_{Up Learn – A Level maths (edexcel) – Parametric Equations}

_{Up Learn – A Level maths (edexcel) – Parametric Equations}

**Integrating Parametric Equations: Summary**

**At A Level, you may be asked to integrate parametric equations. Sometimes, the best approach is converting to Cartesian form first. For other cases, there’s a way of integrating the parametric equations without converting.**

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### More videos on Parametric Equations:

^{Introduction to Parametric Equations (free trial)}

^{What are Parametric Equations? }

^{Parametric Functions Tables}

^{The Coordinates Given by Parametric Equations}

^{Sketching the Curves of Parametric Equations (free trial)}

^{Sketching Curves within a Restricted Domain (free trial)}

^{What is a Parameter? (free trial)}

^{Turning Parametric Equations into a Cartesian Equation (free trial)}

^{Taking Shortcuts When Finding Cartesian Equations} ^{(free trial)}

^{Turning Cartesian Equations into Parametric Equations} ^{(free trial)}

^{Trigonometric Parametric Equations (free trial)}

^{Converting When the Trig Functions are the Same (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 integrating parametric equations.

If we’re given a curve that’s defined parametrically…[x=t , y=t , 0≤t≤π ]

And asked to integrate y with respect to x…

It might be possible to just convert the parametric equations into cartesian form first.[y=1x ]

And then integrate as normal.[∫y dx=ln x +c](pause for 1 sec)

If we want to avoid converting parametric equations to cartesian form before integrating…[x=3t+4, y=4cos t , 0≤t≤2]

We can just take our equation for y…[highlight y=4cos t ]

And take the integral of both expressions with respect to x…[y dx=4cos t dx]

Then, take our equation for x…[x=3t+4]

Differentiate… [dxdt=3]

And make this the subject, so we can convert “dx”.[dx=3dt, plug it into the integral so we get:y dx=4cos t (3) dt ]

Finally, expand and integrate as normal.

[y dx=12cos t dt]

[y dx=12sin t +c](pause for 1 sec)

And if we now want to find an area under the curve…[between 4 and ]

We could rewrite our integral in terms of x. [t=4x-π12, plug into the result to get y dx=12sin 4x-π12 ]

…And evaluate it between the boundaries of our area. [4y dx=[12sin 4x-π12 ]4=62](pause for 1 sec)

However, the smartest thing to do will almost definitely be to convert the boundaries [highlight 4 and ] into t values instead. [take the formula t=4x-π12 and replace the boundaries with t=4(4)-π12 and t=4()-π12, and morph into 0 and 4]

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