1. Trigonometrical Identities
and Equations

2. Introduction
This Chapter involves learning 2 Trigonometrical Identities
We will be using these to rewrite expressions
We will also be looking at solving Trigonometrical Equations

3. Teachings for Exercise 10A

4. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
You do not need to be able to prove either of these Identities, but it is useful to see where they come from.
You should remember the ‘SOHCAHTOA’ rule from GCSE Maths. S H C H T A
Replacing these in the original Equation…
Cancel the H’s
10A

5. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
You do not need to be able to prove either of these Identities, but it is useful to see where they come from.
You should remember the ‘SOHCAHTOA’ rule from GCSE Maths.
You should also remember Pythagoras’ Theorem S H C H T A
Hyp θ
Opp 1
Adj
Replace a, b and c
This is how it is written
10A

6. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
The Identities are unchanged if there is a value in front of θ.
10A

7. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
You will need to spend a lot of time on this topic, and develop your own understanding of how to manipulate these Identities
Example Question
Simplify the following Expression:
The value in front of θ does not affect the identity
10A

8. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
You will need to spend a lot of time on this topic, and develop your own understanding of how to manipulate these Identities
Example Question
Simplify the following Expression:
Subtract Sin²θ
10A

9. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
You will need to spend a lot of time on this topic, and develop your own understanding of how to manipulate these Identities
Example Question
Simplify the following Expression:
Replace the bottom using the 2nd Identity
Square root the bottom
Looks a bit like the first Identity?
10A

10. Trigonometrical Identities and Equations
You need to be able to use the Trigonometrical identities
You will need to spend a lot of time on this topic, and develop your own understanding of how to manipulate these Identities
Example Question
Simplify the following Expression:
Expand like a Quadratic
10A

11. Trigonometrical Identities and EquationsH S
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
Example Question
Given that Cosθ is -3/5 and θ is reflex, find the value of Sinθ and Tanθ
Draw a Right Angled Triangle θ 5 4 3
You were effectively told A and H in the question. IGNORE the negative for now…
y = Sinθ 90
180
270
360
The other side should be worked out using Pythagoras’ Theorem…
y = Cosθ
y = Tanθ
Put in the values from the Triangle
Consider the region on the diagram
10A

12. Trigonometrical Identities and EquationsH S
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
Example Question
Given that Cosθ is -3/5 and θ is reflex, find the value of Sinθ and Tanθ
Draw a Right Angled Triangle θ 5 4 3
You were effectively told A and H in the question. IGNORE the negative for now…
y = Sinθ 90
180
270
360
The other side should be worked out using Pythagoras’ Theorem…
y = Cosθ
y = Tanθ
Put in the values from the Triangle
Consider the region on the diagram
10A

13. Trigonometrical Identities and EquationsH S
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
Example Question
Given that Sinθ is 2/5 and θ is obtuse, find the value of Cosθ and Tanθ
Draw a Right Angled Triangle θ 5 2
√21
You were effectively told O and H in the question.
y = Sinθ 90
180
270
360
The other side should be worked out using Pythagoras’ Theorem…
y = Cosθ
y = Tanθ
Put in the values from the Triangle
Consider the region on the diagram
10A

14. Trigonometrical Identities and EquationsH S
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
You need to be able to use the Trigonometrical identities
You also need to be able to work out exact vales of Sinθ, Cosθ or Tanθ, having been given one of the others.
You will also need to use whether θ is Acute, Obtuse, or Reflex…
Example Question
Given that Sinθ is 2/5 and θ is obtuse, find the value of Cosθ and Tanθ
Draw a Right Angled Triangle θ 5 2
√21
You were effectively told O and H in the question.
y = Sinθ 90
180
270
360
The other side should be worked out using Pythagoras’ Theorem…
y = Cosθ
y = Tanθ
Put in the values from the Triangle
Consider the region on the diagram
10A

15. Teachings for Exercise 10B

16. Trigonometrical Identities and Equations
You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k
This is similar to the work covered in Chapter 8, and involves using your calculator and Trigonometrical Graphs to solve equations with multiple solutions.
One thing you should pay careful attention to is the range the answers can be within, eg)
0 > x > 360
Example Question
Use Sin-1
This will give you one answer
0.5
y = Sinθ 90
180
270
360 30
150
10B

17. Trigonometrical Identities and Equations
You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k
This is similar to the work covered in Chapter 8, and involves using your calculator and Trigonometrical Graphs to solve equations with multiple solutions.
One thing you should pay careful attention to is the range the answers can be within, eg)
0 > x > 360
Example Question
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
10B

