To determine the correct expressions that represent the length of side $\overline{KL}$ in the right triangle $\triangle KJL$, we need to apply the Law of Sines.

The Law of Sines states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

where $a$, $b$, and $c$ are the lengths of the sides opposite angles $A$, $B$, and $C$, respectively.

Given the triangle $\triangle KJL$:

• Angle $L$ is $60^\circ$.
• Angle $J$ is $90^\circ$ (right angle).
• Side $JL$ (opposite angle $K$) is given as 3.
• Side $KJ$ (opposite angle $L$) is given as 5.2.

We need to find the expressions for the length of side $\overline{KL}$.

1. Using the angle $L$: $\frac{\overline{KL}}{\sin 60^\circ} = \frac{3}{\sin 30^\circ}$ Simplifying, we get: $\overline{KL} = \frac{3}{\sin 30^\circ} \times \sin 60^\circ = \frac{3 \times \sin 60^\circ}{\sin 30^\circ}$ However, we are directly interested in expressing $\overline{KL}$ using the known angles.

2. Using the angle $J$: Since $\sin(90^\circ - 60^\circ) = \sin 30^\circ$: $\overline{KL} = \frac{3}{\sin 30^\circ} \times \sin 60^\circ$ Alternatively, the expression $\frac{3}{\sin(90^\circ - 60^\circ)}$ simplifies to: $\overline{KL} = \frac{3}{\sin 30^\circ}$

3. Considering the angle $K$: Similarly, we can use side $KJ$ to find $\overline{KL}$: $\frac{\overline{KL}}{\sin 30^\circ} = \frac{5.2}{\sin 60^\circ}$ Rearranging gives: $\overline{KL} = \frac{5.2 \times \sin 30^\circ}{\sin 60^\circ}$ Which matches the form: $\overline{KL} = \frac{5.2}{\sin(90^\circ - 60^\circ)}$

Based on these analyses, the correct expressions that represent the length of side $\overline{KL}$ are:

• Choice C: $\dfrac{3}{\sin(90^\circ-60^\circ)}$
• Choice D: $\dfrac{5.2}{\sin(90^\circ-60^\circ)}$

Would you like more details on how this is derived, or do you have any further questions?

Related Questions:

1. How does the Law of Sines apply to non-right triangles?
2. What is the relationship between the angles in any triangle?
3. How do you determine which trigonometric ratio to use in different triangle problems?
4. What are some real-world applications of the Law of Sines?
5. How does the Law of Cosines differ from the Law of Sines?

Tip:

Remember that the Law of Sines is particularly useful when dealing with non-right triangles, or when you know an angle and the side opposite it.