Math  /  Geometry

Question3. (15 pts) Given is that A=38A=38^{\circ} and b=19 cmb=19 \mathrm{~cm} and c=22 cmc=22 \mathrm{~cm}. Solve the triangle ABCA B C. Round measures to 1 decimal place if necessary.

Studdy Solution

STEP 1

1. We are given a triangle ABC \triangle ABC with angle A=38 A = 38^\circ .
2. The sides b=19cm b = 19 \, \text{cm} and c=22cm c = 22 \, \text{cm} are opposite angles B B and C C , respectively.
3. We need to solve the triangle, which means finding all unknown sides and angles.

STEP 2

1. Use the Law of Sines to find one of the unknown angles.
2. Use the sum of angles in a triangle to find the remaining angle.
3. Use the Law of Sines again to find the remaining side.

STEP 3

Apply the Law of Sines to find angle B B . The Law of Sines states:
asinA=bsinB=csinC\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
We can use:
bsinB=csinC\frac{b}{\sin B} = \frac{c}{\sin C}
Substitute the known values:
19sinB=22sin38\frac{19}{\sin B} = \frac{22}{\sin 38^\circ}

STEP 4

Solve for sinB\sin B:
sinB=19sin3822\sin B = \frac{19 \cdot \sin 38^\circ}{22}
Calculate sinB\sin B:
sinB190.6157220.5319\sin B \approx \frac{19 \cdot 0.6157}{22} \approx 0.5319
Find B B using the inverse sine function:
Bsin1(0.5319)32.1B \approx \sin^{-1}(0.5319) \approx 32.1^\circ

STEP 5

Use the sum of angles in a triangle to find angle C C :
A+B+C=180A + B + C = 180^\circ
Substitute the known angles:
38+32.1+C=18038^\circ + 32.1^\circ + C = 180^\circ
Solve for C C :
C=1803832.1109.9C = 180^\circ - 38^\circ - 32.1^\circ \approx 109.9^\circ

STEP 6

Use the Law of Sines to find side a a :
asinA=csinC\frac{a}{\sin A} = \frac{c}{\sin C}
Substitute the known values:
asin38=22sin109.9\frac{a}{\sin 38^\circ} = \frac{22}{\sin 109.9^\circ}
Solve for a a :
a=22sin38sin109.9a = \frac{22 \cdot \sin 38^\circ}{\sin 109.9^\circ}
Calculate a a :
a220.61570.945514.3cma \approx \frac{22 \cdot 0.6157}{0.9455} \approx 14.3 \, \text{cm}
The solved triangle has the following measures: - Angle A=38 A = 38^\circ - Angle B32.1 B \approx 32.1^\circ - Angle C109.9 C \approx 109.9^\circ - Side a14.3cm a \approx 14.3 \, \text{cm} - Side b=19cm b = 19 \, \text{cm} - Side c=22cm c = 22 \, \text{cm}

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