Sine, Cosine & Area Rules (Grade 11) — Worksheet & Memo

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Grade 11 · Trigonometry

Sine, Cosine & Area Rules (Grade 11)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 27

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
In △ABC, Â = 40°, B̂ = 75° and AC = 12 units. Calculate the length of BC.
(3)
BC/(sin 40°) = 12/(sin 75°) → BC = 7,99 units✓✓✓ (3)
2
In △ABC, AC = 8 units, AB = 5 units and Â = 60°. Calculate the length of BC.
(3)
BC² = 64 + 25 − 80cos60° = 49 → BC = 7 units✓✓✓ (3)
3
Calculate the area of △ABC if AC = 8 units, AB = 5 units and Â = 60°.
(2)
1/2(8)(5)sin 60° = 17,32 square units✓✓ (2)
4
Which rule would you use given three sides and no angles?
(1)
The cosine rule, rearranged as cos A = (b² + c² − a²)/2bc✓ (1)
5
Which rule would you use given two angles and one side?
(1)
The sine rule✓ (1)
6
Write down the area rule for △PQR using sides p and q.
(1)
Area = 1/2·pq·sin R✓ (1)
7
In △ABC, a = 7, b = 8, c = 5. Calculate Â to one decimal.
(3)
cos A = (64 + 25 − 49)/(2(8)(5)) = 40/80 = 0,5 → Â = 60,0°✓✓✓ (3)
8
Why can SOHCAHTOA not be used in △ABC above?
(2)
It is not right-angled — SOHCAHTOA is only defined for right-angled triangles.✓✓ (2)
9
Prove that the area of △ABC = ½ab sin C.
(4)
Draw AD ⊥ BC, AD = h ✓; area = ½ × a × h ✓; in △ADC, sin C = h/b so h = b sin C ✓; ∴ area = ½ab sin C ✓(4)
10
Prove the sine rule for △ABC.
(5)
Draw CD ⊥ AB, CD = h ✓; in △ACD, h = b sin A ✓; in △BCD, h = a sin B ✓; ∴ b sin A = a sin B ✓; dividing by sin A sin B gives a/(sin A) = b/(sin B) ✓(5)
11
What single construction begins the proof of all three rules?
(2)
Drop a perpendicular from one vertex to the opposite side ✓, creating two right-angled triangles ✓(2)

TOTAL: 27 marks