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
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
