THE UNIVERSITY OF AKRON
Theoretical and Applied Mathematics
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c 2003 teprice@uakron.edu
Last Revision Date: April 12, 2003
AcroTEX eDucation Bundle
Katie Jones
and
Tom Price
WebTrig Flash Cards
Flash Cards
Right triangle trigonometry
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Version 1.0
AcroTEX eDucation Bundle
WebTrig Flash Cards
A right triangle contains a 50◦ angle that has an adjacent side of
length 6.3 units. Find the length
of the opposite side and the hypotenuse.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
A right triangle contains an 18◦
angle that has an opposite side of
length 1.7 units. Find the length
of the adjacent side and the hypotenuse.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
A right triangle contains a 62◦
angle that has a hypotenuse of
length 8.3 units. Find the length
of the adjacent side and the opposite side.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
6
. Without
Suppose sin α =
11
using a calculator find cos α and
tan α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
3
Suppose cos α = . Without
5
using a calculator find sin α and
tan α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
5
. Without
Suppose tan α =
12
using a calculator find sin α and
cos α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Let α denote a non-right angle of
a right triangle. If the side adjacent α is 9 units long and the
hypotenuse is 18 units, find sin α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Let α denote a non-right angle of
a right triangle. If the side adjacent α is 12.2 units long and the
hypotenuse is 30 units, find cot α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Let α denote a non-right angle of
a right triangle. If the side opposite α is 12 units long and the
hypotenuse is 24 units, find cos α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Let α denote a non-right angle of
a right triangle. If the side adjacent α is 8 units long and the
hypotenuse is 21 units, find csc α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Let α denote a non-right angle of
a right triangle. If the side opposite α is 15 units long and the
hypotenuse is 31 units, find sec α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Let α denote a non-right angle
of a right triangle. If the side
adjacent α is 4.6 units long and
the hypotenuse is 9.2 units, find
tan α.
Hint
Soln
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9
10
67
α
AcroTEX eDucation Bundle
γ
WebTrig Flash Cards
Let T be the triangle depicted in
the figure. Find sin α.
c
Hint
Soln
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14
b
β
72
AcroTEX eDucation Bundle
γ
WebTrig Flash Cards
Let T be the triangle depicted in
the figure. Find sin γ.
10
Hint
Soln
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26
c
AcroTEX eDucation Bundle
a
7 75
α
WebTrig Flash Cards
Let T be depicted in the figure
below. Find c.
Hint
Soln
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a
112
27
c
AcroTEX eDucation Bundle
γ
19.4
WebTrig Flash Cards
Let T be the triangle in the figure. Find a.
Hint
Soln
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18
α
β
9
AcroTEX eDucation Bundle
γ
11
WebTrig Flash Cards
Let T be the triangle in the figure
below. Find cos β.
Hint
Soln
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γ
α
45
β
70
AcroTEX eDucation Bundle
42
WebTrig Flash Cards
Let T be the triangle depicted in
the figure below. Find cos γ.
Hint
Soln
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γ
a
β
47
24
AcroTEX eDucation Bundle
14
WebTrig Flash Cards
Let T be the triangle given in the
figure below. Find a.
Hint
Soln
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b
2
α
92
1.7
AcroTEX eDucation Bundle
γ
WebTrig Flash Cards
Determine b in the figure.
Hint
Soln
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find the length of the opposite side and
the hypotenuse of a right triangle with a 50◦
angle that has an adjacent side of length 6.3
units, use the triangular definition of the cosine of an angle which is the length of the
adjacent side divided by the hypotenuse.
Hint
Soln
Next
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Answer: hyp = 9.8 and opp = 7.51
Hence,
opp =
√
56.372
= 7.5081
AcroTEX eDucation Bundle
6.3
cos 50◦
= 9.801 1
hyp =
WebTrig Flash Cards
Solution: First, find the Next, the Pythagorean theohypotenuse using the cosine rem1 suggests that
ratio:
2
2
2
(9.801 1) = (6.3) + (opp) .
6.3
.
cos 50◦ =
So,
hyp
2
(opp) = 56.372
So,
1 (hyp)2
= (adj)2 + (opp)2
Hint
Soln
Next
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find the length of the adjacent side and
the hypotenuse of a right triangle with an
18◦ angle that has an adjacent side of length
1.7 units, use the triangular definition of the
sine of an angle which is the length of the
opposite side divided by the hypotenuse.
