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$A=\\var{AA0}$, $B=\\var{BB0}$, $a=\\var{a0}$

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Side length $b=$ [[0]]

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Angle $C=$ [[1]]

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Side length $c=$ [[2]]

", "steps": [{"type": "information", "prompt": "

Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.

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$B=\\var{BB3}$, $C=\\var{CC3}$, $c=\\var{c3}$

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Side length $b=$ [[0]]

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Angle $A=$ [[1]]

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Side length $a=$ [[2]]

\n \n \n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "

Suppose that $\\Delta ABC$ is a triangle with $C> \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).

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Given the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.

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", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "Solving triangles", "Triangle", "Two angles and a side"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

I want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2<c^2<(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$. 

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", "licence": "Creative Commons Attribution 4.0 International", "description": "

Two questions testing the application of the Sine Rule when given two angles and a side. In this question the triangle is obtuse. In one question, the two given angles are both acute. In the second, one of the angles is obtuse.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

a) We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t0}}{\\var{s0}}=\\var{a0*t0/s0}$. The closest integer is then $\\var{b0}$.

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Since $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC1}$. To 3dp, this gives $\\var{CC2}$.

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We use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u2}}{\\var{s0}}=\\var{a0*u2/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC1}$ would give a slightly different long decimal value of $c$, but the integer value would be the same.

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b) We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t3}}{\\var{u3}}=\\var{c3*t3/u3}$. The closest integer is then $\\var{b3}$.

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Since $A+B+C=\\pi$, we calculate $A=\\pi-B-C=\\var{AA4}$. To 3dp, this gives $\\var{AA5}$.

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We use the Sine Rule to find $a$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $a=\\dfrac{c \\sin A}{\\sin C}=\\dfrac{\\var{c3}* \\var{s5}}{\\var{u3}}=\\var{c3*s5/u3}$. The closest integer is then $\\var{a3}$. Note that this solution uses the 3dp value of $A$; the answer using $\\var{AA4}$ would give a slightly different long decimal value of $a$, but the integer value would be the same.

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