18. Trigonometrical Identities and Equations
You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k
This is similar to the work covered in Chapter 8, and involves using your calculator and Trigonometrical Graphs to solve equations with multiple solutions.
One thing you should pay careful attention to is the range the answers can be within, eg)
0 > x > 360
Example Question
Divide by Cosθ
Use Trig Identities
Use Tan-1 2
y = Tanθ 90
180
270
360
63.4
243.4
10B

19. Trigonometrical Identities and Equations
You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k
This is similar to the work covered in Chapter 8, and involves using your calculator and Trigonometrical Graphs to solve equations with multiple solutions.
One thing you should pay careful attention to is the range the answers can be within, eg)
0 > x > 360
Example Question
Use Cos-1 in RADIANS
0.5
y = Cosθ
1/2π π
3/2π 2π
1/3π
5/3π
10B

20. Teachings for Exercise 10C

21. Trigonometrical Identities and Equations
You need to be able to solve equations in the form Sin/Cos/Tan(aθ + b) = k
This can be a confusing process. Ensure you set your work out as done in the examples, you will start to understand better after a few practice questions.
Example Question
Multiply by 2
Solve using Cos-1
1) Work out the acceptable interval for 2θ
y = Cosθ
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range 90
180
270
360 -1
180
Adding 360 to the value we worked out (staying within the range)
3) Add/Subtract 360 to these values until you have all the answers within the 2θ range
Divide by 2
4) These answers are for 2θ. Undo them to find values for θ itself
10C

22. Trigonometrical Identities and Equations
You need to be able to solve equations in the form Sin/Cos/Tan(aθ + b) = k
This can be a confusing process. Ensure you set your work out as done in the examples, you will start to understand better after a few practice questions.
Example Question
Multiply by 2. Subtract 35
Solve using Sin-1
1) Work out the acceptable interval for (2θ – 35)
y = Sinθ
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range 90
180
270
360 -1
270
Adding/Subtracting 360 to the value we worked out (staying within the range)
3) Add/Subtract 360 to these values until you have all the answers within the (2θ - 35) range
Add 35, Divide by 2
4) These answers are for (2θ – 35). Undo this to find values for θ itself
10C

23. Trigonometrical Identities and Equations
You need to be able to solve equations in the form Sin/Cos/Tan(aθ + b) = k
This can be a confusing process. Ensure you set your work out as done in the examples, you will start to understand better after a few practice questions.
Example Question
Multiply by -1
Solve using tan-1
Add 20
‘Turn round’
1) Work out the acceptable interval for (20 – θ)
71.6
251.6 3
y = Tanθ
2) Work out one possible answer as before. Find all values in the standard 0 – 360 range 90
180
270
360
Adding/Subtracting 180 to the values we worked out (staying within the range)
3) Add/Subtract 180 to these values until you have all the answers within the (20 - θ) range
Subtract 20
4) These answers are for (20 – θ). Undo this to find values for θ itself
Multiply by -1
10C

24. Teachings for Exercise 10D

25. Trigonometrical Identities and Equations
You need to be able to solve Quadratic Equations given to you using Sin, Cos or Tan. The process is identical to standard Quadratics, but there are even more answers (usually!)
Solve the following Equation
Factorise
Work out what value would make either bracket 0
10D

26. Trigonometrical Identities and Equations
You need to be able to solve Quadratic Equations given to you using Sin, Cos or Tan. The process is identical to standard Quadratics, but there are even more answers (usually!)
Solve the following Equation
Factorise
Work out what value would make either bracket 0 2
Sinθ = 2 has no solutions 90 1
Sinθ = 1 has 1 solution
y = Sinθ 90
180
270
360
10D

27. Trigonometrical Identities and Equations
You need to be able to solve Quadratic Equations given to you using Sin, Cos or Tan. The process is identical to standard Quadratics, but there are even more answers (usually!)
Solve the following Equation
Factorise
Work out what value would make either bracket 0
360 1
Cosθ = 1 has 2 solutions
y = Cosθ
-0.5
Cosθ = -0.5 has 2 solutions 90
180
270
360
120
240
10D

28. Trigonometrical Identities and Equations
You need to be able to solve Quadratic Equations given to you using Sin, Cos or Tan. The process is identical to standard Quadratics, but there are even more answers (usually!)
Solve the following Equation in the range 0 ≤ θ ≤ 360
Work out the acceptable range. Subtract 30
Square root both sides. On fractions root top and bottom separately. Can be positive or negative. 45
135
1/√2
y = Sinθ
-1/√2 90
180
270
360
225
315
360 added to get a value in the range
10D

29. Summary We have learnt 2 important Trigonometrical identities
We have looked at solving Trigonometrical Equations under various circumstances