Practice – Non Right-Angled Triangle Trigonometry

1. The sides of a triangle are in arithmetic sequence and the greatest angle is double the smallest angle. Prove that the ratio of its sides is 4:5:6.

2. If  acosA=bcosB , prove that  ABC  is either isosceles, or right angled. 

3. The angles of a triangle are in the ratio 1:2:7, then show that the ratio of the greatest side to the least side is  5+1 : 51 .

4. If the angles of a triangle are  30° and  45° and the included side is  3+1 cm, the area of the triangle is  123+1 cm2.

5. The sides of a triangle are  2, 3, 19 then find the value of the greatest angle of the triangle.

6. The diagram shows a sector of a circle OAB. C is the midpoint of OB. Angle AOB is  θ radians.

Given that the area of the triangle OAC is equal to one quarter of the area of the sector OAB, show that  θ=2sinθ .     [4 marks]


The diagram shows a circle with centre O and radius r cm. The points A and B lie on the circle and AT is a tangent to the circle. Angle AOB = 1 radians and OBT is a straight line.
(i) Express the area of the shaded region in terms of  r and l . [3 marks]

(ii) In the case where r=3 and θ=1.2 , find the perimeter of the shaded region. [4 marks]