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Approximate a triangular area inside a square

The diagram shows a square ABCD of side AB=a cm. E is the point on BC such that angle CAE=θ\theta. Show that the area of △ACE\triangle ACE is tan⁡θ1+tan⁡θa2\dfrac{\tan\theta}{1+\tan\theta}a^2.

If θ\theta is sufficiently small for θ3\theta^3 and higher powers of θ\theta to be neglected, show that area⁡(△ACE)≈a2θ(1−θ)\operatorname{area}(\triangle ACE)\approx a^2\theta(1-\theta).

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Square ABCD with A bottom left, B bottom right, C top right and D top left, AB=a cm. E lies on BC below C. Segments AC and AE enclose angle θ at A. The shaded triangle ACE is the difference between right triangles ABC and ABE. Diagram not to scale.
Square ABCD with A bottom left, B bottom right, C top right and D top left, AB=a cm. E lies on BC below C. Segments AC and AE enclose angle θ at A. The shaded triangle ACE is the difference between right triangles ABC and ABE. Diagram not to scale.
Square ABCD with A bottom left, B bottom right, C top right and D top left, AB=a cm. E lies on BC below C. Segments AC and AE enclose angle θ at A. The shaded triangle ACE is the difference between right triangles ABC and ABE. Diagram not to scale.

CJC · Lecture · Example 11

Rewrite with exact identities first, then retain enough terms to survive any multiplication, cancellation or division.

Read the worked solution