Worksheet: Applications of Laws of Sines and Cosines

In this worksheet, we will practice solving triangles using the sine law, the cosine law, or both of them.

Q1:

𝐴 𝐵 𝐶 is a triangle, where 𝐷 is the midpoint of 𝐵 𝐶 , 𝑚 𝐵 = 2 8 , 𝑚 𝐴 = 6 9 , and 𝑎 = 2 0 c m . Find the length of 𝐴 𝐷 giving the answer to two decimal places.

Q2:

𝐷 is the midpoint of 𝐵 𝐶 and 𝑚 𝐵 𝐴 𝐷 = 3 7 . Find t a n 𝐷 𝐴 𝐶 giving the answer to three decimal places.

Q3:

𝐴 𝐵 𝐶 is a triangle where s i n s i n s i n 𝐴 2 5 = 𝐵 2 9 = 𝐶 1 5 . Find the largest angle in 𝐴 𝐵 𝐶 giving the answer to the nearest degree.

Q4:

𝐴 𝐵 𝐶 is a triangle where 𝑎 = 4 4 c m , 𝑏 = 3 1 c m and 𝑚 𝐶 = 6 9 . Point 𝐷 lies on 𝐴 𝐵 such that 𝐶 𝐷 𝐴 𝐷 . Find the length of 𝐶 𝐷 giving the answer to two decimal places.

Q5:

𝐴 𝐵 𝐶 is a triangle, where 𝑎 = 1 9 cm, 𝑏 = 9 cm, and 𝑚 𝐶 = 4 5 . Find the radius of the circumcircle giving the answer to two decimal places.

Q6:

𝐴 𝐵 𝐶 is a triangle where 𝑎 = 2 6 cm, 𝑏 = 2 2 cm and 𝑐 = 6 cm. Find the radius of the circumcircle giving the answer to three decimal places.

Q7:

𝐴 𝐵 𝐶 is a triangle where 𝑚 𝐴 = 3 0 , the ratio between 𝑏 and 𝑐 is 3 2 and the area of the circumcircle is 2 2 5 𝜋 cm2. Find the perimeter of the triangle 𝐴 𝐵 𝐶 giving the answer to the nearest centimetre.

Q8:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝐴 𝐵 = 1 6 c m , 𝑚 𝐴 𝐷 𝐵 = 4 0 , 𝑚 𝐷 𝐵 𝐴 = 1 0 0 , 𝐵 𝐶 = 2 1 c m and 𝐷 𝐶 = 9 c m . Find 𝑚 𝐵 𝐶 𝐷 giving the answer to the nearest second and the area of 𝐵 𝐶 𝐷 giving the answer to three decimal places.

  • A 1 1 1 1 0 6 , 88.123 cm2
  • B 2 3 3 3 2 3 , 37.767 cm2
  • C 6 9 2 3 5 8 , 134.283 cm2
  • D 4 5 1 6 3 0 , 67.142 cm2

Q9:

𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝑚 𝐴 𝐵 𝐶 = 9 0 , 𝑚 𝐵 𝐴 𝐷 = 4 1 , 𝐴 𝐵 = 𝐴 𝐷 = 3 0 . 9 c m and 𝐵 𝐷 = 𝐵 𝐶 . Find the area of 𝐴 𝐵 𝐶 𝐷 giving the answer to two decimal places.

Q10:

Find the area of 𝐴 𝐵 𝐶 𝐷 given 𝐸 is the point of intersection of 𝐴 𝐶 and 𝐵 𝐷 , 𝐴 𝐸 = 5 c m , 𝐸 𝐶 = 8 . 9 c m , 𝐸 𝐷 = 7 . 7 c m , and 𝑚 𝐴 𝐸 𝐵 = 8 0 . Give the answer to the nearest square centimetre.

Q11:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝐴 𝐵 = 4 4 c m and 𝐵 𝐶 = 6 8 c m . Find the length of 𝐴 𝐶 to two decimal places and then the measure of angles 𝐴 and 𝐶 to the nearest second.