Hint
Soln
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Answer: hyp = 5.5 and adj = 5.23
= 5. 232
AcroTEX eDucation Bundle
Next, the Pythagorean theo-
WebTrig Flash Cards
Solution: Find the hy- rem2 suggests that
potenuse using the sine ratio:
2
2
2
(5. 501 3) = (adj) + (1.7) .
1.7
sin 18◦ =
So,
hyp
2
so,
(adj) = 27. 374
1.7
hyp =
Hence,
sin 18◦
√
= 5.501 3
adj = 27. 374
2 (hyp)2
= (adj)2 + (opp)2
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find the length of the adjacent side and
the opposite side of a right triangle with a
62◦ angle that has an adjacent side of length
8.3 units, use the triangular definition of the
sine of an angle which is the length of the
opposite side divided by the hypotenuse.
Hint
Soln
Next
Home
Answer: opp = 7.33 and adj = 3.9
opp
.
8.3
So,
2
2
2
(8.3) = (adj) + (7. 328 5)
so,
2
(adj) = 15. 183.
opp = 8.3 · sin 62◦
= 7.328 5
Hence,
adj =
√
15. 183 = 3. 896 5
AcroTEX eDucation Bundle
sin 62◦ =
WebTrig Flash Cards
Solution: Using the sine Next, the Pythagorean theoratio we get
rem3 suggests that
3 (hyp)2
= (adj)2 + (opp)2
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find cos α and tan α when
6
sin α =
11
use the triangular definition of the sine of an
angle which is sin α = opp
.
hyp
Hint
Soln
Next
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2
So, (adj) = 85,or
√
adj = 85.
√
6
opp
6 85
=√ =
.
tan α =
adj
85
85
AcroTEX eDucation Bundle
Solution: Given sin α = Hence,
6
√
, use the Pythagorean the85
adj
11
=
cos α =
orem4 to get
hyp
11
2
2
2
(11) = (adj) + (6) .
and
WebTrig Flash Cards
√ Answer:
√
85
6 85
cos α =
and tan α =
11
85
4 (hyp)2
= (adj)2 + (opp)2
Hint
Soln
Next
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find sin α and tan α when
3
cos α = ,
5
use the triangular definition
adj
cos α =
.
hyp
Hint
Soln
Next
Home
Answer: sin α =
AcroTEX eDucation Bundle
WebTrig Flash Cards
4
4
and tan α =
5
3
Solution: Given
suggesting that
adj
3
,
cos α = =
4
opp
5
hyp
=
sin α =
hyp
5
and using the Pythagorean
theorem5 we get
2
2
2
and
(5) = (3) + (opp)
Consequently,
opp =
√
16 = 4
tan α =
4
opp
= .
adj
3
5 (hyp)2
= (adj)2 + (opp)2
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find sin α and cos α when
5
tan α =
12
recall the triangular definitions of these functions.
Hint
Soln
Next
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Answer: sin α =
AcroTEX eDucation Bundle
WebTrig Flash Cards
12
5
and tan α =
13
13
Solution: Recall that
Consequently,
√
opp
5
opp = 169 = 13
=
.
tan α =
12
adj
so that
Next, find the hypotenuse us5
opp
ing the Pythagorean theo=
sin α =
hyp
13
rem6
2
2
2
and
(hyp) = (12) + (5)
adj
12
cos α =
=
.
= 169.
hyp
13
6 (hyp)2
= (adj)2 + (opp)2
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find sin α if the side adjacent α is 9 units
long and the hypotenuse is 18 units remember the triangular definition of the sine of an
angle uses the length of the side opposite α
and the length of the hypotenuse.
Hint
Soln
Next
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Answer: sin α = .87
2
2
2
=⇒
(opp) = 243
√
opp = 243 = 15. 588
=⇒
Then we know,
opp
=
sin α =
hyp
√
243
= 0.866 03.
18
AcroTEX eDucation Bundle
2
(18) = (9) + (opp)
WebTrig Flash Cards
Solution: Find the opposite side using the pythagorean
2
2
2
theorem (hyp) = (adj) + (opp)
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find cot α if the side adjacent α is 12.2
units long and the hypotenuse is 30 units recall that
adj
cot α =
opp
.
