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3.7 Pythagoras and trigonometry

Pythagoras and trigonometry

Everything on pythagoras and trigonometry that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.

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A worked example

Distance between points · [3 marks]

A is the point (0, 0) and B is the point (9, 2). Work out the length of AB. Give your answer to 1 decimal place.

  1. M1Across 9, up 2
  2. M19² + 2² = 85
  3. A19.2

The exam skills (9)

Pythagoras

  1. The hypotenuse is the longest side, opposite the right angle.
  2. Square the two shorter sides and ADD.
  3. Square root the answer.

Watch out: Forgetting to square root at the end.

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Distance between points

  1. Find how far across and how far up between the points.
  2. Use Pythagoras: across² + up².
  3. Square root.

Watch out: Getting a negative difference and squaring it wrongly. A negative squared is positive.

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Trigonometry

  1. Label the sides: hypotenuse, opposite and adjacent to the angle.
  2. Choose SOH, CAH or TOA using the side you have and the side you want.
  3. Rearrange and use your calculator (in degrees).

Watch out: Calculator in radians. Check you see a D on the screen.

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Exact trig values

  1. Learn the table for 0°, 30°, 45°, 60°, 90°.
  2. sin 30° = cos 60° = 1/2. tan 45° = 1.
  3. sin 60° = cos 30° = √3/2, sin 45° = cos 45° = √2/2.

Watch out: Mixing up sin 30° and sin 60°.

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Elevation and depression

  1. Angles of elevation are measured up from the horizontal; depression is down from the horizontal.
  2. Draw the right-angled triangle and label O, A and H.
  3. Use SOH CAH TOA. Depression from the top equals elevation from the bottom (alternate angles).

Watch out: Measuring the angle from the vertical instead of the horizontal.

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Sine rule

  1. Use the sine rule when you have a side and its opposite angle.
  2. a/sin A = b/sin B.
  3. Put in the three things you know and solve.

Watch out: Pairing a side with an angle that is not opposite it.

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Cosine rule

  1. Use it for a side when you have two sides and the angle between them.
  2. a² = b² + c² − 2bc cos A.
  3. Work out the right-hand side, then square root.

Watch out: Typing it in so the calculator does b² + c² − 2bc first and then multiplies by cos A. Use brackets or do it in parts.

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Area of a triangle

  1. Area = ½ab sin C.
  2. C is the angle between the two sides a and b.
  3. Work it out and give units squared.

Watch out: Using an angle that is not between the two sides.

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3D Pythagoras

  1. Find the diagonal of the base first with Pythagoras.
  2. Use that diagonal and the height in Pythagoras again.
  3. Or in one go: √(a² + b² + c²).

Watch out: Rounding the base diagonal too early.

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