GCSE Trigonometry 2 SineCosine Rule Dr J Frost

GCSE: Trigonometry 2 – Sine/Cosine Rule Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 8 th December 2016

RECAP: Right-Angled Triangles We’ve previously been able to deal with right-angled triangles, to find the area, or missing sides and angles. 5 6 4 3? Area = 15 ? 5 3 30. 96° ? 5

Labelling Sides of Non-Right Angle Triangles Right-Angled Triangles: Non-Right-Angled Triangles: ? ? ?

OVERVIEW: Finding missing sides and angles You have You want #1: Two angle-side opposite pairs Missing angle or Sine rule side in one pair #2 Two sides known and a missing side opposite a known angle Remaining side Cosine rule #3 All three sides An angle Cosine rule #4 Two sides known Remaining side and a missing side not opposite known angle Use Sine rule twice

The Sine Rule b 5. 02 For this triangle, try calculating each side divided by the sin of its opposite angle. What do you notice in all three cases? c 65° A 10 85° C B 30° 9. 10 ? a You have You want Use #1: Two angle-side opposite pairs Missing angle or Sine rule side in one pair

Examples 8 Q 2 Q 1 85° 45° 8 50° 100° 30° ? 11. 27 ? 15. 76 You have You want Use #1: Two angle-side opposite pairs Missing angle or Sine rule side in one pair

Examples 5 Q 3 Q 4 126° 40. 33° ? 85° 56. 11° ? 6 10 8

Test Your Understanding ? ?

Exercise 1 Find the missing angle or side. Give answers to 3 sf. Q 1 Q 2 Q 6 Q 5 ? ? ? Q 4 Q 3 ?

Cosine Rule The sine rule could be used whenever we had two pairs of sides and opposite angles involved. However, sometimes there may only be one angle involved. We then use something called the cosine rule. How are sides labelled ? Calculation?

Sin or Cosine Rule? If you were given these exam questions, which would you use? Sine Cosine Sine Cosine Sine Cosine

Test Your Understanding e. g. 1 e. g. 2 ? ? You have You want Use Two sides known and a missing side opposite a known angle Remaining side Cosine rule

Dealing with Missing Angles You have You want Use All three sides An angle Cosine rule ? ? ? ? Label sides then substitute into formula. Simplify each bit of formula. Rearrange (I use ‘swapsie’ trick to swap thing you’re subtracting and result)

Test Your Understanding ? ?

Exercise 2 Use the cosine rule to determine the missing angle/side. Quickly copy out the diagram first. 1 2 ? 3 ? 5 4 ? 6 ? ? ?

Exercise 2 9 8 7 ? ? 10 ? 11 ? ?

Using sine rule twice You have You want #4 Two sides known Remaining side and a missing side not opposite known angle Use Sine rule twice ?

Using sine rule twice You have You want #4 Two sides known Remaining side and a missing side not opposite known angle ! Use Sine rule twice 2: Which means we would then know this angle. ? 1: We could use the sine rule to find this angle. ? ?

Test Your Understanding ? ?

Area of Non Right-Angled Triangles 3 cm Area = 0. 5 x 3 x 7 x sin(59) = 9. 00 cm 2? 59° 7 cm !

Test Your Understanding ? ?

Harder Examples Q 1 (Edexcel June 2014) ? Q 2 ?

Exercise 3 Find the area of each of these shapes. Q 3 Q 2 Q 1 Q 4 3. 6 100° 3. 8 5 75° 5. 2 Area = 7. 39? Area = 9. 04? ? Q 5 70° 2 cm Q 7 Q 6 Area = 8. 03? 3 cm Q 8 ? 4. 2 m 3 m ? 5. 3 m ? ?

Segment Area ? ? ?

Test Your Understanding ? ?

Where it gets more Further Maths-ey… You will frequently encounter either algebraic or surd sides. The approach is exactly the same as before. ?

Another Example Jan 2013 Paper 2 Q 20 ? Can now use cosine rule.

Test Your Understanding AQA Set 4 Paper 1 Frost Special ? ?

Exercise 4 1 3 ? 2 ? ?

Exercise 4 4 5 ? ?

Exercise 4 6 ?

Exercise 4 7 June 2013 Paper 2 Q 23 ?

Exercise 5 - Mixed Exercises Q 1 Q 2 Q 4 Q 3 ? ? Q 5 ? ? Q 8 Q 7 Q 6 ? ? ?