3. solve the following right triangles for the parts not given:
a. ( a = 42^circ 23 12 ), hypotenuse = 437.25 ft
b. ( a = 61^circ 28 47 ), opposite side = 345.51 m
c. ( a = 35^circ 46 17 ), adjacent side = 358.17 ft
d. hypotenuse = 432.89 m, opposite side = 398.24 m
e. hypotenuse = 471.65 ft, adjacent side = 270.46 ft
f. opposite side = 368.47 m, adjacent side = 274.61 m

Question

Accepted Answer

### Part a:
# Explanation:
## Step1: Convert angle A to decimal degrees
First, convert the minutes and seconds to degrees. $ 23' = \frac{23}{60} \approx 0.3833^\circ $, $ 12'' = \frac{12}{3600} = 0.0033^\circ $. So $ A = 42 + 0.3833 + 0.0033 = 42.3866^\circ $. The other angle $ B = 90^\circ - 42.3866^\circ = 47.6134^\circ $ (or convert back to DMS if needed).
## Step2: Find opposite side (a)
Using $ \sin A=\frac{a}{c} $, so $ a = c\sin A $. $ c = 437.25 $ ft, $ \sin(42.3866^\circ)\approx\sin(42^\circ23'12'')\approx0.6747 $. Then $ a = 437.25\times0.6747\approx295.0 $ ft (approximate, more precise calculation: $ \sin(42.3866^\circ)=\sin(42 + 23/60+12/3600)=\sin(42.386666...^\circ)\approx0.67469 $, so $ a = 437.25\times0.67469\approx295.0 $ ft).
## Step3: Find adjacent side (b)
Using $ \cos A=\frac{b}{c} $, so $ b = c\cos A $. $ \cos(42.3866^\circ)\approx0.7381 $. Then $ b = 437.25\times0.7381\approx322.7 $ ft (more precise: $ \cos(42.3866^\circ)\approx0.73809 $, $ b = 437.25\times0.73809\approx322.7 $ ft). Also, angle $ B = 90^\circ - 42^\circ23'12'' = 47^\circ36'48'' $ (since $ 90^\circ = 89^\circ59'60'' $, subtract $ 42^\circ23'12'' $: $ 89 - 42 = 47^\circ $, $ 59 - 23 = 36' $, $ 60 - 12 = 48'' $).

# Answer (for part a):
- Angle $ B = 47^\circ36'48'' $ (or $ 47.6134^\circ $)
- Opposite side (a) $ \approx 295.0 $ ft
- Adjacent side (b) $ \approx 322.7 $ ft

### Part b:
# Explanation:
## Step1: Convert angle A to decimal degrees
$ 28'=\frac{28}{60}\approx0.4667^\circ $, $ 47''=\frac{47}{3600}\approx0.0131^\circ $, so $ A = 61 + 0.4667 + 0.0131 = 61.4798^\circ $.
## Step2: Find hypotenuse (c)
Using $ \sin A=\frac{a}{c} $, so $ c=\frac{a}{\sin A} $. $ a = 345.51 $ m, $ \sin(61.4798^\circ)\approx0.8788 $. Then $ c=\frac{345.51}{0.8788}\approx393.1 $ m.
## Step3: Find adjacent side (b)
Using $ \tan A=\frac{a}{b} $, so $ b=\frac{a}{\tan A} $. $ \tan(61.4798^\circ)\approx1.857 $. Then $ b=\frac{345.51}{1.857}\approx186.1 $ m. Also, angle $ B = 90^\circ - 61^\circ28'47'' = 28^\circ31'13'' $ ( $ 90^\circ = 89^\circ59'60'' $, $ 89 - 61 = 28^\circ $, $ 59 - 28 = 31' $, $ 60 - 47 = 13'' $ ).

# Answer (for part b):
- Angle $ B = 28^\circ31'13'' $ (or $ 28.5202^\circ $)
- Hypotenuse (c) $ \approx 393.1 $ m
- Adjacent side (b) $ \approx 186.1 $ m

### Part c:
# Explanation:
## Step1: Convert angle A to decimal degrees
$ 46'=\frac{46}{60}\approx0.7667^\circ $, $ 17''=\frac{17}{3600}\approx0.0047^\circ $, so $ A = 35 + 0.7667 + 0.0047 = 35.7714^\circ $.
## Step2: Find opposite side (a)
Using $ \tan A=\frac{a}{b} $, so $ a = b\tan A $. $ b = 358.17 $ ft, $ \tan(35.7714^\circ)\approx0.7213 $. Then $ a = 358.17\times0.7213\approx258.3 $ ft.
## Step3: Find hypotenuse (c)
Using $ \cos A=\frac{b}{c} $, so $ c=\frac{b}{\cos A} $. $ \cos(35.7714^\circ)\approx0.8138 $. Then $ c=\frac{358.17}{0.8138}\approx440.1 $ ft. Also, angle $ B = 90^\circ - 35^\circ46'17'' = 54^\circ13'43'' $ ( $ 90^\circ = 89^\circ59'60'' $, $ 89 - 35 = 54^\circ $, $ 59 - 46 = 13' $, $ 60 - 17 = 43'' $ ).

