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Welcome to GCSE Edexcel Maths revision.

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Topic M 32: Pythagoras and right-angle trigonometry.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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In a right-angled triangle, a squared plus b squared equals c squared , where c is the hypotenuse opposite the right angle.

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The theorem does not apply directly to a non-right-angled triangle.

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A three four five right triangle

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To find the hypotenuse,

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add the other squared sides and square root: for legs 3 centimetres and 4 centimetres,

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c equals square root of open bracket 9 plus 16 close bracket equals 5 centimetres.

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To find a shorter side, subtract before taking the square root: a equals square root of open bracket c squared minus b squared close bracket .

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If c equals 13 centimetres and b equals 5 centimetres, a equals square root of 144 equals 12 centimetres.

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Pythagoras also gives distance between coordinates: length equals square root of open bracket open bracket x subscript open bracket 2 close bracket minus x subscript open bracket 1 close bracket close bracket squared plus open bracket y subscript open bracket 2 close bracket minus y subscript open bracket 1 close bracket close bracket squared close bracket .

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From open bracket 1, 2 close bracket to open bracket 4, 6 close bracket , the distance is 5 units.

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Relative to the chosen angle theta , label opposite, adjacent and hypotenuse.

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Opposite and adjacent swap when the angle changes; the hypotenuse stays opposite the right angle.

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S O H,

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C A H,

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T O A means sine theta equals the fraction with numerator open bracket opposite close bracket and denominator open bracket hypotenuse close bracket ,

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end fraction ,

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cosine theta equals the fraction with numerator open bracket adjacent close bracket and denominator open bracket hypotenuse close bracket ,

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end fraction ,

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and tangent theta equals the fraction with numerator open bracket opposite close bracket and denominator open bracket adjacent close bracket ,

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end fraction .

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Choose the ratio using the known and required sides.

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Worked example: Opposite x,

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hypotenuse 10 centimetres and angle 30 degrees give sine 30 degrees equals the fraction with numerator open bracket x close bracket and denominator open bracket 10 close bracket ,

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end fraction ,

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so x equals 5 centimetres.

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Worked example: Adjacent 7 centimetres and angle 40 degrees with hypotenuse h give cosine 40 degrees equals the fraction with numerator open bracket 7 close bracket and denominator open bracket h close bracket ,

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end fraction ,

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so h equals the fraction with numerator open bracket 7 close bracket and denominator open bracket cosine 40 degrees close bracket ,

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end fraction is approximately equal to 9.14 centimetres.

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To find an angle, use the inverse trig key.

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Opposite 3 and adjacent 4 give theta equals inverse tangent open bracket 3 over 4 close bracket is approximately equal to 36.9 degrees.

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Here the inverse tangent key returns an angle; it does not calculate the reciprocal of tangent.

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Set the calculator to degrees.

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Check that a hypotenuse is the longest side and that each acute angle in a right triangle is between 0 degrees and 90 degrees.

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Exact sine values for 0 degrees,

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30 degrees,

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45 degrees,

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60 degrees,

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90 degrees are 0,

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1 over 2 ,

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the fraction with numerator open bracket square root of 2 close bracket and denominator open bracket 2 close bracket ,

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end fraction ,

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the fraction with numerator open bracket square root of 3 close bracket and denominator open bracket 2 close bracket ,

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end fraction ,

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1.

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Cosine values in the same order are 1,

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the fraction with numerator open bracket square root of 3 close bracket and denominator open bracket 2 close bracket ,

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end fraction ,

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the fraction with numerator open bracket square root of 2 close bracket and denominator open bracket 2 close bracket ,

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end fraction ,

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1 over 2 ,

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0.

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Exact tangent values for 0 degrees,

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30 degrees,

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45 degrees,

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60 degrees are 0,

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the fraction with numerator open bracket square root of 3 close bracket and denominator open bracket 3 close bracket ,

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end fraction ,

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1,

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square root of 3. tangent 90 degrees is undefined.

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The 45 degrees and 30 degrees and 60 degrees special triangles explain these values.

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Exact-value right triangles

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Angles of elevation or depression are measured from horizontal lines, not vertical ones.

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Sketch a right triangle, mark the angle and include any observer height in the final total if needed.

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Find a useful right triangle inside the solid, often using Pythagoras once for a face diagonal and again for a space diagonal.

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Keep the face diagonal exact during the calculation.

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Worked example: A cuboid 3 centimetres by 4 centimetres by 12 centimetres has base diagonal 5 centimetres and space diagonal square root of open bracket 5 squared plus 12 squared close bracket equals 13 centimetres.

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The angle the space diagonal makes with the base is inverse tangent open bracket 12 over 5 close bracket is approximately equal to 67.4 degrees.

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The space diagonal, its 5 centimetres projection on the base and the 12 centimetres vertical edge form the right triangle used in this example.

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Label that triangle separately from the solid if the perspective view is confusing.

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Right triangle for a cuboid space diagonal

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For an angle between a line and a plane, use the angle between the line and its projection onto the plane.

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A three-dimensional sketch alone may make the relevant right angle hard to see.

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That completes Pythagoras and right-angle trigonometry.

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Revisit the notes and test yourself on the revision website.