1. GCSE Exact Trigonometric Ratios
Dr J Frost
Objectives:
Determine the exact values of sin/cos/tan of ‘common’ angles, e.g. sin 45° and 𝑡𝑎𝑛 60°.
Last modified: 1st January 2019

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3. RECAP :: Trigonometry of Right-Angled Triangles
60 ° 𝒙 12 ?
𝒔𝒊𝒏 𝟔𝟎 = 𝟏𝟐 𝒙 𝒙= 𝟏𝟐 𝒔𝒊𝒏 𝟔𝟎 =𝟏𝟑.𝟖𝟔

4. sin/cos/tan of 30°, 45°, 60°, 90°
You will frequently encounter angles of 30°, 60°, 45° in geometric problems. Why?
We see these angles in equilateral triangles and right-angled isosceles triangles. ?
You need to be able to calculate these in non-calculator exams.
All you need to remember:
! Draw half a unit square and half an equilateral triangle of side 2.
sin 30° = cos 30° = tan 30° = sin 60° = cos 60° = tan 60° = 3 ?
sin 45° = cos 45° = tan 45° =1 ? 2 ? ? ? 1 ? ?
30° ? 2 ? ?
45° 3 ? ? ? ? 1
60° ? 1 ? ? ?

5. Example ‘Show That’ Exam Questions? 2 3
60° 1
cos 60° = 𝑎𝑑𝑗 ℎ𝑦𝑝 = 1 2
𝒉𝒚𝒑= 𝟏 𝟐 + 𝟏 𝟐 = 𝟐 𝐬𝐢𝐧 𝟒𝟓° = 𝒐𝒑𝒑 𝒉𝒚𝒑 = 𝟏 𝟐 ? ?

6. Further Example
You may have a standard trigonometry question, but without the aid of a calculator:
Without use of a calculator, determine the value of 𝑥. ?
60° 𝟖 𝒙
sin 60° = 𝑥 8 𝑥=8 sin 60° =8× =4 3

7. Test Your Understanding
Without use of a calculator, determine the value of 𝑥. ?
tan 30° = 5 𝑥 𝑥= 5 tan 30° =5÷ = 5 1 × =5 3
30° 𝟓 𝒙

8. Exercise ? ? ? ? ? ? ? ? (questions on provided sheet) 1
Using an appropriate diagram, determine the exact value of:
tan 45 =𝟏 cos 60° = 𝟏 𝟐 sin 30°= 𝟏 𝟐
cos 30°= 𝟑 𝟐 sin 45° = 𝟏 𝟐 a ? b ? c ? d ? e ?
[Edexcel GCSE(9-1) Nov H Q20]
The table shows some values of 𝑥 and 𝑦 that satisfy the equation 𝑦=𝑎 cos 𝑥 °+𝑏
Find the value of 𝑦 when 𝑥=45
Solution: 𝟏+ 𝟐 2 3
30° 𝟑 𝒙
Determine the exact value of 𝑥 𝟑 𝟑 ? ? 4
[OCR GCSE(9-1) Nov F Q19b, Nov H Q8b Edited] The angles in a triangle are in the ratio 1 : 2 : 3. It can be shown that the triangle is right-angled.
The hypotenuse of the triangle is 15cm long.
Calculate the length of the shortest side in the triangle.
7.5 cm ?

9. Exercise ? ? (questions on provided sheet) 5
[OCR GCSE(9-1) SAM 5H Q17b]
Find the exact area of this triangle. 6
(Knowledge of sine/cosine rules required)
[Edexcel GCSE(9-1) June H Q22] The diagram shows a hexagon 𝐴𝐵𝐶𝐷𝐸𝐹.
𝐴𝐵𝐸𝐹 and 𝐶𝐵𝐸𝐷 are congruent parallelograms where 𝐴𝐵=𝐵𝐶=𝑥 cm.
𝑃 is the point on 𝐴𝐹 and 𝑄 is the point on 𝐶𝐷 such that 𝐵𝑃=𝐵𝑄=10 cm.
Given that angle 𝐴𝐵𝐶=30°, prove that  cos 𝑃𝐵𝑄=1− 2− 𝑥 2  .
Using cosine rule on triangle 𝑨𝑩𝑪:
𝑷 𝑸 𝟐 = 𝒙 𝟐 + 𝒙 𝟐 −𝟐 𝒙 𝟐 𝐜𝐨𝐬 𝟑𝟎° 𝑷 𝑸 𝟐 =𝟐 𝒙 𝟐 − 𝟑 𝒙 𝟐 = 𝟐− 𝟑 𝒙 𝟐
Using cosine rule on triangle 𝑷𝑩𝑸:
𝑷 𝑸 𝟐 = 𝟏𝟎 𝟐 + 𝟏𝟎 𝟐 −𝟐𝟎𝟎 𝐜𝐨𝐬 𝑷𝑩𝑸 𝟐− 𝟑 𝒙 𝟐 =𝟐𝟎𝟎−𝟐𝟎𝟎 𝐜𝐨𝐬 𝑷𝑩𝑸 𝟐𝟎𝟎 𝐜𝐨𝐬 𝑷𝑩𝑸 =𝟐𝟎𝟎− 𝟐− 𝟑 𝒙 𝟐 𝐜𝐨𝐬 𝑷𝑩𝑸 = 𝟐𝟎𝟎− 𝟐− 𝟑 𝒙 𝟐 𝟐𝟎𝟎 =𝟏− 𝟐− 𝟑 𝟐𝟎𝟎 𝒙 𝟐 ? ?
𝟐𝟒 𝟑

10. Exercise ? ? (questions on provided sheet) 𝐸 𝐴 8 7 𝐷 𝐵 𝐶 Solution: 𝟔𝟎°
[SMC 2011 Q24]
Three circles and the lines 𝑃𝑄 and 𝑄𝑅 touch as shown. The distance between the centres of the smallest and the biggest circles is 16 times the radius of the smallest circle.
What is the size of ∠𝑃𝑄𝑅? 7 8
[Edexcel GCSE(9-1) Mock Set 3 Autumn H Q20]
The diagram shows three right-angled triangles. Given that 𝑦=𝑘𝑛, find the value of 𝑘.
Using triangle 𝑨𝑩𝑪:
𝐬𝐢𝐧 𝟑𝟎° = 𝒚 𝑨𝑪 → 𝑨𝑪= 𝒚 𝐬𝐢𝐧 𝟑𝟎° = 𝒚 𝟎.𝟓 =𝟐𝒚 Using triangle 𝑨𝑪𝑫:
𝐜𝐨𝐬 𝟑𝟎° = 𝟐𝒚 𝑨𝑫 → 𝑨𝑫= 𝟐𝒚 𝐜𝐨𝐬 𝟑𝟎° = 𝟐𝒚 𝟑 /𝟐 = 𝟒𝒚 𝟑
Using triangle 𝑨𝑬𝑫:
𝐭𝐚𝐧 𝟑𝟎° = 𝒏 𝑨𝑫 → 𝒏=𝑨𝑫 𝐭𝐚𝐧 𝟑𝟎° = 𝟒𝒚 𝟑 × 𝟏 𝟑 = 𝟒𝒚 𝟑
𝟑𝒏=𝟒𝒚 𝒚= 𝟑 𝟒 𝒏 ? ?
Solution: 𝟔𝟎°