Non-Right-Angled Triangles — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)Sine rule: a/sin A = b/sin B, so b = a × sin B / sin A.

b = 8 × sin65° / sin40° = 11.3 cm (3 s.f.)

Question 2

(a)Cosine rule: c² = a² + b² − 2ab cos C.

c² = 7² + 9² − 2(7)(9)cos55° = 57.7…; c = 7.60 cm (3 s.f.)

Question 3

(a)Area = ½ab sin C.

½ × 7 × 9 × sin55° = 25.8 cm² (3 s.f.)

Question 4

(a)Angles sum to 180°: A = 180−50−70=60°. Sine rule: b = a×sinB/sinA.

A = 60°; b = 10×sin50°/sin60° ≈ 8.85 cm (3 s.f.)

Question 5

(a)cos C = (a²+b²−c²)/(2ab).

cosC = (64+100−169)/160 = −0.03125; C ≈ 91.8° (3 s.f.)

Question 6

(a)Area = ½ab sinC.

½(8)(10)sin(91.8°) ≈ 40.0 cm² (3 s.f.)

Question 7

(a)The sine rule is used when you know two angles and a side, or two sides and a non-included angle.

When two angles and one side are known (or two sides and a non-included angle)

Question 8

(a)c² = a² + b² − 2ab cos C.

c² = 36+81−2(6)(9)cos40° ≈ 34.3; c ≈ 5.85 cm (3 s.f.)

Question 9

(a)cos A = (b²+c²−a²)/(2bc).

cosA = (64+81−49)/144 = 0.667; A ≈ 48.2° (3 s.f.)

Question 10

(a)This is the standard cosine rule.

a² = b² + c² − 2bc cos A

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