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Welcome to GCSE Edexcel Maths revision.

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Topic M 34: Trigonometry in general triangles.

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This video covers Higher tier.

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For a general triangle, label side a opposite angle A, b opposite B and c opposite C.

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Matching each side to its opposite angle is essential.

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Opposite side and angle labels

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The sine rule is the fraction with numerator open bracket a close bracket and denominator open bracket sine A close bracket ,

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end fraction equals the fraction with numerator open bracket b close bracket and denominator open bracket sine B close bracket ,

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end fraction equals the fraction with numerator open bracket c close bracket and denominator open bracket sine C close bracket ,

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end fraction .

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It is useful when an opposite side-angle pair and another side or angle are known.

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Worked example: A equals 30 degrees, B equals 45 degrees, a equals 6 centimetres.

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Then b equals the fraction with numerator open bracket 6 sine 45 degrees close bracket and denominator open bracket sine 30 degrees close bracket ,

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end fraction equals 6 multiplied by square root of 2 centimetres is approximately equal to 8.49 centimetres.

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The larger angle faces the longer side.

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To find an angle using the sine rule,

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rearrange sine B equals the fraction with numerator open bracket b sine A close bracket and denominator open bracket a close bracket ,

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end fraction and use inverse sine.

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Check whether the supplementary angle also fits the information.

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The sine-rule ambiguous case can give two triangles when two sides and a non-included angle are known.

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An acute calculator answer is not always the only valid angle; use the triangle's angle sum and context.

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The cosine rule is a squared equals b squared plus c squared minus 2 b c cosine A.

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Use it for two sides and their included angle, or to find an angle from three sides.

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Worked example: Sides b equals 5 centimetres, c equals 7 centimetres enclose A equals 60 degrees.

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Then a squared equals 25 plus 49 minus 70 multiplied by 1 over 2 equals 39, so a equals square root of 39 centimetres is approximately equal to 6.24 centimetres.

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For an angle,

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cosine A equals the fraction with numerator open bracket b squared plus c squared minus a squared close bracket and denominator open bracket 2 b c close bracket ,

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end fraction .

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Use the side opposite A in the subtracted term.

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Worked example: For a equals 7,

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b equals 5,

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c equals 6,

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cosine A equals the fraction with numerator open bracket 25 plus 36 minus 49 close bracket and denominator open bracket 60 close bracket ,

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end fraction equals 0.2,

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so A is approximately equal to 78.5 degrees.

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Pythagoras is the special case A equals 90 degrees, where cosine A equals 0.

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The cosine rule extends it to acute and obtuse angles.

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Area of a triangle is 1 over 2 a b sine C, where C is the angle included between sides a and b.

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A non-included angle cannot be substituted in this form.

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Worked example: Two sides 8 centimetres and 5 centimetres enclose 30 degrees.

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Area equals 1 over 2 multiplied by 8 multiplied by 5 multiplied by sine 30 degrees equals 10 square centimetres .

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To find an angle from the area formula,

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sine C equals the fraction with numerator open bracket 2 multiplied by area close bracket and denominator open bracket a b close bracket ,

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end fraction .

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Consider C and 180 degrees minus C if both fit the stated triangle.

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For bearing problems, first derive the interior triangle angles from north lines.

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Then use a suitable sine or cosine rule, and convert back to a three-digit bearing if requested.

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In three-dimensional problems, first find relevant lengths or plane angles using right triangles, then apply a general-triangle rule if necessary.

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Keep intermediate values unrounded and use degrees.

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That completes Trigonometry in general triangles.

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Revisit the notes and test yourself on the revision website.