Question

the cosine law:
$c^{2}=a^{2}+b^{2}-2ab(\cos c)$ or $\cos c = \frac{a^{2}+b^{2}-c^{2}}{2ab}$
knowledge

1. calculate the length of side using the cosine law. show your work. 3 marks

(there is a triangle with sides 6.5, 5 and angle 72° between them, and side x opposite? or as per the handwritten and diagram: sides 6.5, 8? wait, the diagram has 6.5, 5 and angle 72°, and side x. then handwritten work: $x^{2}=6.6^{2}+8^{2}-2(6.6)(8)\cos72°$ etc. then 2. calculate the measure of angle x using the cosine law. show your work. marks? diagram: triangle with sides 7 cm (maybe), 9 cm, 10 cm, angle x between 7 and 9 cm. handwritten: $\cos x = \frac{7^{2}+9^{2}-10^{2}}{2(7)(9)}$, $\cos x = \frac{49 + 81 - 100}{126}$, $\cos x = \frac{30}{126}$, then $x = \cos^{-1}(\frac{30}{...})$

Explanation:

Step1: Identify the formula

We use the Cosine Law formula \( c^{2}=a^{2}+b^{2}-2ab(\cos C) \), where \( a = 6.5 \), \( b = 5 \), and \( C = 72^{\circ} \), and \( c=x \).

Step2: Substitute the values

Substitute \( a = 6.5 \), \( b = 5 \), and \( C = 72^{\circ} \) into the formula:
\( x^{2}=6.5^{2}+5^{2}-2\times6.5\times5\times\cos(72^{\circ}) \)
First, calculate \( 6.5^{2}=42.25 \), \( 5^{2} = 25 \), and \( 2\times6.5\times5=65 \).
\( \cos(72^{\circ})\approx0.3090 \)
So \( x^{2}=42.25 + 25-65\times0.3090 \)
\( x^{2}=67.25 - 20.085 \)
\( x^{2}=47.165 \)

Step3: Solve for \( x \)

Take the square root of both sides: \( x=\sqrt{47.165}\approx6.87 \) (Wait, there was a mistake in the original problem's side length, the side should be 5 instead of 8? Let's re - check. If the two sides are 6.5 and 5, and the included angle is \( 72^{\circ} \))
Wait, maybe the original problem had a typo, but following the correct Cosine Law:
If we assume the two sides are \( a = 6.5 \), \( b = 5 \), included angle \( C=72^{\circ} \)
\( x^{2}=6.5^{2}+5^{2}-2\times6.5\times5\times\cos72^{\circ} \)
\( 6.5^{2}=42.25 \), \( 5^{2}=25 \), \( 2\times6.5\times5 = 65 \), \( \cos72^{\circ}\approx0.3090 \)
\( x^{2}=42.25 + 25-65\times0.3090=67.25 - 20.085 = 47.165 \)
\( x=\sqrt{47.165}\approx6.87 \)

For the second problem (angle \( X \)):

Step1: Identify the formula

Use the Cosine Law formula for angle: \( \cos X=\frac{a^{2}+b^{2}-c^{2}}{2ab} \), where \( a = 7 \), \( b = 9 \), \( c = 10 \)

Step2: Substitute the values

\( \cos X=\frac{7^{2}+9^{2}-10^{2}}{2\times7\times9}=\frac{49 + 81-100}{126}=\frac{30}{126}=\frac{5}{21}\approx0.2381 \)

Step3: Find the angle

\( X=\cos^{-1}(0.2381)\approx76.2^{\circ} \)

Answer:

For the first problem (length of \( x \)): \( x\approx6.87 \) (if sides are 6.5 and 5 with included angle \( 72^{\circ} \)); For the second problem (measure of angle \( X \)): \( X\approx76.2^{\circ} \)