_{Up Learn – A Level maths (edexcel) – The Double and Half Angle Identities}

_{Up Learn – A Level maths (edexcel) – The Double and Half Angle Identities}

**Double Angle and Half Angle Formulae: Summary**

**Here’s a summary of everything you need to know about the double and half angle identities – otherwise known as the double and half angle formulae – for A Level.**

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### More videos on The Double and Half Angle Identities:

^{Introduction to the Double and Half Angle Identities (free trial)}

^{What is a Double Angle?(free trial)}

^{The Sine Double Angle Identity (free trial)}

^{Using the Sine Double Angle Identity Part 1 (free trial)}

^{Using the Sine Double Angle Identity Part 2 (free trial)}

^{The Cosine Double Angle Identity Part 1 (free trial)}

^{The Cosine Double Angle Identity Part 2 (free trial)}

^{Using the Cosine Double Angle Identity Part 1 (free trial)}

^{Using the Cosine Double Angle Identity Part 2 (free trial)}

^{Using the Cosine Double Angle Identity Part 3 (free trial)}

^{Using the Cosine Double Angle Identity Part 4(free trial)}

^{Using the Cosine Double Angle Identity Part 5 (free trial)}

## Trigonometry

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2. The Sine Rule

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3. Using the Sine rule to calculate unknown side lengths

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4. Using the Sine rule to calculate unknown angles

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5. When the Sine Rule Identifies Two Possible Angles

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6. When the Sine Rule Identifies Two Possible Angles

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7. Why does subtracting from 180 give us the size of obtuse angles?

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8. The Cosine Rule

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9. Using the Cosine Rule to calculate unknown angles

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10. Choosing between Soh Cah Toa, Sine Rule and Cosine Rule

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11. Finding Triangle Area Using the Sine Function

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2. Sine and Cosine’s First Outputs – Part 1

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3. Sine and Cosine’s First Outputs – Part 2

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4. Tangent’s First Outputs

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5. Where Does Sine Get Its Name From?

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6. All the Other Outputs For Sine – Part 1

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7. All the Other Outputs For Sine – Part 2

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8. All the Other Outputs For Cosine

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9. All the Other Outputs For Tangent

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9. Negative Outputs from the Tangent Function

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9. Why is the Output from Tangent Opposite over Adjacent? – Part 1

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9. Why is the Output from Tangent Opposite over Adjacent? – Part 2

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9. The First Trigonometric Identities

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2. Substituting Trig Functions – Part 1

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3. The CAST Diagram

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4. Substituting Trig Functions – Part 2

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5. Acute Angles and The Sine Function – Part 1

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6. Acute Angles and The Sine Function – Part 2

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7. Acute Angles and The Sine Function – Part 3

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8. Acute Angles and the Cosine Function

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9. Acute Angles and the Tangent Function

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10. Solving Linear Equations with Trigonometric Functions – Part 1

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11. Solving Linear Equations with Trigonometric Functions – Part 2

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12. Solving Linear Equations with Trigonometric Functions – Part 3

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13. Solving Trig Equations Using the Tan Identity – Part 1

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14. Solving Trig Equations Using the Tan Identity – Part 2

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15. Solving Trig Equations Using the Tan Identity – Part 3

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16. When the Tan Identity Can’t Be Used

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17. Solving Trig Equations Using the Pythagorean Identity – Part 1

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18. Solving Trig Equations Using the Pythagorean Identity – Part 2

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19. Solving Trig Equations When the Input Isn’t Theta – Part 1

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20. Solving Trig Equations When the Input Isn’t Theta – Part 2

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21. Solving Trig Equations When the Input Isn’t Theta – Part 3

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22. Recognising quadratic equations with trigonometric functions – Part 1

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23. Recognising quadratic equations with trigonometric functions – Part 2

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24. Solving Quadratic Equations Involving Trig Functions

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2. Measuring Arc Length Part 1

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3. Measuring Arc Length Part 2

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4. Measuring the Area of a Sector Part 1

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5. Measuring the Area of a Sector Part 2

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6. Converting Between Different Units

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7. Converting from Degrees to Radian Measure

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8. Converting from Radian Measure to Degrees

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9. What is a Radian?

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10. Arc Length in Radian Measure

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11. Area of a Sector in Radian Measure

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12. Radian Mode and Trigonometry

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13. Area of a Segment in Radian Measure

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14. Hipparchus’ Triangles in Radians

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15. The CAST Diagram in Radians

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16. Trigonometric Curves in Radians

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17. Solving Trigonometric Equations in Radian Measure

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18. Degrees or Radians?

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2. What is a ‘Small Angle’?

