Gr 12 Maths BOOKWORK - All the proofs you need to know! - Paper 1 & Paper 2 Proofs

Q: What topics are covered in Paper 1?

Paper 1 focuses on proofs for arithmetic and geometric series, including expressions for the sum of n terms and related identities.

Q: Which trigonometry formula derivations are included?

The document derives compound angle formulae (cos and sin of sums and differences) and double angle formulae for sin 2A and cos 2A.

Q: Are there recommendations for practice and exam preparation?

Yes. It states that practice is essential and directs readers to graded exercises and copious exam questions in TAS Grade 12 study guides.

Oct 14, 2026

Upload by Margaret

“The Answer Series” Gr 12 Maths Bookwork provides proof-focused extracts aligned to the TAS Grade 12 Maths structure for 2023/2024. It covers Paper 1 proofs on arithmetic and geometric series with sum-to-n term derivations, including infinite series conditions, and Paper 2 proofs related to geometry theorems and trigonometry proofs plus formulae derivations. It also includes calculator instructions and emphasizes understanding through essential practice with graded exercises and exam questions.

LanguageEnglish

CategoryPaper Templates

FormatPDF

Pages9

Reading Time3 min

AccessFree to download

Arithmetic & Geometric Series

Arithmetic series sum derivations

Geometric series sum derivations

Derivation of Compound Angle Formulae

cos(A - B) and cos(A + B)

sin(A + B) and sin(A - B)

Derivation of Double Angle Formulae

sin 2A

cos 2A

Practice and exam preparation

The Answer Series Gr 12 Maths BOOKWORK 2023/2024 (Extracts from TAS Gr 12 Maths 2-in-1 and Toolkit) All the proofs you need to know! (also available in Afrikaans) Paper 1: (a maximum of 6 marks) page 1 Arithmetic & Geometric Series Paper 2: (a maximum of 12 marks) page 2 to 7 Geometry (7 theorems) and Trigonometry (4 proofs & 5 formulae derivations) Plus Calculator Instructions PAPER 1: PROOFS Arithmetic and Geometric Series Sn : the sum of n terms 􀀂 Arithmetic Series: Sn = a + (a + d) + (a + 2d) + + Tn n terms & Sn = Tn + (Tn – d) + (Tn – 2d) + + a n terms ∴ 2 Sn = (a + Tn) + (a + Tn) + (a + Tn) + + (a + Tn) n terms ∴ 2 Sn = n (a + Tn)∴ Sn = n2 (a + Tn) 􀀂 But T n = a + (n – 1)d ∴ Sn = n2 [ a + a + (n – 1)d ] ∴ Sn = n2 [2a + (n – 1)d] 􀀂 􀀂 Geometric Series: Sn = a + ar + ar2 + ar3 + + arn – 2 + arn – 1 ➊ % r) ∴ rS n = ar + ar2 + ar3 + + arn – 1 + arn ➋ the 'middle bit'falls away. ➊ – ➋: ∴ Sn – rS n = a – arn ∴ Sn(1 – r) = a(1 – rn ) ∴ Sn = a(11−~~ ~~rr~~ ~~n~~ ~~)~~ ~~ . . . for r < 1 or ×~~ ~~((~~ ~~−~~ ~~11)) : Sn = a(rr1) . . . for r > 1 Also remember: 􀀂 S∞ = Sn as n 􀀃 ∞ if –1 < r < 1 = a~~ ~~(11~~ ~~0r)~~ ~~ . . . rn 􀀃 0 if –1 < r < 1 = 1~~ ~~~~ ~~r 􀀂 k∑ 1Tk = Sn and k1Tk = S ∞ if –1 < r < 1 1 Copyright © The Answer Series Derivation of Compound Angle Formulae We accept and use the formula: cos(A-- B) = cos A cos B + sin A sin B The cos of the difference 􀀂 cos(A + B) = cos [A-- (–B)] of the two angles = cos A cos(–B) + sin A sin(–B) = cos A cos B + sin A (–sin B) = cos A cos B -- sin A sin B 􀀂 􀀂 sin(A + B) = cos[90º – (A + B)] = cos[90º – A – B] The cos of the difference = cos[(90º– A) -- B] of the two angles = cos(90º– A) cos B + sin(90º – A) sin B = sin A cos B + cos A sin B 􀀂 􀀂 sin(A--B) = cos[90º–(A – B)] = cos[90º – A + B] The cos of the difference = cos[(90º– A) -- (–B)] of the two angles = cos(90º– A) cos(–B) + sin(90º– A) sin(–B) = sin A cos B + cos A(–sin B)= sin A cos B -- cos A sin B 􀀂 Practice is essential ! See our graded exercises and copious exam questions in TAS Grade 12 study guides. 7 Derivation of Double Angle Formulae 􀀂 sin 2A sin 2A = sin(A + A) . . . see sin(A + B) above = sin A cos A + cos A sin A . . . add like terms: xy + y x = 2xy = 2 sin A cos A 􀀂 􀀂 cos 2A cos 2A = cos(A + A) . . . see cos(A + B) above = cos A cos A – sin A sin A = cos2A--sin2A 􀀂 But: sin2A + cos2A = 1 ∴ cos2A = 1 – sin2A & sin2A = 1 – cos2A â cos 2A = (1 – sin2A)– sin2A = 1--2 sin2A 􀀂 OR cos 2A = cos2A –(1 – cos2A)= cos2A – 1 + cos2A = 2 cos2A-- 1 􀀂 It is easier to memorise formulae which we understand. Copyright © The Answer Series

FAQs

What topics are covered in Paper 1?

Paper 1 focuses on proofs for arithmetic and geometric series, including expressions for the sum of n terms and related identities.

Which trigonometry formula derivations are included?

The document derives compound angle formulae (cos and sin of sums and differences) and double angle formulae for sin 2A and cos 2A.

Are there recommendations for practice and exam preparation?

Yes. It states that practice is essential and directs readers to graded exercises and copious exam questions in TAS Grade 12 study guides.

Oct 14, 2026

Upload by Margaret