// Numbas version: exam_results_page_options {"name": "Trigonometry: Cosine Rule 1", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry: Cosine Rule 1", "tags": [], "metadata": {"description": "

Calculating the missing side-length of a triangle using the cosine rule.

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Find the length of $c$:

{geogebra_applet('https://www.geogebra.org/m/pnfanbze',defs)}

", "advice": "

Since we know two side-lengths and the angle between them, we can use the cosine rule to find the third length, $c$. The cosine rules states:

\\[ c^2=a^2+b^2-2ab\\cos(C) \\]

where $a$, $b$, and $c$ are the side-lengths, and $C$ is the angle between $a$ and $b$:

{geogebra_applet('https://www.geogebra.org/m/pg7d3hvz')}

So, for the triangle

{geogebra_applet('https://www.geogebra.org/m/pnfanbze',defs)}

we know that side $a=\\var{sidea}$, side $b=\\var{sideb}$, and angle $C=\\var{anglecr}^\\circ$. Substituting these values into the cosine rule we can calculate the length of $c$:

\\[ \\begin{split} c^2 &\\,= \\var{sidea}^2+\\var{sideb}^2-2 \\times \\var{sidea}\\times\\var{sideb} \\times \\cos(\\var{anglecr}) \\\\ &\\,=\\var{sidecr^2} \\\\ \\implies c &\\,=\\var{sidecr} \\text{ (2 d.p.)}. \\end{split} \\]

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$c=$[[0]]

(Give your answer to 2 decimal places when necessary)

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