MATH / A2 level

Advanced trigonometry and parametric curves

Addition and double-angle formulae transform expressions and solve equations. Radian measure is essential for calculus involving trigonometric functions.

Learning objectives

Explain advanced trigonometry and parametric curves using a concrete example.
Apply the worked method and justify each step.
Recognise the limitation described in the final check.

The big idea

Addition and double-angle formulae transform expressions and solve equations. Radian measure is essential for calculus involving trigonometric functions.

Parametric equations specify x and y using a third variable. The gradient is (dy/dt)/(dx/dt) where dx/dt is nonzero; vertical tangents need separate consideration.

Equations and relationships

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

Parametric gradient where defined.

Worked example

For x=t² and y=t³, find dy/dx at t=2.

Follow the solution

dy/dx=3t²/(2t)=3t/2 for t≠0; at t=2 it is 3.

Investigate and explain

Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.

Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.

Open the interactive investigation

Check your understanding

1. Which equals sin(2x)?

2 sin x
2 sin x cos x
sin²x−cos²x

Check the answer and reasoning

2 sin x cos x

The sine double-angle identity contains both sine and cosine.

2. For x=t² and y=t³, find dy/dx at t=2.

Check the answer and reasoning

dy/dx=3t²/(2t)=3t/2 for t≠0; at t=2 it is 3.

3. Can division by sin x lose solutions of a trig equation?

Yes
No
Only for complex x

Check the answer and reasoning

Yes

Any solutions where sin x=0 must be checked before dividing.