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3. Approximating Sin Part 1

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4. Approximating Sin Part 2

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5. Approximating Tan Part 1

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6. Approximating Tan Part 2

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7. Approximating Cos Part 1

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8. Approximating Cos Part 2

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2. Arcus Functions

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3. The Inverses of Trig Functions

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4. The Problem with Inverse Trig Functions

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5. Restricting the Domains of Trig Functions (Part 1)

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6. Restricting the Domains of Trig Functions (Part 2)

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7. Memorising the Graph of the Inverse Sin Function

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8. Memorising the Graph of the Inverse Cos Function

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9. Memorising the Graph of the Inverse Tan Function

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2. What are Compound Angles?

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3. The Compound Angle Identity for Sine

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4. Finding More Outputs from the Sine Function

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5. The Compound Angle Identity for Cosine

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6. Finding More Outputs from the Cosine Function

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7. Compound Angles with One Unknown

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8. Solving More Trigonometric Equations

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9. How Do We Simplify asinx + bcosx?

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10. Dividing Equationss

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11. Dividing and Simplifying Equations

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12. Solving Simultaneous Trigonometric Equations Part 1

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13. Solving Simultaneous Trigonometric Equations Part 2

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14. Finding the Exact Value of y Part 1

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15. Finding the Exact Value of y Part 2

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16. Coefficients in Identities Are Always the Same

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17. Equating Coefficients in Conditional Identities

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18. Simplifying asinx + bcosx Part 1

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19. Simplifying asinx + bcosx Part 2

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20. Simplifying asinx + bcosx Part 3

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21. Simplifying asinx + bcosx Part 4

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22. Solving Harder Trigonometric Equations

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23. The Compound Angle Identity for Tangent

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24. Finding More Outputs from the Tangent Function

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25. Solving More Trigonometric Equations Part 3

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26. Solving More Trigonometric Equations Part 4

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27. Proving the Compound Angle Identities

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28. Proof Part 1: Side Lengths for the Three Triangles

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29. Proof Part 2: The Secret Triangle

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30. Proof Part 3: Proving the Sine and Cosine Identities

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31. Proving the Tangent Identity

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2. What is a Double Angle?

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3. The Sine Double Angle Identity

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4. Using the Sine Double Angle Identity Part 1

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5. Using the Sine Double Angle Identity Part 2

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6. The Cosine Double Angle Identity Part 1

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7. The Cosine Double Angle Identity Part 2

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8. Using the Cosine Double Angle Identity Part 1

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9. Using the Cosine Double Angle Identity Part 2

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10. Using the Cosine Double Angle Identity Part 3

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11. Using the Cosine Double Angle Identity Part 4

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12. Using the Cosine Double Angle Identity Part 5

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13. The Tangent Double Angle Identity

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14. Using the Tangent Double Angle Identity Part 1

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15. Using the Tangent Double Angle Identity Part 2

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16. Using the Double Angle Identities with Other Angles

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17. Finding the Half Angle Identities

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2. The Cosecant Function

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3. Outputs from the Cosecant Function

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4. Solving Equations with Cosecant Functions

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5. The Secant Function

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6. Outputs from the Secant Function

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7. Solving Equations with Secant Functions

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8. The Cotangent Function

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9. Outputs from the Cotangent Function

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10. Solving Equations with Cotangent Functions Part 1

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11. Solving Equations with Cotangent Functions Part 2

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12. Solving Equations with Multiple Reciprocal Functions

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13. More Pythagorean Identities Part 1

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14. Solving Equations with the Pythagorean Identities Part 1

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15. More Pythagorean Identities Part 2

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16. Solving Equations with the Pythagorean Identities Part 2

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17. Memorising the Graph of the Cosecant Function

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18. Memorising the Graph of the Secant Function

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19. Memorising the Graph of the Cotangent Function

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20. Where Do the Reciprocal Functions Get Their Names?

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21. What Are Complementary Angles?

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22. Tangents and Secants

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23. Where Does Tangent Get Its Name?

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24. Where Does Secant Get Its Name?

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25. Where Does Cotangent Get Its Name?

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26. Where Does Cosecant Get Its Name?

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27. Proving the Pythagorean Identities

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2. What are Parametric Equations?

3. Parametric Functions Tables

4. The Coordinates Given by Parametric Equations

5. Sketching the Curves of Parametric Equations

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6. Sketching Curves within a Restricted Domain

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7. What is a Parameter?

