Trigonometry Task 1: Knowledge Quiz
Time Limit: 1:15:00
Time Left:1:14:22
Zahree Campbell: Attempt 2
Page 1:
$13014 \quad 15$
\begin{tabular}{|l|l|l|} \hline 16 & 17 & 18 \\ -- & -- \\ \hline \end{tabular}
$19 \quad 20 \quad 21$
A student has been asked to solve for $\theta$ as indicated in the following diagram:
The student has written the following on their paper:
\[
\cos \theta=\frac{12^{2}+8^{2}-11^{2}}{2(12)(8)}
\]
Is the student correct?
Yes
No

Step 1 ：The student is trying to use the law of cosines to solve for the angle \(\theta\). The law of cosines states that for any triangle with sides of lengths a, b, and c, and an angle \(\theta\) opposite the side of length c, the following equation holds: \(c^2 = a^2 + b^2 - 2ab\cos\theta\)

Step 2 ：In this case, the student has rearranged the equation to solve for \(\cos\theta\): \(\cos\theta = \frac{a^2 + b^2 - c^2}{2ab}\)

Step 3 ：The student has substituted 12 for a, 8 for b, and 11 for c. Therefore, the student's equation is correct if and only if the sides of the triangle are 12, 8, and 11, and \(\theta\) is the angle opposite the side of length 11.

Step 4 ：The value of \(\cos\theta\) calculated using the student's equation is approximately 0.453125. This is a valid value for \(\cos\theta\), as it falls within the range of -1 to 1. Therefore, the student's equation appears to be correct.

Step 5 ：Final Answer: \(\boxed{\text{Yes, the student is correct.}}\)