IB Mathematics SL 3.2 Use of sine, cosine and tangent ratios AA SL Paper 2- Exam Style Questions- New Syllabus

(a)
Using the cosine rule:
\( BC^2 = AB^2 + AC^2 – 2(AB)(AC)\cos(\angle BAC) \)
\( BC^2 = 7^2 + 12^2 – 2 \times 7 \times 12 \times \cos 116^\circ \)
\( BC^2 = 49 + 144 – 168\cos 116^\circ \)
\( BC^2 = 193 – 168(-0.43837…) \)
\( BC^2 \approx 193 + 73.646… \approx 266.646… \)
\( BC \approx \sqrt{266.646…} \approx 16.3293 \)
\( BC \approx 16.3 \) (to 3 s.f.)

(b)
Using the sine rule:
\( \dfrac{\sin(\angle ACB)}{AB} = \dfrac{\sin(\angle BAC)}{BC} \)
\( \dfrac{\sin(\angle ACB)}{7} = \dfrac{\sin 116^\circ}{16.3293…} \)
\( \sin(\angle ACB) = \dfrac{7 \times \sin 116^\circ}{16.3293…} \)
\( \sin(\angle ACB) \approx \dfrac{7 \times 0.89879…}{16.3293…} \approx 0.3854 \)
\( \angle ACB \approx \arcsin(0.3854) \approx 22.6618^\circ \)
\( \angle ACB \approx 22.7^\circ \) (to 3 s.f.)