Use the cosine rule to find the two possible values of x in the diagramto the righ

It appears that you are referring to a diagram that involves a triangle. However, since I cannot see the diagram, I’ll provide a step-by-step guideline on how to use the cosine rule to find the two possible values of $x$ given a generic triangle situation.

1. Break Down the Problem

Identify the sides and angles of the triangle, particularly:

• The lengths of two sides.
• The angle between the sides.
• The side opposite the angle, which is denoted as $x$.

2. Relevant Concepts

The cosine rule states:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

Where:

• $c$ is the length of the side opposite angle $C$,
• $a$ and $b$ are the lengths of the other two sides.

3. Analysis and Detail

1. Substitute the known values into the cosine rule formula to find the length of the side $x$ (if $x$ corresponds to $c$ in the formula).
2. Rearrange and solve for $x$:

$x = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)}$

3. This yields one possible value for $x$. The cosine function has two angles that can yield the same cosine value, therefore:

$x_1 = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)}$ $x_2 = \sqrt{a^2 + b^2 - 2ab \cdot \cos(180^\circ - C)} \quad \text{(to find the second possible value)}$

4. Verify and Summarize

Check both values by using the cosine rule again to confirm they work in the context of the triangle. Generally, both values should satisfy the triangle inequality property.

Final Answer

The two possible values of $x$ can be expressed using the cosine rule as derived above:

• $x_1 = \sqrt{a^2 + b^2 - 2ab \cdot \cos(C)}$
• $x_2 = \sqrt{a^2 + b^2 + 2ab \cdot \cos(C)}$

You will need to plug in the specific values for $a$, $b$, and $C$ from your diagram to obtain the numerical results for $x_1$ and $x_2$.