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Question 3

From Math Analysis HL Paper 1 (May 2025, TZ2)

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  1. [Maximum mark: 7]

The following diagram shows a non-right angled triangle ABC.

Diagram showing a non-right angled triangle ABC with side AB=5, side BC=6√2, angle ACB=θ, and angle BAC=2θ.
Diagram showing a non-right angled triangle ABC with side AB=5, side BC=6√2, angle ACB=θ, and angle BAC=2θ.

AB=5AB = 5, BC=62BC = 6\sqrt{2}, AC^B=θA\hat{C}B = \theta and BA^C=2θB\hat{A}C = 2\theta, where 0<θ<π20 < \theta < \frac{\pi}{2}.

(a) Using the sine rule, show that cos⁡θ=325\cos \theta = \frac{3\sqrt{2}}{5}. [3]

(b) Hence, find sin⁡θ\sin \theta. [2]

Point D is located on [AC] such that the area of triangle BCD is 2142\sqrt{14}.

(c) Find DC. [2]