Introduction This Chapter involves learning 2 Trigonometrical Identities

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

Trigonometrical Identities and Equations You need to be able to use the Trigonometrical identities O S H A C H O T A 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. Replacing these in the original Equation… Cancel the H’s 10 A

Trigonometrical Identities and Equations You need to be able to use the Trigonometrical identities O A C H S H Hyp 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 1 O T A Opp θ Adj Replace a, b and c This is how it is written 10 A

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. 10 A

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 10 A

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

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

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

Trigonometrical Identities and Equations O 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… 90 180 270 360 y = Sinθ y = Cosθ y = Tanθ S H C A H O T A Example Question Given that Cosθ is -3/5 and θ is reflex, find the value of Sinθ and Tanθ Draw a Right Angled Triangle 5 θ 4 3 You were effectively told A and H in the question. IGNORE the negative for now… The other side should be worked out using Pythagoras’ Theorem… Put in the values from the Triangle Consider the region on the diagram 10 A

Trigonometrical Identities and Equations O 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… 90 180 270 360 y = Sinθ y = Cosθ y = Tanθ S H C A H O T A Example Question Given that Cosθ is -3/5 and θ is reflex, find the value of Sinθ and Tanθ Draw a Right Angled Triangle 5 θ 4 3 You were effectively told A and H in the question. IGNORE the negative for now… The other side should be worked out using Pythagoras’ Theorem… Put in the values from the Triangle Consider the region on the diagram 10 A

Trigonometrical Identities and Equations O 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… 90 180 270 360 y = Sinθ y = Cosθ y = Tanθ S H C A H O T A Example Question Given that Sinθ is 2/5 and θ is obtuse, find the value of Cosθ and Tanθ Draw a Right Angled Triangle 5 θ 2 √ 21 You were effectively told O and H in the question. The other side should be worked out using Pythagoras’ Theorem… Put in the values from the Triangle Consider the region on the diagram 10 A

Trigonometrical Identities and Equations O 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… 90 180 270 360 y = Sinθ y = Cosθ y = Tanθ S H C A H O T A Example Question Given that Sinθ is 2/5 and θ is obtuse, find the value of Cosθ and Tanθ Draw a Right Angled Triangle 5 θ 2 √ 21 You were effectively told O and H in the question. The other side should be worked out using Pythagoras’ Theorem… Put in the values from the Triangle Consider the region on the diagram 10 A

Trigonometrical Identities and Equations You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k Example Question 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 Use Sin-1 This will give you one answer 0. 5 90 30 180 270 360 y = Sinθ 150 10 B

Trigonometrical Identities and Equations You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k Example Question This is similar to the work covered in Chapter 8, and involves using your calculator and Trigonometrical Graphs to solve equations with multiple solutions. Divide by 5 Use Sin-1 Not within the range. You can add 360° to obtain an equivalent value One thing you should pay careful attention to is the range the answers can be within, eg) 0 > x > 360 203. 6 90 -0. 4 180 336. 4 270 360 y = Sinθ 10 B

Trigonometrical Identities and Equations You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k Example Question This is similar to the work covered in Chapter 8, and involves using your calculator and Trigonometrical Graphs to solve equations with multiple solutions. Divide by Cosθ Use Trig Identities One thing you should pay careful attention to is the range the answers can be within, eg) 0 > x > 360 Use Tan-1 2 90 63. 4 180 270 360 y = Tanθ 243. 4 10 B

Trigonometrical Identities and Equations You need to be able to solve Trigonometrical Equations of the form Sin/Cos/Tanθ = k Example Question 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 Use Cos-1 in RADIANS 0. 5 1/ π 2 1/ 3π π 3/ π 2 5/ 2π y = Cosθ 3π 10 B

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 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range 3) Add/Subtract 360 to these values until you have all the answers within the 2θ range 4) These answers are for 2θ. Undo them to find values for θ itself -1 90 180 270 360 y = Cosθ 180 Divide by 2 Adding 360 to the value we worked out (staying within the range) 10 C

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) 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range 3) Add/Subtract 360 to these values until you have all the answers within the (2θ - 35) range 4) These answers are for (2θ – 35). Undo this to find values for θ itself -1 90 180 270 360 y = Sinθ 270 Adding/Subtracting 360 to the value we worked out (staying within the range) Add 35, Divide by 2 10 C

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) 2) Work out one possible answer as before. Find all values in the standard 0 – 360 range 3) Add/Subtract 180 to these values until you have all the answers within the (20 - θ) range 4) These answers are for (20 – θ). Undo this to find values for θ itself 3 71. 6 251. 6 90 180 270 360 y = Tanθ Adding/Subtracting 180 to the values we worked out (staying within the range) Subtract 20 Multiply by -1 10 C

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 10 D

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 1 Sinθ = 2 has no solutions 90 90 Sinθ = 1 has 1 solution 180 270 360 y = Sinθ 10 D

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 1 -0. 5 0 360 90 120 180 270 360 Cosθ = 1 has 2 solutions y = Cosθ = -0. 5 has 2 solutions 240 10 D

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. 1/ 45 135 √ 2 -1/√ 2 90 180 225 270 360 y = Sinθ 315 360 added to get a value in the range 10 D

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