smc-4553-10-Cosine Rule

Trigonometry, SMB-067

Find the value of \(\theta\), correct to the nearest minute. (3 marks)

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\(\theta=112^{\circ} 12^{′}\)

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\(\cos A\)	\(= \dfrac{b^2+c^2-a^2}{2bc} \)
\(\cos \theta\)	\(= \dfrac{7.8^2+10.2^2-15.0^2}{2 \times 7.8 \times 10.2}\)
	\(= -0.3778…\)
\(\therefore \theta\)	\(= \cos^{-1} (-0.3778…) \)
	\(=112.199…^{\circ} \)
	\(=112^{\circ} 12^{′}\ \ \text{(nearest minute)} \)

Trigonometry, SMB-066

Find the value of \(\theta\), correct to the nearest degree. (2 marks)

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\(\theta=35^{\circ}\)

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\(\cos A\)	\(= \dfrac{b^2+c^2-a^2}{2bc} \)
\(\cos \theta\)	\(= \dfrac{16.2^2+18.1^2-10.5^2}{2 \times 16.2 \times 18.1}\)
	\(= 0.818…\)
\(\therefore \theta\)	\(= \cos^{-1} (0.818…) \)
	\(=35.09…^{\circ} \)
	\(=35^{\circ}\ \ \text{(nearest degree)} \)

Trigonometry, SMB-065

Find the value of \(\theta\), correct to 1 decimal place. (2 marks)

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\(\theta=38.9^{\circ}\)

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\(\cos A\)	\(= \dfrac{b^2+c^2-a^2}{2bc} \)
\(\cos \theta\)	\(= \dfrac{9^2+5^2-6^2}{2 \times 9 \times 5}\)
	\(= 0.777…\)
\(\therefore \theta\)	\(= \cos^{-1} (0.777…) \)
	\(=38.94…^{\circ} \)
	\(=38.9^{\circ}\ \ \text{(1 d.p.)} \)

Trigonometry, SMB-064

Find the value of \(d\), correct to 2 decimal places. (3 marks)

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\(d=12.9\ \text{km}\)

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\(a^2\)	\(=b^2+c^2-2bc\ \cos A \)
\(d^2\)	\(= 8.9^2+5.2^2-2 \times 8.9 \times 5.2 \times \cos 130^{\circ}\)
	\(= 165.746…\)
\(\therefore d\)	\(=12.87…\)
	\(=12.9\ \text{km (1 d.p.)} \)

Trigonometry, SMB-063

Find the value of \(x\), correct to 1 decimal place. (3 marks)

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\(x=15.4\ \text{cm}\)

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\(a^2\)	\(=b^2+c^2-2bc\ \cos A \)
\(x^2\)	\(= 15^2+25^2-2 \times 15 \times 25 \times \cos 35^{\circ}\)
	\(= 235.63…\)
\(\therefore x\)	\(=15.35…\)
	\(=15.4\ \text{cm (1 d.p.)} \)

Trigonometry, SMB-062

Find the value of \(x\), correct to 1 decimal place. (3 marks)

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\(x=8.8\ \text{m}\)

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\(a^2\)	\(=b^2+c^2-2bc\ \cos A \)
\(x^2\)	\(= 5^2+9^2-2 \times 5 \times 9 \times \cos 72^{\circ}\)
	\(= 78.188…\)
\(\therefore x\)	\(=8.842…\)
	\(=8.8\ \text{m (1 d.p.)} \)

Measurement, STD2 M6 2018 HSC 12 MC

The diagram shows a triangle with side lengths 8 m, 9 m and 10m.

What is the value of `theta`, marked on the diagram, to the nearest degree?

49°
51°
59°
72°

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`text(D)`

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`text(Using the cosine rule:)`

`costheta`	`= (8^2 + 9^2 – 10^2)/(2 xx 8 xx 9)`
	`= 0.3125`
`:.theta`	`= cos^(−1)(0.3125)`
	`= 71.790…^@`

`=>D`

Measurement, STD2 M6 2006 HSC 24b

A 130 cm long garden rake leans against a fence. The end of the rake is 44 cm from the base of the fence.

If the fence is vertical, find the value of `theta` to the nearest degree. (2 marks)

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The fence develops a lean and the rake is now at an angle of 53° to the ground. Calculate the new distance (`x` cm) from the base of the fence to the head of the rake. Give your answer to the nearest centimetre. (2 marks)

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