Answer :
Sure, let's find the length of [tex]\( j \)[/tex] given the values [tex]\( k = 7.2 \)[/tex] inches, [tex]\( i = 1.3 \)[/tex] inches, and [tex]\( LJ = 58^\circ \)[/tex]. We will use the Law of Sines to solve this.
### Step-by-Step Solution:
1. Understanding the Law of Sines:
The Law of Sines states that in any triangle:
[tex]\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides opposite angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
2. Setting Up the Problem:
Given the values:
- [tex]\( k = 7.2 \)[/tex] inches (one side of the triangle),
- [tex]\( i = 1.3 \)[/tex] inches (another side of the triangle),
- [tex]\( LJ = 58^\circ \)[/tex] (angle opposite the side [tex]\( j \)[/tex]).
We need to find the length of [tex]\( j \)[/tex].
3. Applying the Law of Sines:
According to the given relationship:
[tex]\[ \frac{j}{\sin(LJ)} = \frac{k}{\sin(90^\circ)} \][/tex]
Because [tex]\( \sin(90^\circ) = 1 \)[/tex], the equation simplifies to:
[tex]\[ j = k \cdot \sin(LJ) \][/tex]
4. Converting the Angle to Radians:
Since the sine function typically uses radians in most calculators and programming languages, we need to convert [tex]\( 58^\circ \)[/tex] to radians.
[tex]\[ \text{Radians} = \frac{58 \times \pi}{180} \][/tex]
Calculating this,
[tex]\[ \text{Radians} \approx 1.01229 \][/tex]
5. Calculating the Length of [tex]\( j \)[/tex]:
Now, we substitute the values into the equation:
[tex]\[ j = 7.2 \cdot \sin(1.01229) \][/tex]
Evaluating [tex]\( \sin(1.01229) \)[/tex] (using a calculator or trigonometric table),
[tex]\[ \sin(1.01229) \approx 0.8480 \][/tex]
Thus,
[tex]\[ j \approx 7.2 \cdot 0.8480 \approx 6.1059462923262675 \][/tex]
6. Rounding the Result:
Finally, we round [tex]\( j \)[/tex] to the nearest tenth of an inch.
[tex]\[ j \approx 6.1 \][/tex]
### Final Answer:
The length of [tex]\( j \)[/tex], to the nearest tenth of an inch, is [tex]\( \boxed{6.1} \)[/tex] inches.
### Step-by-Step Solution:
1. Understanding the Law of Sines:
The Law of Sines states that in any triangle:
[tex]\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the lengths of the sides opposite angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.
2. Setting Up the Problem:
Given the values:
- [tex]\( k = 7.2 \)[/tex] inches (one side of the triangle),
- [tex]\( i = 1.3 \)[/tex] inches (another side of the triangle),
- [tex]\( LJ = 58^\circ \)[/tex] (angle opposite the side [tex]\( j \)[/tex]).
We need to find the length of [tex]\( j \)[/tex].
3. Applying the Law of Sines:
According to the given relationship:
[tex]\[ \frac{j}{\sin(LJ)} = \frac{k}{\sin(90^\circ)} \][/tex]
Because [tex]\( \sin(90^\circ) = 1 \)[/tex], the equation simplifies to:
[tex]\[ j = k \cdot \sin(LJ) \][/tex]
4. Converting the Angle to Radians:
Since the sine function typically uses radians in most calculators and programming languages, we need to convert [tex]\( 58^\circ \)[/tex] to radians.
[tex]\[ \text{Radians} = \frac{58 \times \pi}{180} \][/tex]
Calculating this,
[tex]\[ \text{Radians} \approx 1.01229 \][/tex]
5. Calculating the Length of [tex]\( j \)[/tex]:
Now, we substitute the values into the equation:
[tex]\[ j = 7.2 \cdot \sin(1.01229) \][/tex]
Evaluating [tex]\( \sin(1.01229) \)[/tex] (using a calculator or trigonometric table),
[tex]\[ \sin(1.01229) \approx 0.8480 \][/tex]
Thus,
[tex]\[ j \approx 7.2 \cdot 0.8480 \approx 6.1059462923262675 \][/tex]
6. Rounding the Result:
Finally, we round [tex]\( j \)[/tex] to the nearest tenth of an inch.
[tex]\[ j \approx 6.1 \][/tex]
### Final Answer:
The length of [tex]\( j \)[/tex], to the nearest tenth of an inch, is [tex]\( \boxed{6.1} \)[/tex] inches.
