_{Up Learn – A Level maths (edexcel) – The Compound Angle Identities}

_{Up Learn – A Level maths (edexcel) – The Compound Angle Identities}

**Compound Angle Formulae: Summary**

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

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### More videos on The Compound Angle Identities:

^{Introduction to the Compound Angle Identities (free trial)}

^{What are Compound Angles? (free trial)}

^{The Compound Angle Identity for Sine (free trial)}

^{Finding More Outputs from the Sine Function (free trial)}

^{The Compound Angle Identity for Cosine (free trial)}

^{Finding More Outputs from the Cosine Function (free trial)}

^{Compound Angles with One Unknown (free trial)}

^{Solving More Trigonometric Equations (free trial)}

^{How Do We Simplify asinx + bcosx? (free trial)}

^{Dividing Equations (free trial)}

^{Dividing and Simplifying Equations (free trial)}

^{Solving Simultaneous Trigonometric Equations Part 1 (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 compound angle identities.

A compound angle is made up of multiple angles… either added together, or subtracted from each other.

And whenever we have a trig function with a compound angle for an input…

… It’s possible to rewrite that function, without the compound angle, using an identity.

[Stack the 3 identities vertically. Maybe show the LHSs on the first line of above and the RHSs on the 2nd..

sin(A±B)≡ABAB

cos(AB)cos(A)cos(B)sin(A)sin(B)

tan(AB)=tan A tan B 1tan A tan B ]

The compound angle identity for sine is this… [emphasise sin(A±B)≡

ABAB ]

The compound angle identity for cosine is this. [emphasise cos(AB)cos(A)cos(B)sin(A)sin(B)]

And the compound angle identity for tan, is this. [emphasise tan(AB)=tan A tan B 1tan A tan B ]

These identities are useful if we want to find new outputs from the trig functions, using outputs we already know.(wait for animation and pause for 2 sec) [Show this process using dynamic “drag and copy, then morph” approach with colour coding on the 30 and 45.sin(75)=sin 30+45 sin 30 cos 45 +cos 30 sin(45)=1222+3222=2+64]

They’re also useful for rewriting trig functions like this one, where the input is a compound angle and one part is unknown.(wait for animation and pause for 2 sec) [cos (x-45°)=cos(x)cos(45)+sin(x)sin(45)= 22cos (x) +22sin (x)]

For example, rewriting this function using a compound angle identity eventually allows us to solve the equation.. [Show sin (x+3) -cos (x) =0 from the start of the line and pulse box sin(x+pi/3) on “this”. Make sure the compound angle identity is also visible: sin(A±B)≡ABAB ]

We rewrite… [→ sin(x)cos(3) + cos(x)sin(3)-cos(x)=0 ]

… simplify… [12sin(x)+32cos (x)-cos(x)=0 ][12 sin(x)+3-22cos(x)=0]

And then use the tangent identity.(click once and wait for animation) [sin(x)cos(x)tan(x)] [Then show the steps to solving but can just bring them up one by one, no fancy animations needed: 12 sin(x)=-3-22cos(x) –>sin(x)=-(3-2)cos(x) –>sin(x)=(2-3)cos(x) –> sin(x)cos(x)=(2-3) –> tan(x)=(2-3) –> x=tan-1(2-3)=12 ]

And many equations can be solved this same way.

Next, there’s a whole set of other equations we can solve by using the compound angle identities in a different way.

In this case, this pesky term prevents us from dividing sin by cos and getting just one tan function like before. [Show trying to do this and show that we get 6tan(x) + 7 = 5/cos(x). Show that the 5/os(x) is a problem maybe with a wobbling ? over that term or a big cross or something]

Instead, we use a crafty trick: we use a compound angle identity in reverse.

[6sin x +7cos x =5, 0≤x≤180°] [Full working:

First, whenever we have some amount of sin(x) add some amount of cos(x), it’s possible to rewrite that as a single trig function with a compound input.

So we do that, putting an unknown here and here.

Second, we use this identity to rewrite this again.

Third, we equate the coefficients of sine x (wait 1 sec) and cos x.(wait 1 sec) [Step 3]

Fourth, we solve these simultaneous equations. [Step 4]

To find alpha, we divide the equations and then use the tangent identity. [tan(θ)=sin(θ)/cos(θ)]

And to find an exact value for R, we square these two values, add them and square root. [The 6 and the 7 in step 3]

Fifth, we put our values for R and alpha back into this equation. [Rsin(x+alpha)=5]

And now we’ve rewritten the left hand side in this form as planned. [Rsin(x+alpha)]

To finish, we solve as normal using the inverse trig function, making sure to find all the values in the range.(click once and pause for 4 sec) [Can just put the steps up one at a time but fairly quickly – this step is prereq knowledge]

Finally, in steps 1 and 2, we could instead have written this [Rcos(xα)] and used this identity… [cos(xα)cos(x)cos(α)sin(x)sin(α)]

But the rest of the process remains the same.

And actually, if you need to use this method in the exam, most of the time, they’ll tell you whether they want a sine function [emphasise Rsin(xα)] or a cos function. [emphasise Rcos(xα)]

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