The diagram shows a right-angled triangle AOB where AO = 15 cm and BO = 8 cm. P lies on AB such that OP is an arc of a circle with centre A. Q lies on AB such that OQ is an arc of a circle with centre B.

  1. (a)

    Show that angle ABO is 1.0808 radians, correct to 4 decimal places.

    Answer

  2. (b)

    Find the area of the shaded region.

    Answer ... cm^2

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Worked solution

23(a)

1. Use the tangent ratio to find angle ABO

In right-angled triangle AOBAOB,

tan⁡∠ABO=AOBO=158.\tan \angle ABO = \frac{AO}{BO} = \frac{15}{8}.

Hence

∠ABO=tan⁡−1(158)=1.080839… radians.\angle ABO = \tan^{-1}\left(\frac{15}{8}\right) = 1.080839\ldots \text{ radians}.

2. Round the angle to 4 decimal places

1.080839…=1.0808 radianscorrect to 4 decimal places.1.080839\ldots = 1.0808 \text{ radians} \quad \text{correct to 4 decimal places}.

Answer

∠ABO=1.0808 radians(correct to 4 d.p.)\angle ABO = 1.0808\text{ radians} \quad \text{(correct to 4 d.p.)}

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