$$ \left. \begin{array} { l } { 10 ^ { 2 } = ( 13 - 15 ) ^ { 2 } + ( 13 - 5 ) ^ { 2 } - 2 ( 8 ) ^ { 2 } \cos \angle ( 0 ) } \ { \cos \angle C O D ) } \end{array} ight. $$

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$$ \left. \begin{array} { l } { 10 ^ { 2 } = ( 13 - 15 ) ^ { 2 } + ( 13 - 5 ) ^ { 2 } - 2 ( 8 ) ^ { 2 } \cos \angle ( 0 ) } \ { \cos \angle C O D ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

To solve the problem, we need to analyze the equation given:

$$
10^2 = (13 - 15)^2 + (13 - 5)^2 - 2(8)^2 \cos \angle (0)
$$

Let's break this down step by step.

1. **Calculate $$10^2$$**:
   $$
   10^2 = 100
   $$

2. **Calculate $$(13 - 15)^2$$**:
   $$
   (13 - 15)^2 = (-2)^2 = 4
   $$

3. **Calculate $$(13 - 5)^2$$**:
   $$
   (13 - 5)^2 = (8)^2 = 64
   $$

4. **Calculate $$2(8)^2$$**:
   $$
   2(8)^2 = 2 \times 64 = 128
   $$

5. **Substituting these values into the equation**:
   $$
   100 = 4 + 64 - 128 \cos \angle (0)
   $$

6. **Combine the constants on the right side**:
   $$
   100 = 68 - 128 \cos \angle (0)
   $$

7. **Rearranging the equation to isolate $$\cos \angle (0)$$**:
   $$
   100 - 68 = -128 \cos \angle (0)
   $$
   $$
   32 = -128 \cos \angle (0)
   $$

8. **Dividing both sides by -128**:
   $$
   \cos \angle (0) = -\frac{32}{128} = -\frac{1}{4}
   $$

Thus, we find that:

$$
\cos \angle (0) = -\frac{1}{4}
$$

This is the value of $$\cos \angle C O D$$.
$$ \cos \angle C O D = -\frac{1}{4} $$

\( \left. \begin{array} { l } { 10 ^ { 2 } = ( 13 - 15 ) ^ { 2 } + ( 13 - 5 ) ^ { 2 } - 2 ( 8 ) ^ { 2 } \cos \angle ( 0 ) } \\ { \cos \angle C O D ) } \end{array} \right. \)

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