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mathematics UK Year 11 Mathematics (KS4)
 
Solved problems for the chapter:
 

Sine and cosine rules

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:sine_and_cosine_rules/5351/


Example 1.

The diagram below shows triangle ABC:




Given that the length of side a is 6 cm, side b is 3 cm and side c is 6 cm. Calculate angle .

    
Example 2.

The ratio between sides of the parallelogram is 7:8. The diagonal e measures 14.44cm, and one of its angle measures 58°. Calculate the sides of the parallelogram.

    
3.

The diagram below shows a triangle ABC:




Given that the lengths of side b is 4 cm, side c is 4 cm and angle is 59°, calculate the length of side a.

    
4.

The diagram below shows a triangle ABC:




Given that the length of side a is 5 cm, side c is 4 cm and angle is 108°. Calculate angle .

    
5.

The length of the longer diagonal of a trapezium measures 7 cm, the parallel sides are 7.93 cm and 3 cm with one of its interior angle as 120 °. Calculate the length of the other diagonal.

    
6.

The length of the longer diagonal of a trapezium measures 5.80 cm, the two parallel sides are 7 cm and 5 cm while one of the non-parallel sides is 4 cm. Calculate the length of the other diagonal.

    
7.

The diagram below shows a triangle ABC:




Given that the angle of is 44°, is 81° and the length of side c is 4 cm. Calculate the length of side a.

    
8.

The diagram below shows a triangle ABC. Given that the side a is 3.35 cm, and the angle is 47.15°. The angle bisector intersects side c at point D. The quotient is 0.90. Calculate the missing interior angles and sides in a triangle.


    

9.

The diagonals of a trapezium measure 9.47 and 7 cm. One of the parallel side of the trapezium is 7 cm and a non parallel side is 4.49 cm. If one of its interior angle is 71.32 °, calculate the remaining angles and sides of the trapezium.

    
10.

The height of trapezium measures 5 cm, its two parallel sides are 6 cm and 3 cm with an interior angle measuring 66 °. Calculate the length of the diagonals in the trapezium.

    
11.

The length ratio of the non parallel sides of a trapezium is 1.00 and one of its interior angle measures 61 °. Calculate the remaining interior angles in the trapezium.

    
12.

Two non-parallel sides of a trapezium measure 6.39 and 8.80 cm, and height 6 cm. If its perimeter is 31.81 cm. Calculate all interior angles and the remaining sides of the trapezium.

    
13.

The diagram below shows an isosceles triangle ABC. The perimeter of triangle ABC is 16.97 cm, the length of side c is 8 cm and side a is 3 cm. Calculate for .


    
14.

The diagram below shows a triangle ABC.




Given that is 48°, is 73.53°, side a is 6.97 cm and the area of the triangle is 26.75 , calculate the perpendicular height for side b.

    
15.

The diagram below shows a triangle ABC. The quotient of triangle ABC is 1.07 and the length of side c is 6 cm. The angle is 76.78° and the angle bisector intersects side c at point D. Calculate the height of the base and the perimeter of the triangle.


    
16.

The perimeter of a rhombus measures 16 cm and one of its interior angle is 109 °. Calculate both diagonals of the rhombus.

    
17.

The longer of the parallel sides of an isosceles trapezium measures 12.40 cm and its height is 6 cm. If one of its interior angle is 55 °, calculate its diagonal.

    
18.

Two parallel sides of an isosceles trapezium are 17.31 cm and 4 cm while the length of its diagonal and the perimeter are 11.38 cm and 36.85 cm respectively. Calculate all the interior angles of the trapezium.

    

19.

The legs of an isosceles trapezium measures 4 cm each with height 3.87 cm and perimeter 18 cm. Calculate the other sides of the trapezium.

    
20.

The sides of a kite is given as 8 cm and 12 cm. Calculate its interior angles and the length of the longer diagonal if the shorter diagonal measures 8 cm.

    
21.

Diagonals e and f in a trapezium are 5.32 and 10 cm, and sides c and d are 5 and 1.26 cm and an internal angle measures 165.81 °.


Calculate the area of the figure bounded by the angle bisector , the base a, and the bisector of the base a.

    
22.

  • Calculate the area of triangle with sides and and angle .


  • Calculate the side length and area of a dodecagon inscribed in a circle of radius 2.


  • The line PR is the diameter of a circle, and a point T lie on the circumference bounding this circle. As measures , and the length of the line RT is . Calculate the radius and area of this circle.


Take .

    
23.

Given a triangle with sides cm, cm and angle .


  • Calculate the length of side a.


  • Calculate the angle of size .


  • Calculate the length of the centre of gravity of each side c.

    
24.

In triangle ABC, the length of side AB is 6 cm. Calculate the length of side AC and the area of triangle ABC. Round the results to two decimal places.

    
25.

In triangle ABC, the following sides and angle are given:








Calculate the length of side a and the area of triangle ABC.

    
26.

In a triangle ABC, side c = 10 cm and b = 16 cm. The angle between them is . Calculate the length of side a to the nearest centimeter.

    
27.

Given triangle ABC with sides b = 5 cm, c = 9 cm and angle . Calculate the length of the side a to two decimal places.

    
28.

The sides and angle of triangle ABC are given below:








  • Make a sketch of the triangle and calculate its area.

  • Calculate the length of the center of gravity of each side c.

  • Proof if the given triangle is isosceles.

    

29.

In isosceles triangle ABC, the base and the angle at the vertex C is 78°. Make a sketch and calculate the angle at the base and the length of the sides.

    
30.

In triangle ABC, the following data are given:








Calculate:

  • The length of side c calculated to the nearest centimeter.

  • Calculate the angle to the nearest degree.

  • The area of the triangle and the area of the inscribed circle.

    
31.

In a given triangle , angle , and . Calculate the length of side to the nearest millimeter. Then calculate the area of the triangle.