Porisma

A hint to get started

Approximate a triangle side using radians

In triangle ABCABC, AB=1AB=1, ∠BAC=θ\angle BAC=\theta radians and ∠ABC=3π/4\angle ABC=3\pi/4 radians (see diagram).

(i) Show that AC=1cos⁡θ−sin⁡θAC=\dfrac1{\cos\theta-\sin\theta}.

(ii) For sufficiently small θ\theta, show that AC≈1+aθ+bθ2AC\approx1+a\theta+b\theta^2, determining a,ba,b.

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Triangle ABC: A is at the lower left, B above and to its right, and C at the lower right. AB has length 1, angle BAC is theta, and angle ABC is 3 pi/4 radians. The third angle is pi/4 minus theta.
Triangle ABC: A is at the lower left, B above and to its right, and C at the lower right. AB has length 1, angle BAC is theta, and angle ABC is 3 pi/4 radians. The third angle is pi/4 minus theta.
Triangle ABC: A is at the lower left, B above and to its right, and C at the lower right. AB has length 1, angle BAC is theta, and angle ABC is 3 pi/4 radians. The third angle is pi/4 minus theta.

[N2012/I/4]

CJC · Discussion · Question 12

Find the third angle, apply the sine rule and then expand the reciprocal.

Read the worked solution