Hint
Soln
Next
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Answer: cot α = .45
2
2
=⇒
=⇒
2
(opp) = 751. 16
√
opp = 751. 16 = 27. 407
We know
cot α =
12.2
adj
=
= 0.445 14
opp
27.407
AcroTEX eDucation Bundle
2
(30) = (12.2) + (opp)
WebTrig Flash Cards
Solution: Find the opposite side using the pythagorean
2
2
2
theorem (hyp) = (adj) + (opp)
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find cos α if the side opposite α is 12 units
long and the hypotenuse is 24 units use the
triangular definition of the cosine of an angle
which uses the length of the hypotenuse and
side adjacent to α.
Hint
Soln
Next
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Answer: cos α = .87
2
2
=⇒
=⇒
2
(adj) = 432
√
adj = 432 = 20. 785
We know,
adj
=
cos α =
hyp
√
√
432
3
=
= 0.866 03
24
2
AcroTEX eDucation Bundle
2
(24) = (adj) + (12)
WebTrig Flash Cards
Solution: Find the adjacent side using the pythagorean
2
2
2
theorem (hyp) = (adj) + (opp)
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find csc α if the side adjacent α is 8 units
long and the hypotenuse is 21 units use
hyp
csc α =
opp
Hint
Soln
Next
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Answer: csc α = 1.08
2
2
=⇒
=⇒
2
(opp) = 377
√
opp = 377 = 19. 416
We know,
csc α =
21
hyp
=√
= 1. 081 6
opp
377
AcroTEX eDucation Bundle
2
(21) = (8) + (opp)
WebTrig Flash Cards
Solution: Find the opposite side using the pythagorean
2
2
2
theorem (hyp) = (adj) + (opp)
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find sec α if the side opposite α is 15 units
long and the hypotenuse is 31 units you will
need the hypotenuse and the side adjacent to
α.
Hint
Soln
Next
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Answer: sec α = 1.14
2
2
=⇒
=⇒
2
(adj) = 736
√
adj = 736 = 27. 129
We know,
sec α =
31
hyp
=√
= 1. 142 7
adj
736
AcroTEX eDucation Bundle
2
(31) = (adj) + (15)
WebTrig Flash Cards
Solution: Find the adjacent side using the pythagorean
2
2
2
theorem (hyp) = (adj) + (opp)
Hint
Soln
Next
Home
HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find tan α if the side adjacent α is 4.6
units long and the hypotenuse is 9.2 units
you will need the length of the side opposite
α and the hypotenuse.
Hint
Soln
Next
Home
Answer: tan α = 1.73
2
2
=⇒
2
2
2
=⇒
=⇒
(9.2) = (4.6) + (opp)
2
(opp) = 63. 48
√
opp = 63.48 = 7. 967 4
We know,
tan α =
7.9674
opp
=
= 1. 732
adj
4.6
AcroTEX eDucation Bundle
2
(hyp) = (adj) + (opp)
WebTrig Flash Cards
Solution: Find the opposite side using the pythagorean
theorem.
Hint
Soln
Next
Home
HINT
9
10
67
α
AcroTEX eDucation Bundle
γ
WebTrig Flash Cards
To find sin α if a = 9, b =
10, and β = 67◦ as depicted
in the figure, use the law of
sines.
c
Hint
Soln
Next
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Answer: sin α = 0.83
67
α
c
sin α = 9 (0.092 05)
= 0.828 45
AcroTEX eDucation Bundle
9
10
WebTrig Flash Cards
γ
Solution: By the law of sines
sin β
sin α
=
a
b
sin 67◦
sin α
=
9
10
so
Hint
Soln
Next
Home
HINT
β
72
10
AcroTEX eDucation Bundle
14
b
WebTrig Flash Cards
γ
To find sin γ in the given triangle recall that the law of sines
sin β
sin γ
sin α
=
=
.
a
b
c
Hint
Soln
Next
Home
Answer: sin γ = 0.07
β
72
10
AcroTEX eDucation Bundle
14
b
WebTrig Flash Cards
γ
Solution: Use the law of sines
sin α
sin γ
=
a
c
to obtain
sin 72◦
sin γ
=
14
10
sin γ = 10 (.00679)
= 0.0679
Hint
Soln
Next
Home
HINT
26
c
AcroTEX eDucation Bundle
a
7 75
α
WebTrig Flash Cards
To find c if β = 26◦ , b = 7, and γ = 75◦ as
depicted in the figure below use the law of
sines.
