"},{"text":"AB = 11","location":"specification"},{"text":"ACB = 25°","location":"specification"}]},{"type":"text","order":1,"title":"Set up the equation","content":"Substitute our known values into the Cosine Rule formula:\n\n$$11^2 = 18^2 + b^2 - 2(18)(b) \\cos(25^\\circ)$$\n\nNow, let's simplify the terms:\n- $11^2 = 121$\n- $18^2 = 324$\n- $2 \\times 18 \\times \\cos(25^\\circ) \\approx 36 \\times 0.9063 = 32.627$\n\nSubstituting these values gives the quadratic equation:\n\n$$121 = 324 + b^2 - 32.627b$$\n\nRearranging into standard quadratic form $b^2 - 32.627b + 203 = 0$:\n\n$$b^2 - 32.627b + 203 = 0$$"},{"type":"text","order":2,"title":"Solve the quadratic equation","content":"We can solve the quadratic equation $b^2 - 32.627b + 203 = 0$ using the quadratic formula:\n\n$$b = \\frac{-(-32.627) \\pm \\sqrt{(-32.627)^2 - 4(1)(203)}}{2(1)}$$\n\n$$b = \\frac{32.627 \\pm \\sqrt{1064.52 - 812}}{2}$$\n\n$$b = \\frac{32.627 \\pm \\sqrt{252.52}}{2}$$\n\n$$b \\approx \\frac{32.627 \\pm 15.891}{2}$$\n\nThis gives us two possible lengths for side $AC$:\n- $b_1 \\approx \\frac{32.627 + 15.891}{2} \\approx 24.26$ cm\n- $b_2 \\approx \\frac{32.627 - 15.891}{2} \\approx 8.37$ cm"},{"type":"text","order":3,"title":"Select the correct length","content":"We are given that angle $BAC$ is **obtuse** (greater than $90^\\circ$).\n\nIn a triangle, the larger the angle opposite a side, the longer that side must be. However, since the angle $BAC$ is obtuse, the side opposite to it ($BC = 18$ cm) must be the longest side of the triangle. \n\nTherefore, the side $AC$ ($b$) must be shorter than $18$ cm. This means we choose the smaller value of $b$:\n\n$$b = AC \\approx 8.37\\text{ cm}$$","highlights":[{"text":"obtuse","location":"specification"}]},{"type":"text","order":4,"title":"Calculate angle BAC","content":"Now we apply the Cosine Rule again, but this time rearranged to find the cosine of angle $BAC$ (angle $A$):\n\n$$\\cos(A) = \\frac{b^2 + c^2 - a^2}{2bc}$$\n\nSubstitute the values $a = 18$, $b = 8.368$, and $c = 11$ into the formula:\n\n$$\\cos(A) = \\frac{8.368^2 + 11^2 - 18^2}{2 \\times 8.368 \\times 11}$$\n\n$$\\cos(A) = \\frac{70.02 + 121 - 324}{184.10}$$\n\n$$\\cos(A) = \\frac{-132.98}{184.10} \\approx -0.7223$$\n\nNow, take the inverse cosine to find angle $A$:\n\n$$A = \\cos^{-1}(-0.7223) \\approx 136.25^\\circ$$"},{"type":"text","order":5,"title":"Round to 3 sig figs","content":"Rounding our angle $136.25^\\circ$ to $3$ significant figures yields:\n\n$$BAC \\approx 136^\\circ$$","highlights":[{"text":"3 significant figures","location":"specification"}]}]}],"generatedAt":"2026-06-13T00:02:15.629Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":67}},{"id":"cc9d49cd-fb23-4f79-a3be-02ef67505b0e","revisionId":"1247e204-1106-4e79-a21f-bdaa1cf5ab15","questionNumber":null,"questionType":"LA","specification":"A surveyor is measuring a triangular plot of land $PQR$. The interior angle at $P$ is $75^\\circ$ and the interior angle at $R$ is $35^\\circ$. The side length $PQ$ is $15\\text{ m}$.","difficulty":4,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":"calc","parts":[{"id":"f03eef3f-885d-4dc4-9188-5c191d25645a","label":"a","content":"Work out the area of the triangular plot $PQR$.\nGive your answer to 1 decimal place.","markscheme":"- Uses sine rule to find side $QR$ ($x$): $\\frac{x}{\\sin(75)} = \\frac{15}{\\sin(35)} \\Rightarrow x \\approx 25.257...$ M1 A1\n- Finds angle $Q$: $180 - 75 - 35 = 70^\\circ$ B1\n- Uses area formula: $\\text{Area} = \\frac{1}{2} \\times 25.257... \\times 15 \\times \\sin(70)$ M1\n- Final answer: $178.0$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitute values correctly into the Sine Rule to find side QR.","code":"M"},{"id":"p2","marks":1,"description":"Calculate side QR correctly (approx 25.3).","code":"A"},{"id":"p3","marks":1,"description":"Calculate the missing interior angle Q (70 degrees)

The Sine Rule

Question 10

The Sine Rule Questions

The Sine Rule

Question 10

The Sine Rule Questions