# Answer (for part c):
- Angle $ B = 54^\circ13'43'' $ (or $ 54.2286^\circ $)
- Opposite side (a) $ \approx 258.3 $ ft
- Hypotenuse (c) $ \approx 440.1 $ ft

### Part d:
# Explanation:
## Step1: Find angle A
Using $ \sin A=\frac{a}{c} $, $ a = 398.24 $ m, $ c = 432.89 $ m. So $ \sin A=\frac{398.24}{432.89}\approx0.9200 $. Then $ A=\arcsin(0.9200)\approx66.93^\circ $ (convert to DMS: $ 0.93^\circ\times60 = 55.8' $, $ 0.8'\times60 = 48'' $, so $ A\approx66^\circ55'48'' $).
## Step2: Find angle B
$ B = 90^\circ - A\approx90^\circ - 66^\circ55'48'' = 23^\circ4'12'' $ (or $ 23.07^\circ $).
## Step3: Find adjacent side (b)
Using $ \cos A=\frac{b}{c} $, so $ b = c\cos A $. $ \cos(66.93^\circ)\approx0.3939 $. Then $ b = 432.89\times0.3939\approx170.6 $ m (or use Pythagoras: $ b=\sqrt{c^{2}-a^{2}}=\sqrt{432.89^{2}-398.24^{2}}\approx\sqrt{187393 - 158605}=\sqrt{28788}\approx170.6 $ m).

# Answer (for part d):
- Angle $ A \approx 66^\circ55'48'' $ (or $ 66.93^\circ $)
- Angle $ B \approx 23^\circ4'12'' $ (or $ 23.07^\circ $)
- Adjacent side (b) $ \approx 170.6 $ m

### Part e:
# Explanation:
## Step1: Find angle A
Using $ \cos A=\frac{b}{c} $, $ b = 270.46 $ ft, $ c = 471.65 $ ft. So $ \cos A=\frac{270.46}{471.65}\approx0.5734 $. Then $ A=\arccos(0.5734)\approx54.93^\circ $ (convert to DMS: $ 0.93^\circ\times60 = 55.8' $, $ 0.8'\times60 = 48'' $, so $ A\approx54^\circ55'48'' $).
## Step2: Find angle B
$ B = 90^\circ - A\approx90^\circ - 54^\circ55'48'' = 35^\circ4'12'' $ (or $ 35.07^\circ $).
## Step3: Find opposite side (a)
Using $ \sin A=\frac{a}{c} $, so $ a = c\sin A $. $ \sin(54.93^\circ)\approx0.8192 $. Then $ a = 471.65\times0.8192\approx386.4 $ ft (or use Pythagoras: $ a=\sqrt{c^{2}-b^{2}}=\sqrt{471.65^{2}-270.46^{2}}\approx\sqrt{222454 - 73148}=\sqrt{149306}\approx386.4 $ ft).

# Answer (for part e):
- Angle $ A \approx 54^\circ55'48'' $ (or $ 54.93^\circ $)
- Angle $ B \approx 35^\circ4'12'' $ (or $ 35.07^\circ $)
- Opposite side (a) $ \approx 386.4 $ ft

### Part f:
# Explanation:
## Step1: Find angle A
Using $ \tan A=\frac{a}{b} $, $ a = 368.47 $ m, $ b = 274.61 $ m. So $ \tan A=\frac{368.47}{274.61}\approx1.342 $. Then $ A=\arctan(1.342)\approx53.33^\circ $ (convert to DMS: $ 0.33^\circ\times60 = 19.8' $, $ 0.8'\times60 = 48'' $, so $ A\approx53^\circ19'48'' $).
## Step2: Find angle B
$ B = 90^\circ - A\approx90^\circ - 53^\circ19'48'' = 36^\circ40'12'' $ (or $ 36.67^\circ $).
## Step3: Find hypotenuse (c)
Using Pythagoras: $ c=\sqrt{a^{2}+b^{2}}=\sqrt{368.47^{2}+274.61^{2}}\approx\sqrt{135770 + 75391}=\sqrt{211161}\approx459.5 $ m.

# Answer (for part f):
- Angle $ A \approx 53^\circ19'48'' $ (or $ 53.33^\circ $)
- Angle $ B \approx 36^\circ40'12'' $ (or $ 36.67^\circ $)
- Hypotenuse (c) $ \approx 459.5 $ m

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QUESTION IMAGE

Question

Explanation:

Part a:

Step1: Convert angle A to decimal degrees

Step2: Find opposite side (a)

Step3: Find adjacent side (b)

Step1: Convert angle A to decimal degrees

Step2: Find hypotenuse (c)

Step3: Find adjacent side (b)

Step1: Convert angle A to decimal degrees

Step2: Find opposite side (a)

Step3: Find hypotenuse (c)

Answer:

Part b: