.
3~
F
6~/~
4
E
D
~
3.
E
.~~~14
D
48
F
F
Approximate tan 10° to four decimal places.
Calculator keystrokes
10 or N 10
Exemises far Example 2
Display
0.1763269807
Rounded value
0.1763
"
Use a ~a~u~ator to approximate the va~ue’ to four de~ima~ p~a~es.
4. tan 34°
5. tan 71°
6. tan 45°
7.. tan 20°
Chapter 10 Resource Book
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, All rights reserved.
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For use with pages
Use a tangent ratio tq find the ;¢alue of x. Round your answer
to the nearest tenth.
29
tan 65° = opposite leg
adjacent leg
tan 65° = 2_~9
x
x ¯ tan 65° = 29
29
tan 65°
29
2.1445
x -~ 13.5
Write the tangent ratio.
Substitute.
Multiply each side by x.
Divide each side by tan 65°.
Use a calculator or table to approximate tan 65°.
Simplify.
Find the va~ue of x. Reund yeur answeP te the nearest tenth.
x
8. 10~
Find Leg Length
Use two different tangent ratios to find the value of x to the
nearest tenth.
4O
First, find the measure of the other acute angle: 90° - 58° = 32°.
Method 1
tan 58° = 40
_~x
40 ¯ tan 58° = x
40(1.6003) = x
64.0 ~ x
Exercises for Example 4
Method 2
tan 32° =x40
x. tan 32° = 40
-- 40
x = tan
32°
x = . 40
0.6249
64.0
Write two equations you can use to find the va~ue of x.
Then find the value of x to the nearest tenth.
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For ase with pages 563-568
For any acute angle in a right triangle, the ratio of the leg opposite the
angle to the hypotenuse is the sine of the angle.
For any acute angle in a right triangle, the ratio of the leg adjacent to the
angle to the hypotenuse is the cosine of the angle.
Sine Ratio: sin A = leg opposite ZA
hypotenus e
Cosine Ratio: cos A = leg adjacent to ZA
hypotenuse
Find sin A and cos A. Write your answers as fractions and as B
decimals rounded to four decimal places.
C
sin A = leg opposite ZA
hypotenuse
5
sin A = -13
Write ratio for sine.
sinA ~ 0.3846
Use a calculator.
cos A = leg adjacent to ZA
hypotenuse
12
cos A = -13
Write ratio for cosine.
cosA ’ 0.9231
Use a calculator.
12
Substitute.
Substitute.
Find sin A and cos A. Write your answers as fractions and as
70
C
3. A
40 ’
24
B
A
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For use with pages 563=568
Use a calculator to approximate sin 18° and cos 18°. Round your answers
to four decimal places.
Display
0.309016994
,0.951056516
Calculator keystrokes
18
or
18
18
or
18
Rounded value
0.3090
0.9511
Exercises for Example 2
4. sin 33°
5. cos 33°
8. sin 85°
9. cos 13°
6. sin 8°
10. sin 0°
7. cos 67°
1~. cos 0°
Find £eO £enotk~s
Find the length of the legs of the triangle.
Round your answers to the nearest tenth.
sin A = leg Opposite ZA
hypotenuse
sin 25° = a
34
cos A = leg adjacent to ZA
hypotenuse
cos 25° = b
34
34(cos 25°) = b
34(sin 25°) = a
34(0.9063) = b
34(0.4226) ~ a
30.8 ~ b
14.4 ~ a
Answer: In the triangle, BC is about 14.4 and AC is about 30.8.
Exemises for Example 3
12. C
Y
A
C ........
x
C
Chapter 10 Resource Book
14. B
13. B
y
A
Y
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To solve a right triangle means to find the measures of both acute
angles and the lengths of all three sides.
Inverse Tangent:
Inverse Sine:
Inverse Cosine:
For any acute angle A of a right triangle, if
tan A = z then tan-1 z = mZA.
For any acute angle A of a right triangle, if
sin A = y then sin-1 y = mZ A.
For any acute angle A of a right triangle, if
cos A = x then cos-1 x = mZ A.
Solve the right triangle. Round decimals
to the nearest tenth.
15
C 19
To solve the triangle, you need to find c, mZA, and mZ B.
To find c, use the Pythagorean Theorem.
(hypotenuse)a = (leg)2 ÷ (leg)a
ca = 152 + 19a
ca= 586
Pythagorean Theorem.
Substitute.
Simplify.
Find the positive square root.
c = 5V’f8-~
Use a calculator to approximate.
c ~ 24.2
To find mZA, use a calculator and the inverse tangent.
19 1.2667,
mZA ~ tan-1 1.2667.
Since tan A = 1-~
~
Calculator keystrokes ’
Display
Expression
51.7105701
or
1.2667
tan-~ 1.2667 1.2667
Because tan;1 1.2667 ~ 51.7°, mZA ~ 51.7°. To find mZB, use the fact that
Z A and Z B are complementary. Since Z A and Z B are complementary,
mZB ~ 90° - 51.7° = 38.3°.
Enercises ~or Enampie ~
3. p ........ 15
A
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R
O
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Far use with pages 589-575
Z A is an acute angle. Use a calculator to approximate the measure of Z A
to the nearest tenth of a degree.
a. sinA= 0.3112
b. cosA = 0.4492
a. Since sinA = 0.3112, mZA = sin-a 0.3112.
Expression
Calculator keystrokes
-1
sin 0.3112 0.3112
or
0.3112 ~
-1
°.
Because sin 0.3112 ~ 18.10, mZA ~ 18.1
b. Since cos A = 0.4492, mZ_A = cos-1 0.4492.
Expression
Calculator keystrokes
-1
cos 0.4492 0.4992 ~#N~
or
0.4492 :~
Because cos-1 0.4492 ~ 63.3°, mZA -~ 63.3°.
Display
18.1315629
Display
.3076316
. .....................
Z A is an acute angle. Use a calculator to appro×imate the
4. sinA = 0.9403
5. cosA = 0.7844 6. sinA = 0.2337
7. cosA = 0.1818
Solve the right triangle. Round your answers to the
nearest tenth.
B
To solve the triangle, you need to find a, mZAI and mZB. A
18 C
To find a, use the Pythagorean Theorem.
(hypotenuse)2 = (leg)2 + (leg)2
Pythagorean Theorem
202 = a2 + 182
Substitute.
400 = a2 + 324
Simplify.
76 = a2
Subtract 324 from each side.
8.7 = a
Find the positive square root.
18
__ __
-a
0.9
~ 25.8419328, so mZA ~ 25.8°.
Since cos A = 20 .0.9, mZA = cos
To find mZ B, use the fact that Z A and Z B are complementary. Since Z A and Z B are
complementary, mZB ~ 90° - 25.8° = 64.2°.
E:~emises fer E~amp~e 3
8. A
Y
42
b
~B
C
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R
T
52
x
y
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z

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