GRADE 12 MATHS PAPER 2
TRIGONOMETRY
Compound Angles: Double Angles:
sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 sin 2𝛼 = 2 sin 𝛼 cos 𝛼
sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽 cos 2𝛼 = 1 − 2𝑠𝑖𝑛 2 𝛼
cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽 cos 2𝛼 = 2𝑐𝑜𝑠 2 𝛼 − 1
cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽
𝑂 𝐴 𝑂
𝑆=𝐻 𝐶=𝐻 𝑇=𝐴
Special Angles: Trig Rules:
𝑎 𝑏 𝑐
Sine rule: 𝑠𝑖𝑛𝐴 = 𝑠𝑖𝑛𝐵 = 𝑠𝑖𝑛𝐶
Cosine rule: 𝑎2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 ∙ 𝑐𝑜𝑠𝐴
1
Area rule: 𝐴𝑟𝑒𝑎∆𝐴𝐵𝐶 = 2 ∙ 𝑎𝑏 ∙ 𝑠𝑖𝑛𝐶
Reduction Formulae: Identities:
𝑠𝑖𝑛𝜃
𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃
𝑠𝑖𝑛 2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
Trig Graphs:
Move left/right:
sin(𝑥 ± 1) ; cos(𝑥 ± 1) ; tan (𝑥 ± 1)
Vertical stretch:
𝑎 ∙ 𝑠𝑖𝑛𝑥; 𝑎 ∙ 𝑐𝑜𝑠𝑥; 𝑎 ∙ 𝑡𝑎𝑛𝑥
Move up/down:
𝑠𝑖𝑛𝑥 ± 1; 𝑐𝑜𝑠𝑥 ± 1; 𝑡𝑎𝑛𝑥 ± 1
Change period:
𝑠𝑖𝑛𝑎(𝑥); 𝑐𝑜𝑠𝑎(𝑥); 𝑡𝑎𝑛𝑎(𝑥)
General Solutions:
𝑠𝑖𝑛: 𝑥 = ∠ + 𝑘. 360°
𝑜𝑟
𝑠𝑖𝑛: 𝑥 = 180° + ∠ + 𝑘. 360°; 𝑘 ∈ 𝑍
𝑐𝑜𝑠: 𝑥 = ±∠ + 𝑘. 360°; 𝑘 ∈ 𝑍
𝑡𝑎𝑛: 𝑥 = ∠ + 𝑘. 180°; 𝑘 ∈ 𝑍
TRIGONOMETRY
Compound Angles: Double Angles:
sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 sin 2𝛼 = 2 sin 𝛼 cos 𝛼
sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽 cos 2𝛼 = 1 − 2𝑠𝑖𝑛 2 𝛼
cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽 cos 2𝛼 = 2𝑐𝑜𝑠 2 𝛼 − 1
cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽
𝑂 𝐴 𝑂
𝑆=𝐻 𝐶=𝐻 𝑇=𝐴
Special Angles: Trig Rules:
𝑎 𝑏 𝑐
Sine rule: 𝑠𝑖𝑛𝐴 = 𝑠𝑖𝑛𝐵 = 𝑠𝑖𝑛𝐶
Cosine rule: 𝑎2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 ∙ 𝑐𝑜𝑠𝐴
1
Area rule: 𝐴𝑟𝑒𝑎∆𝐴𝐵𝐶 = 2 ∙ 𝑎𝑏 ∙ 𝑠𝑖𝑛𝐶
Reduction Formulae: Identities:
𝑠𝑖𝑛𝜃
𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃
𝑠𝑖𝑛 2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
Trig Graphs:
Move left/right:
sin(𝑥 ± 1) ; cos(𝑥 ± 1) ; tan (𝑥 ± 1)
Vertical stretch:
𝑎 ∙ 𝑠𝑖𝑛𝑥; 𝑎 ∙ 𝑐𝑜𝑠𝑥; 𝑎 ∙ 𝑡𝑎𝑛𝑥
Move up/down:
𝑠𝑖𝑛𝑥 ± 1; 𝑐𝑜𝑠𝑥 ± 1; 𝑡𝑎𝑛𝑥 ± 1
Change period:
𝑠𝑖𝑛𝑎(𝑥); 𝑐𝑜𝑠𝑎(𝑥); 𝑡𝑎𝑛𝑎(𝑥)
General Solutions:
𝑠𝑖𝑛: 𝑥 = ∠ + 𝑘. 360°
𝑜𝑟
𝑠𝑖𝑛: 𝑥 = 180° + ∠ + 𝑘. 360°; 𝑘 ∈ 𝑍
𝑐𝑜𝑠: 𝑥 = ±∠ + 𝑘. 360°; 𝑘 ∈ 𝑍
𝑡𝑎𝑛: 𝑥 = ∠ + 𝑘. 180°; 𝑘 ∈ 𝑍