Hint
Soln
Next
Home
Answer: c = 15.42
AcroTEX eDucation Bundle
c
WebTrig Flash Cards
7 75
α
Solution: By the law of sines
sin β
sin γ
=
.
c
b
So
sin 26◦
sin 75◦
=
a
c
7
7 sin 75◦ = c sin 26◦
26
7 sin 75◦
c=
= 15.424
sin 26◦
Hint
Soln
Next
Home
HINT
a
112
27
AcroTEX eDucation Bundle
γ
19.4
WebTrig Flash Cards
To find a in the figure use the law of sines
sin α
which equates
to similar ratio.
a
c
Hint
Soln
Next
Home
Answer: α = 9.5
a
AcroTEX eDucation Bundle
γ
19.4
WebTrig Flash Cards
Solution: By the law of sines
sin β
sin α
=
.
a
b
So
sin 112◦
sin 27◦
=
a
19.4
19.4 sin 27◦ = a sin 112◦
19.4 sin 27◦
α=
= 9. 499
sin 112◦
112
27
c
Hint
Soln
Next
Home
HINT
18
α
β
9
AcroTEX eDucation Bundle
γ
11
WebTrig Flash Cards
To find cos β for the angle β in the triangle,
use the law of cosines. Remember this law
can be written in three different forms.
Hint
Soln
Next
Home
Answer: cos β = .88
112 = 405 − 324 cos β
121 − 405
= 0.876 54
=⇒ cos β =
−324
γ
11
18
α
β
AcroTEX eDucation Bundle
112 = 182 + 92 − 2 (18) (9) cos β
WebTrig Flash Cards
Solution: By the law of cosines (b2 = a2 + c2 − 2ac cos β)
9
Hint
Soln
Next
Home
HINT
42
γ
α
45
β
70
AcroTEX eDucation Bundle
c2 = a2 + b2 − 2ab cos γ.
WebTrig Flash Cards
To find cos γ if a = 45, b = 42, and c = 70
recall the law of cosines:
Hint
Soln
Next
Home
Answer: cos γ = −0.29
= 3789 − 3780 cos γ
4900 − 3789
= −0.293 92
=⇒ cos γ =
−3780
42
γ
α
45
β
AcroTEX eDucation Bundle
702 = 452 + 422 − 2 (45) (42) cos γ
WebTrig Flash Cards
Solution: Using the law of cosines c2 = a2 + b2 − 2ab cos γ
we have
70
Hint
Soln
Next
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HINT
to another value.
14
γ
a
β
47
AcroTEX eDucation Bundle
b2 + c2 − 2bc cos α
WebTrig Flash Cards
To find a if α = 47◦ , b = 14, and c = 24 use
the law of cosines which equates
24
Hint
Soln
Next
Home
Answer: a = 17.71
a2 = 772 − 672 cos 47◦
a2 = 313.7
√
=⇒ a = 313.7 = 17.712
14
γ
a
β
47
AcroTEX eDucation Bundle
a2 = 142 + 242 − 2 (14) (24) cos 47◦
WebTrig Flash Cards
Solution: By the law of cosines a2 = b2 + c2 − 2bc cos α we
have
24
Hint
Soln
Next
Home
HINT
b
2
α
92
AcroTEX eDucation Bundle
γ
WebTrig Flash Cards
To find b in the triangle use the form of
the law of cosines that equates b2 to another
value.
1.7
Hint
Soln
Next
Home
Answer: b = 2.67
we have
2
b2 = 22 + (1.7) − 2 (2) (1.7) cos 92◦
b2 = 6.89 − 6.8 cos 92◦
b2 = 7.127 3
√
=⇒ b = 7.1273 = 2.6697
γ
b
2
α
92
1.7
AcroTEX eDucation Bundle
b2 = a2 + c2 − 2ac cos β
WebTrig Flash Cards
Solution: By the law of cosines
Hint
Soln
Next
Home