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O-Level The theorem of Pythagoras and trigonometry
What the O-Level syllabus expects for The theorem of Pythagoras and trigonometry, and how to practise it.
What the syllabus expects
- putting Pythagoras' theorem to use
- deciding, from a triangle's three side lengths, whether one of its angles is a right angle
- applying sine, cosine and tangent, the trigonometric ratios of an acute angle, to determine missing sides and missing angles inside right-angled triangles
- widening the definitions of sine and cosine so that obtuse angles are covered
- finding the area of a triangle by means of the formula ½ ab sin C
- employing both the sine rule and the cosine rule on an arbitrary triangle
- problems in two or three dimensions, some involving bearings or angles of elevation and of depression
Worked examples
Example 1 (3 marks)
With PR = 470 m, QR = 318 m and cos θ = −24/25 (θ = angle PRQ), (c) hence work out the length PQ.
Show the worked answer
Method: PQ is the side opposite the known angle PRQ, and the two sides enclosing that angle (RP and RQ) are both known, so this is the cosine rule. Step 1 - write the cosine rule with the correct side opposite the correct angle. The angle θ = angle PRQ sits at R, between RP = 470 m and RQ = 318 m. The side facing it is PQ, so PQ² = PR² + QR² − 2 × PR × QR × cos θ Step 2 - substitute the given values, including cos θ = −24/25. PQ² = 470² + 318² − 2 × 470 × 318 × (−24/25) Step 3 - work out each piece. 470² = 220900 318² = 101124 220900 + 101124 = 322024 2 × 470 × 318 = 298920 298920 × (24/25) = 286963.2 Step 4 - watch the sign. The cosine is negative, so the term being subtracted is negative, and subtracting a negative adds. − 2 × 470 × 318 × (−24/25) = + 286963.2 PQ² = 322024 + 286963.2 = 608987.2 (This makes sense: θ is obtuse, so PQ must be longer than either enclosing side.) Step 5 - square-root and round. PQ = √608987.2 = 780.37... PQ ≈ 780 m (3 s.f.)
More O-Level Elementary Maths (E-Maths) topics
Whole numbers and the operations performed on them · Ratio and proportion · Percentage · Rate and speed · Algebraic expressions and formulae · Functions together with their graphs · all of O-Level Elementary Maths (E-Maths)