question 2: find the lengths of the sides labelled x below.
(a) a right-angled triangle with vertical side 10cm, horizontal side x, and angle 35 degrees adjacent to side x.
(b) a right-angled triangle with hypotenuse x, horizontal side 8cm, and angle 40 degrees opposite to side 8cm.
(c) a right-angled triangle with vertical side x, hypotenuse 6cm, and angle 70 degrees adjacent to side x.
(d) a right-angled triangle with horizontal side 18cm, vertical side x, and angle 22 degrees adjacent to side 18cm.
(e) a right-angled triangle with horizontal side 2.8cm, vertical side x, and angle 31 degrees opposite to side 2.8cm.
(f) a right-angled triangle with hypotenuse 75cm, horizontal side x, and angle 55 degrees opposite to side x.
(g) a right-angled triangle with vertical side x, hypotenuse 50cm, and angle 60 degrees adjacent to side x.
(h) a right-angled triangle with hypotenuse x, side 28cm, and angle 48 degrees adjacent to side 28cm.
(i) a right-angled triangle with vertical side 17cm, horizontal side x, and angle 28 degrees opposite to side x.
(j) a right-angled triangle with hypotenuse 4km, horizontal side x, and angle 60 degrees opposite to side x.
(k) a right-angled triangle with vertical side 15cm, hypotenuse x, and angle 75 degrees adjacent to side 15cm.
(l) a right-angled triangle with hypotenuse 1.3m, horizontal side x, and angle 45 degrees adjacent to side x.

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{ "quality": "clear", "question_count": 12, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Right-angled Trigonometry", "SOH CAH TOA" ], "new_concepts": [], "current_concepts": [ "Right-angled Trigonometry", "SOH CAH TOA" ] } ## Identify trigonometric ratios for (a) to (d) $$ \begin{aligned} &\text{(a) } \cos(35^\circ) = \frac{x}{10} \implies x = 10 \cos(35^\circ) \approx 8.19\text{ cm}\ &\text{(b) } \sin(40^\circ) = \frac{8}{x} \implies x = \frac{8}{\sin(40^\circ)} \approx 12.45\text{ cm}\ &\text{(c) } \cos(70^\circ) = \frac{x}{6} \implies x = 6 \cos(70^\circ) \approx 2.05\text{ cm}\ &\text{(d) } \tan(22^\circ) = \frac{x}{18} \implies x = 18 \tan(22^\circ) \approx 7.27\text{ cm} \end{aligned} $$ ## Identify trigonometric ratios for (e) to (h) $$ \begin{aligned} &\text{(e) } \cos(31^\circ) = \frac{x}{2.8} \implies x = 2.8 \cos(31^\circ) \approx 2.40\text{ cm}\ &\text{(f) } \sin(55^\circ) = \frac{x}{75} \implies x = 75 \sin(55^\circ) \approx 61.44\text{ cm}\ &\text{(g) } \sin(60^\circ) = \frac{x}{50} \implies x = 50 \sin(60^\circ) \approx 43.30\text{ cm}\ &\text{(h) } \cos(48^\circ) = \frac{28}{x} \implies x = \frac{28}{\cos(48^\circ)} \approx 41.84\text{ cm} \end{aligned} $$ ## Identify trigonometric ratios for (i) to (l) $$ \begin{aligned} &\text{(i) } \tan(28^\circ) = \frac{x}{17} \implies x = 17 \tan(28^\circ) \approx 9.04\text{ cm}\ &\text{(j) } \sin(60^\circ) = \frac{x}{4} \implies x = 4 \sin(60^\circ) \approx 3.46\text{ km}\ &\text{(k) } \cos(75^\circ) = \frac{15}{x} \implies x = \frac{15}{\cos(75^\circ)} \approx 57.96\text{ cm}\ &\text{(l) } \cos(45^\circ) = \frac{x}{1.3} \implies x = 1.3 \cos(45^\circ) \approx 0.92\text{ m} \end{aligned} $$ | Part | Equation | Length $x$ (to 2 d.p.) | |---|---|---| | (a) | $x = 10 \cos(35^\circ)$ | $8.19\text{ cm}$ | | (b) | $x = \frac{8}{\sin(40^\circ)}$ | $12.45\text{ cm}$ | | (c) | $x = 6 \cos(70^\circ)$ | $2.05\text{ cm}$ | | (d) | $x = 18 \tan(22^\circ)$ | $7.27\text{ cm}$ | | (e) | $x = 2.8 \cos(31^\circ)$ | $2.40\text{ cm}$ | | (f) | $x = 75 \sin(55^\circ)$ | $61.44\text{ cm}$ | | (g) | $x = 50 \sin(60^\circ)$ | $43.30\text{ cm}$ | | (h) | $x = \frac{28}{\cos(48^\circ)}$ | $41.84\text{ cm}$ | | (i) | $x = 17 \tan(28^\circ)$ | $9.04\text{ cm}$ | | (j) | $x = 4 \sin(60^\circ)$ | $3.46\text{ km}$ | | (k) | $x = \frac{15}{\cos(75^\circ)}$ | $57.96\text{ cm}$ | | (l) | $x = 1.3 \cos(45^\circ)$ | $0.92\text{ m}$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Right-angled Trigonometry" ] }

Part	Equation	Length \(x\) (to 2 d.p.)
(b)	\(x = \frac{8}{\sin(40^\circ)}\)	\(12.45\text{ cm}\)
(c)	\(x = 6 \cos(70^\circ)\)	\(2.05\text{ cm}\)
(d)	\(x = 18 \tan(22^\circ)\)	\(7.27\text{ cm}\)
(e)	\(x = 2.8 \cos(31^\circ)\)	\(2.40\text{ cm}\)
(f)	\(x = 75 \sin(55^\circ)\)	\(61.44\text{ cm}\)
(g)	\(x = 50 \sin(60^\circ)\)	\(43.30\text{ cm}\)
(h)	\(x = \frac{28}{\cos(48^\circ)}\)	\(41.84\text{ cm}\)
(i)	\(x = 17 \tan(28^\circ)\)	\(9.04\text{ cm}\)
(j)	\(x = 4 \sin(60^\circ)\)	\(3.46\text{ km}\)
(k)	\(x = \frac{15}{\cos(75^\circ)}\)	\(57.96\text{ cm}\)
(l)	\(x = 1.3 \cos(45^\circ)\)	\(0.92\text{ m}\)

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Explanation:

Identify trigonometric ratios for (a) to (d)

Identify trigonometric ratios for (e) to (h)

Identify trigonometric ratios for (i) to (l)

Answer: