1. Trig
Equations

2. f(x)= cos 2𝑥, 0°<𝑥≤360° g(x)=tan 𝑥−60 , −360°<𝑥≤0°
Trigonometry
KUS objectives
BAT rearrange and solve trig equations
Starter: sketch these graphs
f(x)= cos 2𝑥, °<𝑥≤360°
g(x)=tan 𝑥−60 , −360°<𝑥≤0°
h x =−sin 𝑥+45 , −180°<𝑥≤180°
Check using Desmos / geogebra

3. Solve sin 𝜃 =0.5 in the interval 0≤ 𝜃 ≤360°
WB21
Solve sin 𝜃 =0.5 in the interval 0≤ 𝜃 ≤360°
One thing you should pay careful attention to is the range the answers can be within,
Use Sin-1
This will give you one answer
0.5
y = Sinθ 90
180
270
360 30
150
The infinite set of solutions
for −∞≤ 𝜃 ≤∞° , would be
𝜃=30±360𝑛
𝜃=150±360𝑛

4. Solve 5sin 𝜃 =−2 in the interval 0≤ 𝜃 ≤360°
WB22
Solve 5sin 𝜃 =−2 in the interval 0≤ 𝜃 ≤360°
Divide by 5
Use Sin-1
Not within the range. You can add 360° to obtain an equivalent value
203.6
336.4 90
180
270
360
y = Sinθ
-0.4
The infinite set of solutions
for −∞≤ 𝜃 ≤∞° , would be … ?
𝜃=203.6±360𝑛
𝜃=336.4±360𝑛

5. tan 𝜃 = 9 2 𝜃=77.5 Solve 2 tan 𝜃 =−9 in the interval −90°≤ 𝜃 ≤90°
WB23
Solve 2 tan 𝜃 =−9 in the interval −90°≤ 𝜃 ≤90°
90º
180º
270º
360º
-90º
-180º
-360º 1 -1
-270º θ
tan 𝜃 = 9 2
Divide by 2
Use Tan-1
𝜃=77.5
There is only one solution if −90°≤ 𝜃 ≤90°
The infinite set of solutions
for −∞≤ 𝜃 ≤∞° , would be … ?
𝜃=77.5±180𝑛

6. The infinite set of solutions
WB24
Solve cos 𝜃 =0.5 in the interval −90≤ 𝜃 ≤ 270 90
y = Cosθ
Y=0.5
270
360
180
-90
cos 𝜃 = 1 2
Use Tan-1
𝜃=…60, 300, …
The infinite set of solutions
for −∞≤ 𝜃 ≤∞° , would be … ?
Now consider −90≤ 𝜃 ≤ 270
𝜃=±60±360𝑛
𝜃=−60, 𝑜𝑟 𝜃=60

7. 2sin 𝜃=1.6, 0≤𝜃≤180 𝑡𝑎𝑛𝜃=2, 0≤𝜃≤360 cos 𝜃 = 2 5 , 0≤𝜃≤360
Practice 1 Solve these equations, give exact answers or round to 3 s.f.
2sin 𝜃=1.6, 0≤𝜃≤180
𝑡𝑎𝑛𝜃=2, 0≤𝜃≤360
cos 𝜃 = 2 5 , 0≤𝜃≤360
𝑡𝑎𝑛𝜃+8=11, −180≤𝜃≤180
cos 𝜃 = 1 5 , −360≤𝜃≤360
3 sin 𝜃 =4, 0≤𝜃≤360
3 tan 𝜃 =1, 0≤𝜃≤360
−3 sin 𝜃 =8, 0≤𝜃≤360
cos 𝜃 =− 2 5 , 0≤𝜃≤360
3 𝑡𝑎𝑛𝜃=−6, 0≤𝜃≤720
sin 𝜃 =− , 0≤𝜃≤360
cos 𝜃 =− 1 2 , −360≤𝜃≤0

8. Solve cos 2𝜃 =−1 in the interval 0≤ 𝜃 ≤360°
WB25 to solve equations in the form Sin/Cos/Tan(aθ + b) = k
Solve cos 2𝜃 =−1 in the interval 0≤ 𝜃 ≤360°
1) Work out the acceptable interval for 2θ
Multiply by 2
Solve using Cos-1
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range
y = Cosθ
3) Add/Subtract 360 to these values until you have all the answers within the 2θ range 90
180
270
360 -1
180
4) These answers are for 2θ. Undo them to find values for θ itself

9. 1) Work out the acceptable interval for (θ-60)
WB26 to solve equations in the form Sin/Cos/Tan(aθ + b) = k
Solve tan (𝜃−60) =−1 in the interval 0≤ 𝜃 ≤360°
1) Work out the acceptable interval for (θ-60)
Subtract 60
−60≤ 𝜃≤300
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range
tan (𝜃−60) =−1
Solve using tan-1
𝜃−60=−45
3) Add/Subtract 180 to these values until you have all the answers within the (20 - θ) range
90º
180º
270º
360º
-90º
-180º
-360º 1 -1
-270º θ
4) These answers are for (20 – θ). Undo this to find values for θ itself
𝜃−60=−45, , ,
𝜃= 15, 195,
Two answers between
0≤ 𝜃 ≤360°

10. 1) Work out the acceptable interval for (2θ – 35)
WB27 to solve equations in the form Sin/Cos/Tan(aθ + b) = k
Solve sin (2𝜃−35) =−1 in the interval−180°≤ 𝜃 ≤180°
Multiply by 2. Subtract 35
1) Work out the acceptable interval for (2θ – 35)
Solve using Sin-1
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range
y = Sinθ
3) Add/Subtract 360 to these values until you have all the answers within the (2θ - 35) range 90
180
270
360 -1
270
Adding/Subtracting 360 to the value we worked out (staying within the range)
4) These answers are for (2θ – 35). Undo this to find values for θ itself
Add 35, Divide by 2

11. 1) Work out the acceptable interval for (20 – θ)
WB28 to solve equations in the form Sin/Cos/Tan(aθ + b) = k
Solve tan (20−𝜃) =3 in the interval−180°≤ 𝜃 ≤180°
1) Work out the acceptable interval for (20 – θ)
Multiply by -1
Solve using tan-1
Add 20
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range
‘Turn round’
71.6
251.6
3) Add/Subtract 180 to these values until you have all the answers within the (20 - θ) range 3
y = Tanθ 90
180
270
360
Adding/Subtracting 180 to the values we worked out (staying within the range)
4) These answers are for (20 – θ). Undo this to find values for θ itself
Subtract 20
Multiply by -1

12. Practice 2 Solve these equations
sin 2𝜃= 1 3 , 0≤𝜃≤180
sin 2𝜃−30 = 1 3 , 0≤𝜃≤180
cos (𝜃+180) = , 0≤𝜃≤360
cos (3𝜃−10) = , 0≤𝜃≤360
tan 𝑥 2 = ≤𝜃≤360
tan 𝜃+20 = , 0≤𝜃≤360
sin (𝜃−40)= ≤𝜃≤360
sin 5𝜃−100 = −90≤𝜃≤180
cos (4𝜃) =0.174 , 0≤𝜃≤90
cos 𝑥−10 =0.174 , −360≤𝜃≤0
tan 𝜃−45 =2− ≤𝜃≤360
tan 2𝑥−15 =0.364 , −90≤𝜃≤90

13. One thing to improve is –
KUS objectives BAT rearrange and solve trig equations
self-assess
One thing learned is –
One thing to improve is –

14. END