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Answer :
We are given a triangle with
- [tex]$A = 46°$[/tex],
- [tex]$B = 73°$[/tex], and
- side [tex]$b = 32.3$[/tex].
Step 1. Find Angle [tex]$C$[/tex]
Since the sum of the angles in a triangle is [tex]$180°$[/tex], we have
[tex]$$
C = 180° - A - B = 180° - 46° - 73° = 61°.
$$[/tex]
Step 2. Apply the Law of Sines
The Law of Sines states that
[tex]$$
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}.
$$[/tex]
We want to find side [tex]$c$[/tex]. Using the relation between [tex]$b$[/tex] and [tex]$c$[/tex], we have
[tex]$$
\frac{b}{\sin(B)} = \frac{c}{\sin(C)}.
$$[/tex]
So, solving for [tex]$c$[/tex] gives
[tex]$$
c = \frac{\sin(C)}{\sin(B)} \cdot b.
$$[/tex]
Step 3. Substitute the Known Values
Substitute [tex]$b = 32.3$[/tex], [tex]$\sin(C)$[/tex] with [tex]$C = 61°$[/tex], and [tex]$\sin(B)$[/tex] with [tex]$B = 73°$[/tex]. This yields
[tex]$$
c = \frac{\sin(61°)}{\sin(73°)} \cdot 32.3.
$$[/tex]
Step 4. Compute the Value of [tex]$c$[/tex]
Evaluating the ratio of sines and then multiplying by [tex]$32.3$[/tex], we obtain a value of approximately
[tex]$$
c \approx 29.54.
$$[/tex]
Rounding this to one decimal place, the length of side [tex]$c$[/tex] is
[tex]$$
c \approx 29.5.
$$[/tex]
Thus, the length of side [tex]$c$[/tex] is [tex]$\boxed{29.5}$[/tex].
- [tex]$A = 46°$[/tex],
- [tex]$B = 73°$[/tex], and
- side [tex]$b = 32.3$[/tex].
Step 1. Find Angle [tex]$C$[/tex]
Since the sum of the angles in a triangle is [tex]$180°$[/tex], we have
[tex]$$
C = 180° - A - B = 180° - 46° - 73° = 61°.
$$[/tex]
Step 2. Apply the Law of Sines
The Law of Sines states that
[tex]$$
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}.
$$[/tex]
We want to find side [tex]$c$[/tex]. Using the relation between [tex]$b$[/tex] and [tex]$c$[/tex], we have
[tex]$$
\frac{b}{\sin(B)} = \frac{c}{\sin(C)}.
$$[/tex]
So, solving for [tex]$c$[/tex] gives
[tex]$$
c = \frac{\sin(C)}{\sin(B)} \cdot b.
$$[/tex]
Step 3. Substitute the Known Values
Substitute [tex]$b = 32.3$[/tex], [tex]$\sin(C)$[/tex] with [tex]$C = 61°$[/tex], and [tex]$\sin(B)$[/tex] with [tex]$B = 73°$[/tex]. This yields
[tex]$$
c = \frac{\sin(61°)}{\sin(73°)} \cdot 32.3.
$$[/tex]
Step 4. Compute the Value of [tex]$c$[/tex]
Evaluating the ratio of sines and then multiplying by [tex]$32.3$[/tex], we obtain a value of approximately
[tex]$$
c \approx 29.54.
$$[/tex]
Rounding this to one decimal place, the length of side [tex]$c$[/tex] is
[tex]$$
c \approx 29.5.
$$[/tex]
Thus, the length of side [tex]$c$[/tex] is [tex]$\boxed{29.5}$[/tex].
Thank you for reading the article Triangle ABC has the following measures tex m angle A 46 circ tex tex m angle B 73 circ tex tex b 32 3 tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
