Question

question
in $\delta hij$, $h = 6.8$ cm, $i = 3.3$ cm and $j = 6.6$ cm. find the measure of $\angle j$ to the nearest 10th of a degree.

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). In \(\triangle HIJ\), to find \(\angle J\), the sides adjacent to \(\angle J\) are \(i = 3.3\) cm and \(h=6.8\) cm, and the side opposite \(\angle J\) is \(j = 6.6\) cm. So we can apply the Law of Cosines: \(j^{2}=h^{2}+i^{2}-2hi\cos(J)\).

Step2: Rearrange the formula to solve for \(\cos(J)\)

\[

$$\begin{align*} 2hi\cos(J)&=h^{2}+i^{2}-j^{2}\\ \cos(J)&=\frac{h^{2}+i^{2}-j^{2}}{2hi} \end{align*}$$

\]

Step3: Substitute the given values

Substitute \(h = 6.8\), \(i=3.3\), and \(j = 6.6\) into the formula:
\[

$$\begin{align*} \cos(J)&=\frac{6.8^{2}+3.3^{2}-6.6^{2}}{2\times6.8\times3.3}\\ &=\frac{46.24 + 10.89- 43.56}{44.88}\\ &=\frac{57.13 - 43.56}{44.88}\\ &=\frac{13.57}{44.88} \end{align*}$$

\]

Step4: Calculate the value of \(\cos(J)\)

\(\frac{13.57}{44.88}\approx0.3024\)

Step5: Find the angle \(J\)

Take the inverse cosine (arccos) of \(0.3024\) to find \(\angle J\): \(\angle J=\arccos(0.3024)\approx72.4^{\circ}\) (rounded to the nearest tenth of a degree)

Answer:

\(72.4^{\circ}\)