Summary Trigonometry, Advanced Programme Mathematics - Grade 12 (IEB)

Trigonometry
Ratios:

Co-ratio Triangle Name Variables Inverse
cos θ 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 sin θ 𝑦 cosec θ
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑟
sin θ 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 cos θ 𝑥 sec θ
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑟
cot θ 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 tan θ 𝑦 cot θ
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑥
sec θ ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 cosec θ 𝑟 sin θ
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑦
cosec θ ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sec θ 𝑟 cos θ
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑥
tan θ 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 cot θ 𝑥 tan θ
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑦

Identities:

1. Quotient
𝑠𝑖𝑛𝜃
tan θ = 𝑐𝑜𝑠𝜃

2. Inverse
1
sin θ =
𝑐𝑜𝑠𝑒𝑐𝜃
1
cos θ = 𝑠𝑒𝑐𝜃
1
tan θ = 𝑐𝑜𝑡𝜃

3. Square
cos2x + sin2x = 1
1 + cot2x = cosec2x
1 + tan2x = sec2x
Reduction formula:



90° + Ɵ 90° - Ɵ

180° - Ɵ 360° + Ɵ




180° + Ɵ 360° - Ɵ - Ɵ

270° - Ɵ 270° + Ɵ

, Radian Measure
Radian measure of an angle is the ratio of the 2 lengths, measures in the same units and is thus
a pure number. (has no unit)
One radian is defined as = the angle subtended at the centre of a circle by an arc equal in
length to the radius of the circle

b
r
θ S
r
a

𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑆
θ= 𝑟𝑎𝑑𝑢𝑖𝑠
= 𝑟

S = θr radius

Angle in radians

Arc length

Notations: 1 radian is written as 1 rad
180°
Radian to Degrees: x 𝜋
𝜋
Degrees to Radian: x 180°


Special angles in radian:
𝜋
30° = 6
𝜋
45° = 4
𝜋
60° =3
𝜋
90° =2
180° = π
360° = 2 π


sin θ and
cosec θ are All positive
positive




tan θ and cos θ and
cot θ are sec θ are
positive positive