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Geometric optimisation with a trigonometric model

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For the variable area below, calculate its coefficient K.

\[A(u)=\frac{1}{2}(5)(6)\sin u,\quad K=\frac{1}{2}(5)(6)\]
Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, Calculator allowedBuild a worksheet

Trigonometric geometric optimisation

A geometric constraint can make the side lengths functions of an angle. Substitute those lengths into an area model, differentiate with respect to the angle and compare the relevant values

\[A(u)=C+K\sin(2u)\]
  1. Relate lengths to the angle Use the given geometry to express each changing side length in terms of the angle before forming the area model \(\displaystyle L_1=8\sin u,\quad L_2=8\cos u\)
  2. Use the area formula For a triangle, take half the product of the two side lengths and the sine of their included angle \(\displaystyle A_{\triangle}=\frac{1}{2}ab\sin u\)
  3. Rewrite the trigonometric product Use a double-angle identity when the model contains the product of sine and cosine \(\displaystyle \sin u\cos u=\frac{1}{2}\sin(2u)\)
  4. Test the optimum Find stationary angles from the derivative and compare their model values with any required endpoints \(\displaystyle A'(u)=0\quad\text{and compare }A(u)\)

Worked example

A shape consists of a fixed region and a right triangle with the dimensions below. Find the maximum total area

\[A_{\mathrm{fixed}}=3\text{ cm}^2,\quad c=8\text{ cm},\quad 0\leq u\leq\frac{\pi}{2}\]
  1. \(\displaystyle a=8\sin u,\quad b=8\cos u\)Resolve the two legs of the right triangle using the hypotenuse and angle u.
  2. \(\displaystyle A(u)=3+\frac{1}{2}(8\sin u)(8\cos u)=3+32\sin u\cos u=3+16\sin(2u)\)Use the triangle area formula and the double-angle identity to form the total area
  3. \(\displaystyle A'(u)=32\cos(2u)=0\quad\Rightarrow\quad 2u=\frac{\pi}{2}\quad\Rightarrow\quad u=\frac{\pi}{4}\)Set the derivative to zero to find the stationary angle in the interval
  4. \(\displaystyle A\left(\frac{\pi}{4}\right)=3+16\sin\left(\frac{\pi}{2}\right)=19\text{ cm}^2\)The stationary value is 19 cm², greater than the endpoint value 3 cm², so it is the maximum

Common mistakes

Dropping the triangle half
✕\(\displaystyle A_{\triangle}=ab\sin u\)✓\(\displaystyle A_{\triangle}=\frac{1}{2}ab\sin u\)
The area of a triangle is half the product of the two side lengths and the sine of their included angle
Setting the original model to zero
✕\(\displaystyle A(u)=0\quad\text{for a maximum}\)✓\(\displaystyle A'(u)=0\quad\text{for a stationary candidate}\)
A stationary candidate is found where the derivative is zero, and the original model is then compared at the candidate and endpoints
Missing the inner factor
✕\(\displaystyle \frac{d}{du}\left(5+8\sin(2u)\right)=8\cos(2u)\)✓\(\displaystyle \frac{d}{du}\left(5+8\sin(2u)\right)=16\cos(2u)\)
The chain rule multiplies by the derivative of the inner angle 2u

Difficulty levels

introVerify a trigonometric area or length formula from a geometric diagram
standardDifferentiate a geometric trigonometric model and evaluate its derivative at a given angle
challengeFind the angle and value that optimise a trigonometric geometric model
examBuild a composite model from side relationships, find stationary values and check the endpoints

Questions about this topic

Why can a geometric area contain sin(2u)?+

Two side lengths may contribute a product such as sin u cos u, which equals one half of sin(2u)

Why must the endpoints be checked?+

For the smooth area models here, an optimum on a closed interval occurs at a stationary point or at an endpoint

What does the angle represent in the model?+

It is the variable geometric angle that determines the changing side lengths and therefore the area

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 5: Differentiation

In this chapter

Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method