1. Trig Graphs
And equations

2. Trigonometry Graphs to equations
KUS objectives
BAT know the key features of trig graphs using radians
BAT solve trig equations using radians
Starter: Simplify (without a calculator)
sin cos 60
2 sin 45 + cos
tan − 3 tan 120

3. These are the Trigonometric graphs, but with radians instead…y
y = sinθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
3π 2
270º
360º 2π -1 y
y = cosθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
270º
3π 2
360º 2π -1
y = tanθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
3π 2
270º
360º 2π -1

4. 𝜋 WB11 Transformations of sin graph Sketch the graph of 𝒔𝒊𝒏 𝒙+ 𝝅 𝟐
y = sinθ 1
𝜋 2 𝜋
3𝜋 2 2𝜋 y -1
Sketch the graph of
a) 𝐬𝐢𝐧 𝒙+ 𝝅 𝟐
b) 𝒔𝒊𝒏 𝒙 −𝟐
c) − 𝒔𝒊𝒏 𝟐𝒙
𝒔𝒊𝒏 𝒙+ 𝝅 𝟐
− 𝒔𝒊𝒏 𝟐𝒙
𝒔𝒊𝒏 𝒙 −𝟐 −𝟑 −𝟐

5. WB12b double transformations (ax+b) 𝑦 =2 cos 𝜃+ 3𝜋 2y 1 -1
y = cosθ
-2π
-3π 2 -π
-π 2 2π
3π 2 π
π 2
𝑦 = cos 𝜃+ 3𝜋 2
Sketch the graph of
a) 2cos 𝑥+ 3𝜋 2
Transformations in order are
i) shift − 3𝜋 2 in the x direction
ii) Stretch ×2 in the y direction

6. 𝜋 WB12b double transformations (ax+b) y = sinθ 𝜋 2 2𝜋
3𝜋 2 2𝜋 y -1
Sketch the graph of
b) sin 2𝑥−𝜋
𝒔𝒊𝒏 𝒙− 𝝅 𝟐
𝒔𝒊𝒏 𝟐 𝒙− 𝝅 𝟐
=sin 2 𝑥− 𝜋 2
Transformations in order are
i) shift + 𝜋 2 in the x direction
ii) Stretch × in the y direction

7. WB12c double transformations (ax+b)
y=tan 𝑥+ 𝜋 2
Sketch the graph of
c) tan 𝑥 2 + 𝜋 4
y=tan 𝑥+ 𝜋 2
y = tanθ 1 -1
-2π
-3π 2 -π
-π 2 2π
3π 2 π
π 2
=tan 𝑥+ 𝜋 2
Transformations in order are
i) shift − 𝜋 2 in the x direction
ii) Stretch ×2 in the x direction

8. For any angle ϴ, sin(- ϴ) = - sinϴ cos(- ϴ) = cosϴ tan(- ϴ) = - tanϴ
Reminder Symmetry of trig graphs 1 -1
𝜋 2 𝜋
3𝜋 2 2𝜋 y
- 𝜋 2
- 𝜋
−2𝜋
- 3𝜋 2
y = sinθ
y = cosθ
y = tanθ
For any angle ϴ,
sin(- ϴ) = - sinϴ
cos(- ϴ) = cosϴ
tan(- ϴ) = - tanϴ
Complete:
sin − 𝝅 𝟐 =
cos − 𝝅 𝟔 =
tan − 𝝅 𝟑 =

9. Trig equations
Solve trig equations in radians the same way as in degrees
Rearrange to sin / cos / tan =
Use arcsin / arccos / arctan on calculator to get a first answer in a list
Use a graph to list all the answers in the given range
Sometimes complete the solution with more inverses ( e.g. when its cos 3𝑥+ 𝜋 you do the operations − 𝜋 2 and ÷3 to your list

10. 𝑥=− 𝜋 6 , 𝜋 6 , …. . What do you notice?
WB13 Solve sin 𝑥+ 𝜋 2 = in the range 0≤𝑥≤2𝜋
y = sinθ 1
𝜋 2 𝜋
3𝜋 2 2𝜋 y -1
𝑥+ 𝜋 2 = arcsin
= 𝜋 3
, 2𝜋 3 , ….
𝜋 3
2𝜋 3
7𝜋 3
𝑥= 𝜋 3 − 𝜋 2 , 2𝜋 3 − 𝜋 2 , ….
𝑥=− 𝜋 6 , 𝜋 6 , … What do you notice?
The other answer in range is 𝜋 3 − 𝜋 2 = 11𝜋 6
The final answers are 𝑥= 𝜋 6 , 11𝜋 check they work with your calculator

11. WB14 Solve each equation for x in the interval 0≤𝑥≤2𝜋 giving your answers in terms of π
𝑎) 𝑥 2 = 𝜋 4 , 3𝜋 4 , …. .
𝑥= 𝜋 2 , 3𝜋 2 ,
𝑏) cos 𝑥+𝜋 =
𝑏) 𝑥+π= 𝜋 4 , 7𝜋 4 , 9𝜋 4 …. .
𝑥= 3𝜋 4 , 5𝜋 4 ,
𝑐) tan 𝑥 2 − 𝜋 4 = 3
c) 𝑥 2 − π 4 = 𝜋 3 , 4𝜋 3 , …. .
𝑥= 7𝜋 6 , only