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8. Turning Parametric Equations into a Cartesian Equation

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9. Taking Shortcuts When Finding Cartesian Equations

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10. Turning Cartesian Equations into Parametric Equations

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11. Trigonometric Parametric Equations

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12. The Problem with Trigonometric Parametric Equations

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13. Converting When the Trig Functions are the Same

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14. Converting When the Trig Functions are Reciprocals

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15. Using the Pythagorean Identity

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16. A Faster Way to Use the Pythagorean Identity

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17. Using the Other Pythagorean Identities

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18. The Secret Power of the Reciprocal Identities Part 1

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19. Parametric Equations We Can Convert So Far

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20. Multiple Trig Functions in One Parametric Equation

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21. Using the Double Angle Identities

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22. Using the Compound Angle Identities

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23. Negative Solutions to Parametric Equations

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24. The Domain and Range of Parametric Equations

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25. Finding Unknown Coordinates

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26. Finding Points of Intersection with Parametric Curves

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27. What if Both Curves Are Defined Parametrically?

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Here’s a reminder of the key points you should know about the double and half angle identities.

A double angle is any compound angle that is the sum of two equal angles.

There are double angle identities for sin, cos and tan which allow us to rewrite trig functions without the double angles. [show them greyed out]

First, the double angle identity for sine is this. [sin(2A) 2sin(A)cos(A)] [zoom in on it or grey out/ remove the other two]

If an equation we’re trying to solve contains sine functions with different multiples of the same input…

… this identity may provide a route to solving the equation.(wait 3 sec) [3sin (2x) -2sin x =0 → 3(2sin(x)cos(x)) – 2sin(x) =0 keep the identity on screen here and make it clear the green bit is what’s changing. Then show the next few lines of working too. 6sin(x)cos(x) -2sin(x) = 0 → 2sin(x)(3cos(x)-1)=0 → 2sin(x)=0 or 3cos(x)=1.]

Or, if we have sine and cosine functions multiplied together, we can use this identity in reverse. [e.g. 8sin(x)cos(x) =3 → 4sin(2x) =3 keep the identity on screen here and make it clear the green bit is what’s changing] (wait 2 sec)

Second, the double angle identity for cosine can be written in 3 ways. [cos(2A) cos2(A)-sin2(A) or cos(2A)1-2sin2(A) or cos(2A)2cos2(A)-1 but can grey out “cos(2A)” for the 2nd and 3rd ones]

If an equation we’re trying to solve contains cos functions with different multiples of the same input… [cos(2x) = 3cos(x) -4]

Any of these identities may provide a route to solving the equation. [Show using the bottom of the 3 identities to change to 2cos^{2}(x)-1 = 3cos(x) -4…. Then 2cos^{2}(x)-3cos(x)+3=0]

And it doesn’t matter which of the three we use: if one of them works, all three of them will work eventually!

But this flowchart can gives us a good idea of which identity is most likely to be useful.(wait 5 sec) [This but include the full identity in each box.]

Alternatively, if we recognise that the equation we want to solve contains a term in one of these forms…(click once) [show all the identities and highlight the RHSs]

Third, the double angle identity for tan is this. [tan(2A) 2tan(A)1-tan2(A)]

If an equation we’re trying to solve contains tan functions with different multiples of the same input…

This identity may provide a route to solving the equation.

[Show this example making it clear that we’re using the identity to change the green bit: 6tan (A) = tan (2A) →6tan (A)= 2tan(A)1-tan2(A) ]

… or when part of the equation is in this form, we can use the identity in reverse. [tan(A)1-tan2(A) And then show this example making it clear that we’re using the identity to change the green bit 4tan(x)1-tan2(x)=3 → 2tan(2A)=3]

And all of the double angle formulae can work with other angles too…

… as long as these angles are always double these angles. [Show sin(2A)2sin(A)cos(A), cos(2A)cos2(A)-sin2(A), tan(2A)2tan(A)1-tan2(A) and highlight all the 2A’s at once followed by all copies of just A at once (not the 2A’s this time).]

So, for instance, these all hold true. (wat for animation and wait 3 sec) [For these 3 examples, the important thing is that the angle on the left is twice the angle on the right so let’s highlight the angles.] [Maybe change one example at a time so they can focus on one at a time. So first for the sin(2A) example, show the equation, highlight 2A and A, then change to 20 and 10. Then same for the other examples.]

[sin(2A)2sin(A)cos(A) → sin(20)2sin(10)cos(10)]

[cos(2A)cos2(A)-sin2(A) → cos(4)cos2(A)-sin2(2)]

[tan(2A)2tan(A)1-tan2(A) → tan(25)2tan(12.5)1-tan2(12.5)]

Finally, here are the half angle identities for sine… [sin(x2)1-cos(x)2]

… and cos. [cos(x2)1+cos(x)2]

And you might find it useful to remember these, but technically the double angle identities can do anything these can, so memorising these is not strictly necessary.

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