TRIGONOMETRY AQA A-Level Maths Summary Sheet

Chapter 14
Reciprocal Trig Functions:
1 1 1
cosecθ= secθ= cotθ=
sinθ cosθ tanθ
sinθ cosθ
Additionally, as tanθ= , cotθ=
cosθ sinθ
Usingsin2 θ+cos 2 θ=1 :
2 2 2 2 2 2 2 2
cosec θ=1+ cot θ cot θ=cosec θ−1 tan θ=sec θ−1 sec θ=tan θ+1
Compound Angle Formulae Double Angle Formulae Half-Angle Formulae
(formed by letting A = B from
(In the formula booklet) Compound Angle Formulae)

θ θ
sin ( A ± B ) =sinAcosB ± cosAsinB sin 2 A=2 sinAcosA sinθ=2 sin cos
2 2
2 2 2θ 2θ
cos ( A ± B )=cosAcosB ∓ sinAsinB cos 2 A=cos A−sin A cosθ=cos −sin
2 2

θ
2 tan
tanA ± tanB 2
tan ( A ± B )= 2
¿ 2 cos A−1 tan θ=
1∓tanAtanB θ
1−tan2
2
2 1
¿ 1−2 sin A
2
cos θ= (cos 2θ+1)
2
2 tanA 2 1
Writing Expressions in Harmonic Form tan2 A= sin θ= (1−cos 2θ)
1−tan 2 A 2
Rcos ( θ ± α ) =Rcosθcosα ∓Rsinθsinα Used to write an expression
containing multiple trigonometric Arcs & Sectors
Rsin ( θ ± α )=Rsinθcosα ± Rcosθsinα variables in terms of only one
variable
Example: Arc Length=rθ
5 sinθ −12 cosθ=Rsin (θ−α ) 1 2
180 Area of a Sector= r θ
5 sinθ −12 cosθ=Rsinθcosα −Rcosθsinα degrees=radians × 2
π
Rsinα=5∧Rcosα =12
12 180
tanα= ∧R= √ 5 +12
2 2 radians=degrees ÷
5 π
π π 4π
α =67.4 ° ¿ R=13 30 °= 90 °= 240 °=
6 2 3
5 sinθ −12 cosθ=13 sin ⁡(θ−67.4 °) π 2π 3π
45 ° = 120 °= 270°=
4 3 2
Graphs:




y=secx y=cosecx y=cotx
(2 k +1) Domain : x ∈ R , x ≠ kπ Domain : x ∈ R , x ≠ kπ
Domain : x ∈ R , x ≠ π
2
Range : y ∈ R , y ≥ 1∨ y ≤−1 Range : y ∈ R , y ≥ 1∨ y ≤−1 Range : y ∈ R
Period=2 π Period=2 π Period=π

Swap x ’s
with y ’s and