1. Trig Graphs
And equations
Revision A2

2. sin 𝜃 =− sin (−𝜃) sin 𝜃 = sin (180−𝜃) 𝑐𝑜𝑠 𝜃 = 𝑐𝑜s (−𝜃)
Radians
These are the Trigonometric graphs, but with radians instead… y
y = sinθ
sin 𝜃 =− sin (−𝜃)
sin 𝜃 = sin (180−𝜃)
cos 𝜃 = cos (360−𝜃)
𝑐𝑜𝑠 𝜃 = 𝑐𝑜s (−𝜃)
tan 𝜃 =− tan (−𝜃)
tan 𝜃 = tan (𝜃+180)
Properties of the graphs
Can you put them in words? 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
90º
180º π
3π 2
270º
360º
π 2 2π -1 y
y = cosθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
90º
π 2
180º π
3π 2
270º
360º 2π -1
y = tanθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
270º
3π 2
360º 2π -1

3. Maxima/Minima at (90,1) and (270,-1) (and every 180 from then)
𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃
𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃
Reciprocal graphs These are the Reciprocal Trigonometric graphs 1
y = Sinθ 90
180
270
360 -1
Maxima/Minima at (90,1) and (270,-1)
(and every 180 from then) 1 90
180
270
360 -1
Asymptotes at 0, 180, 360
(and every 180° from then)
y = Cosecθ

4. Maxima/Minima at (0,1) (180,-1) and (360,1) (and every 180 from then)
𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃
𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃
Reciprocal graphs These are the Reciprocal Trigonometric graphs 1
y = Cosθ 90
180
270
360 -1
Maxima/Minima at (0,1) (180,-1) and (360,1)
(and every 180 from then) 1 90
180
270
360 -1
Asymptotes at 90 and 270
(and every 180° from then)
y = Secθ

5. 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃
𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃
Reciprocal graphs These are the Reciprocal Trigonometric graphs
y = Tanθ 90
180
270
360
Asymptotes at 0, 180 and 360
(and every 180° from then) 90
180
270
360
y = Cotθ

6. 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃
Trig Identities
Can you write ten useful Trig identities
𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃= 1 𝑐𝑜𝑠𝜃
𝑐𝑜𝑡𝜃= 1 𝑡𝑎𝑛𝜃
𝑎 2 = 𝑏 2 + 𝑐 2 −2𝑏𝑐 cos 𝐴
sin 𝐴 𝑎 = sin 𝐵 𝑏
𝑠𝑖𝑛 2 𝑥=1− 𝑐𝑜𝑠 2 𝑥
𝑐𝑜𝑠 2 𝑥=1− 𝑠𝑖𝑛 2 𝑥

7. Trig Identities Can you write ten useful Trig identities
Formula booklet

8. A. Solve, for 0 ≤ x < 2π, 2 cot (𝑥− 𝜋 3 ) =5⁡
Give your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(5)

9. A. Solve, for 0 ≤ x < 2π, 2 cot (𝑥− 𝜋 3 ) =5⁡
Give your answers to two decimal places
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(4)
SOLUTION and MARKSCHEME
cot 𝑥− 𝜋 3 = 5 2
tan 𝑥− 𝜋 3 = 2 5 𝐵1
𝑥− 𝜋 3 = gives 𝑥=1.43
𝑀1 𝐴1 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑎𝑛𝑠𝑤𝑒𝑟
𝑥− 𝜋 3 =0.381, ,, … gives 𝑥=4.57
𝐴1 𝑓𝑜𝑟 2𝑛𝑑 𝑎𝑛𝑠𝑤𝑒𝑟

10. Give your answers to two decimal places
B. Solve, for 0<𝑥< 3𝜋 2 , 𝑐𝑜𝑡 2 𝑥− cot 𝑥 + 𝑐𝑜𝑠𝑒𝑐 2 𝑥=4
Give your answers to two decimal places
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(5)

11. Give your answers to two decimal places
B. Solve, for 0<𝑥< 3𝜋 2 , 𝑐𝑜𝑡 2 𝑥− cot 𝑥 + 𝑐𝑜𝑠𝑒𝑐 2 𝑥=4
Give your answers to two decimal places
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(5)
SOLUTION and MARKSCHEME
𝑐𝑜𝑡 2 𝑥− cot 𝑥 + 1+ 𝑐𝑜𝑡 2 𝑥 =4
𝐵1 𝑢𝑠𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦
𝑡𝑜 𝑚𝑎𝑘𝑒 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
2 𝑐𝑜𝑡 2 𝑥 − cot 𝑥 − 3 = 0
2 cot 𝑥 −3 cot 𝑥 +1 = 0
𝑀1 𝑠𝑜𝑙𝑣𝑖𝑛𝑔 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
𝐴1 𝐴1 𝑒𝑎𝑐ℎ 𝑎𝑛𝑠𝑤𝑒𝑟
cot 𝑥 = rearrange to t𝑎𝑛 𝑥 = gives 𝑥=0.588,
cot 𝑥 =− rearrange to t𝑎𝑛 𝑥 =−1 gives 𝑥= 3𝜋 4 𝑜𝑛𝑙𝑦
𝐴1 𝑓𝑜𝑟 3𝑟𝑑 𝑎𝑛𝑠𝑤𝑒𝑟

12. END