12. WB15 Solve each equation for x in the interval 0≤𝑥≤2𝜋 giving your answers to 2 decimal places
𝑎) sin 2𝑥=0.6
𝑎) 2𝑥= , 2.498, , …. .
𝑥=0.32, 1.25, 3.46, 4.39
𝑏) cos 𝑥−0.3 =0.8
𝑥−0.3=−0.644, , , …. .
𝑥=, 0.94,
𝑏) 2 cos 𝑥−0.3 =1.6
𝑐) 3−2 tan 𝑥− 𝜋 3 =0
c) tan 𝑥− 𝜋 3 = 3 2
𝑥− 𝜋 3 =0.9828, , …. .
𝑥= , 5.17

13. sin 𝑥 =+ 2 2 gives 𝑥= 𝜋 4 , 3𝜋 4 𝑠𝑖𝑛 2 𝑥= 1 2 sin 𝑥 =± 2 2
WB16 Solve each equation for θ in the interval −𝜋≤𝜃≤𝜋 giving your answers in terms of π
𝑎) 𝑠𝑖𝑛 2 𝑥 =1
sin 𝑥 = gives 𝑥= 𝜋 4 , 3𝜋 4
𝑠𝑖𝑛 2 𝑥= 1 2
sin 𝑥 =±
sin 𝑥 =− gives 𝑥=− 𝜋 4 ,− 3𝜋 4
𝑏) 𝑐𝑜𝑠 2 𝑥+4 cos 𝑥 +1=0
cos 𝑥 =−1 gives 𝑥=−𝜋, 𝜋
cos 𝑥 cos 𝑥 −1 =0
cos 𝑥 = −1 , 1 2
sin 𝑥 = gives 𝑥=− 𝜋 3 , 𝜋 3

14. cos 𝑥 = −(−1)± (−1) 2 −4(3)(−1) 2(3)
WB17a Solve each equation for θ in the interval −𝜋≤𝜃≤𝜋 giving your answers to 3 significant figures
𝑎) cos 𝑥 −1 = sin 𝑥 tan 𝑥
cos 𝑥 =−0.4343
gives 𝑥=−2.02, 2.02
2 cos 𝑥 −1=sin x sin x cos x
2 𝑐𝑜𝑠 2 𝑥 − cos 𝑥= 𝑠𝑖𝑛 2 𝑥
sin 𝑥 =−
gives 𝑥=−0.696, 0.696
2 𝑐𝑜𝑠 2 𝑥 − cos 𝑥=1− 𝑐𝑜𝑠 2 𝑥
3 𝑐𝑜𝑠 2 𝑥 − cos 𝑥 −1=0
cos 𝑥 = −(−1)± (−1) 2 −4(3)(−1) 2(3)
cos 𝑥 = −0.4343,

15. 𝑏) 𝑐𝑜𝑠 2 𝑥−3 sin 𝑥 − 𝑠𝑖𝑛 2 𝑥=2 sin 𝑥 =− 1 2 gives 𝑥=− 5𝜋 6 ,− 𝜋 6
WB17b Solve each equation for θ in the interval −𝜋≤𝜃≤𝜋 giving your answers to 3 significant figures
𝑏) 𝑐𝑜𝑠 2 𝑥−3 sin 𝑥 − 𝑠𝑖𝑛 2 𝑥=2
sin 𝑥 =− 1 2
gives 𝑥=− 5𝜋 6 ,− 𝜋 6
(1− 𝑠𝑖𝑛 2 𝑥)−3 sin x− 𝑠𝑖𝑛 2 𝑥=2
0 = 2 𝑠𝑖𝑛 2 𝑥+3 sin 𝑥+1
sin 𝑥 =−1
gives 𝑥=− 𝜋 2
2 sin 𝑥 +1 sin 𝑥 +1 =0
sin 𝑥 − 1 2 , -1

16. WB18 Sketch the curve 𝑦 = 2cos⁡𝑥° for x in the interval 0≤𝑥≤2𝜋
WB18 Sketch the curve 𝑦 = 2cos⁡𝑥° for x in the interval 0≤𝑥≤2𝜋. Sketch on the same diagram the curve 𝑦 = cos⁡(𝑥 − 𝜋 3 ) .
use your graph to find the values of x in the interval 0≤𝑥≤2𝜋 for which 2 cos 𝑥 =cos⁡ 𝑥 − 𝜋 3
From the graph
2 cos 𝑥 =cos⁡ 𝑥 − 𝜋 3
When y = 1 and y=-1
𝐴 𝑖𝑠 𝑝𝑜𝑖𝑛𝑡 𝜋 3 , 1
B 𝑖𝑠 𝑝𝑜𝑖𝑛𝑡 4𝜋 3 , −1

17. One thing to improve is –
KUS objectives BAT know the key features of trig graphs using radians
BAT solve trig equations using radians
self-assess
One thing learned is –
One thing to improve is –

18. END