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Parametric equations

Using trigonometric identities

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

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Modelling with statics

Friction with static particles

Statics of rigid bodies

Dynamics and inclined planes

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Projectiles: Horizontal and vertical components

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Equation of the trajectory of a projectile

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Moments

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Centre of mass

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Motion in two dimensions

Connected particles

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Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

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Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

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The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

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Probability formulae

Conditional probability: Tree diagrams

Correlation and hypothesis testing

Exponential models

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Hypothesis testing I

Hypothesis testing

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One-tailed tests

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Binomial distribution

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The binomial distribution

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Probability

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Venn diagrams

Probability: Mutually exclusive and independent events

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Representation of data

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Box plots

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Sampling techniques

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Vectors II

3D coordinates

Vectors in 3D

Solving geometric problems

Applications to mechanics: 3D vectors

Numerical methods

Locating roots

Iteration

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Newton-Raphson method

Numerical methods: Applications to modelling

Integration II

Integrating standard functions

Integrating f(ax + b)

Integration with trigonometric identities

Reverse chain rule

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Finding areas using integration

The trapezium rule

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Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

Addition formulae

Using the angle addition formulae

Double angle formulae

Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

Modelling with trigonometric functions

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Graphs of sec x, cosec x and cot x

Solving equations with sec x, cosec x and cot x

Further trigonometric identities

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Radians

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Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

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Sum to infinity

Sigma notation

Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Using trigonometric identities

Sketching parametric curves

Points of intersection of parametric curves

Modelling with parametric equations

Functions

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The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

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Modulus inequalities

Algebraic fractions

Operations with algebraic fractions

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Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

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Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

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Integration I

Integrating x^n

Indefinite integrals

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Area between a curve and a line

Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationary points

Sketching gradient functions

Modelling with differentiation

Exponentials and logarithms

Exponential functions

y = e^x

Exponential modelling

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Natural logarithms

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Angles in all four quadrants

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Simple trigonometric equations

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Equations and identities

Trigonometry

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The sine rule

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Trig graphs: sin x, cos x, tan x

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Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

Solving binomial problems

Binomial estimation

Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

Circles and triangles

Intersection of two circles

Straight line graphs

y = mx + c

Equations of straight lines

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Length and area

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Transformations

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Transforming graphs

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Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

Functions

Expanding brackets

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Polynomial division

The factor theorem

Equations and inequalities

Linear simultaneous equations

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Set notation and interval notation

Linear inequalities

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Inequalities on graphs

Inequality regions

Quadratics

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Solving disguised quadratics

Indices and surds

Laws of indices

Negative and fractional indices

Surds

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Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Alice

Summary

Using trigonometric identities

In a nutshell

You can convert trigonometric parametric equations into Cartesian form using trigonometric identities. Make sure that your calculator is set to work with angles in radians.

Example 1

A curve has parametric equations $x(t)=\cos(t) +4$ and $y(t) = \sin(t) -5$ , $0 \lt t \lt 2\pi$ .

i) Find the Cartesian equation of the curve given by the parametric equations.

ii) Hence sketch the curve.

i) Write the expressions for $\sin(t)$ and $\cos(t)$ in terms of $x$ and $y$ .

$\begin{aligned}x& =\cos(t) +4 \\\cos(t) &= x - 4 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \textcircled{1} \\y&=\sin(t) -5 \\\sin(t )&= y + 5 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \textcircled{2} \end{aligned}$ 

Substitute $\textcircled{1}$ and $\textcircled{2}$ into $\sin^2(t) + \cos^2(t) \equiv 1$ .

$\underline{(y+5)^2+(x-4)^2=1}$ 

ii) $(x-a)^2+(y-b)^2=r^2$ is the equation of a circle with centre $(a,b)$ and radius $r$ . Hence this curve is a circle with centre $(4,-5)$ and radius $1$ .

Maths; Parametric equations; KS5 Year 13; Using trigonometric identities

Example 2

A curve has parametric equations $x=2\cot^2(2t)$ and $y = 2\sin^2(2t)$ , $0 \lt t \le \dfrac{\pi}{4}$ .

i) Find a Cartesian equation of the curve in the form $y = f(x)$ .

ii) State the domain on which $f(x)$ is defined.

i) Rewrite $\cot^2(2t)$ in terms of $\sin(2t)$ and $\cos(2t)$ .

$\begin{aligned}y&=2\sin^2(2t) \\x&=2\cot^2(2t) = \dfrac{2\cos^2 (2t)}{\sin^2 (2t)} \end{aligned}$ 

Eliminate the $\sin^2(2t)$ from $x$ by multiplying $x$ and $y$ .

$xy = 2\sin^2(2t) \times \dfrac{2 \cos^2(2t)}{\sin^2(2t)} = 4 \cos^2 (2t)$ 

Use the trigonometric identity $\sin^2 (x)+ \cos^2 (x) \equiv 1$ .

$2\sin^2 (2t) + 2\cos^2(2t) \equiv 2$ 

Substitute in $y$ and $xy$ and manipulate algebraically.

$\begin{aligned} y+\dfrac{xy}{2}&=2 \\2y+xy &= 4 \\y(x+2) &=4 \\y&= \underline{\dfrac{4}{x+2}} \end{aligned}$ 

ii) The domain is the range of $x(t)$ :

$\underline{x \gt 0}$ 

Learn with Basics

Length:

Unit 1

Equations and identities

Unit 2

Trigonometric identities

Jump Ahead

Optional

Unit 3

Using trigonometric identities

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FAQs - Frequently Asked Questions

Which trigonometric identities should you use when working with parametric equations?

You can use any of the trigonometric identities you have learnt, such as sin2x=2sinxcosx.

Should you use degrees or radians when working with trigonometric parametric equations?

You should use radians when working with trigonometric parametric equations.

How do you convert trigonometric parametric equations into Cartesian form?

You can convert trigonometric parametric equations into Cartesian form using trigonometric identities.