  • A 𝐴 𝐶 = 5 1 . 8 5 c m , 𝑚 𝐴 = 3 2 5 4 1 9 , 𝑚 𝐶 = 5 7 5 4 1
  • B 𝐴 𝐶 = 8 0 . 9 9 c m , 𝑚 𝐴 = 5 7 5 4 1 , 𝑚 𝐶 = 3 2 5 4 1 9
  • C 𝐴 𝐶 = 8 0 . 9 9 c m , 𝑚 𝐴 = 4 9 4 0 4 7 , 𝑚 𝐶 = 5 7 5 4 1
  • D 𝐴 𝐶 = 8 0 . 9 9 c m , 𝑚 𝐴 = 3 2 5 4 1 9 , 𝑚 𝐶 = 5 7 5 4 1

Q12:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝑚 𝐶 = 6 2 and 𝐴 𝐶 = 1 7 c m . Find the lengths of 𝐴 𝐵 and 𝐵 𝐶 giving the answer to two decimal places and the measure of 𝐴 giving the answer to the nearest degree.

  • A 𝐴 𝐵 = 1 5 . 0 1 c m , 𝐵 𝐶 = 7 . 9 8 c m , 𝑚 𝐴 = 3 8
  • B 𝐴 𝐵 = 7 . 9 8 c m , 𝐵 𝐶 = 1 5 . 0 1 c m , 𝑚 𝐴 = 2 8
  • C 𝐴 𝐵 = 7 . 9 8 c m , 𝐵 𝐶 = 1 5 . 0 1 c m , 𝑚 𝐴 = 3 8
  • D 𝐴 𝐵 = 1 5 . 0 1 c m , 𝐵 𝐶 = 7 . 9 8 c m , 𝑚 𝐴 = 2 8

Q13:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝐴 𝐵 = 2 7 c m and 𝑚 𝐴 = 6 3 . Find the lengths 𝐴 𝐶 and 𝐵 𝐶 giving the answer to two decimal places and the measure of angle 𝐶 giving the answer to the nearest degree.

  • A 𝐴 𝐶 = 3 0 . 3 0 c m , 𝐵 𝐶 = 1 3 . 7 6 c m , 𝑚 𝐶 = 3 7
  • B 𝐴 𝐶 = 1 3 . 7 6 c m , 𝐵 𝐶 = 3 0 . 3 0 c m , 𝑚 𝐶 = 2 7
  • C 𝐴 𝐶 = 1 3 . 7 6 c m , 𝐵 𝐶 = 3 0 . 3 0 c m , 𝑚 𝐶 = 3 7
  • D 𝐴 𝐶 = 5 9 . 4 7 c m , 𝐵 𝐶 = 5 2 . 9 9 c m , 𝑚 𝐶 = 2 7

Q14:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝑚 𝐶 = 1 . 1 8 8 r a d and 𝐴 𝐶 = 1 2 c m . Find 𝑚 𝐴 in radians and lengths 𝐴 𝐵 and 𝐵 𝐶 giving all answers to three decimal places.

  • A 𝑚 𝐴 = 0 . 5 5 7 r a d , 𝐴 𝐵 = 4 . 4 8 2 c m , 𝐵 𝐶 = 1 1 . 1 3 1 c m
  • B 𝑚 𝐴 = 0 . 3 8 3 r a d , 𝐴 𝐵 = 1 1 . 1 3 1 c m , 𝐵 𝐶 = 2 9 . 8 0 2 c m
  • C 𝑚 𝐴 = 1 . 1 8 8 r a d , 𝐴 𝐵 = 1 1 . 1 3 1 c m , 𝐵 𝐶 = 1 1 . 1 3 1 c m
  • D 𝑚 𝐴 = 0 . 3 8 3 r a d , 𝐴 𝐵 = 1 1 . 1 3 1 c m , 𝐵 𝐶 = 4 . 4 8 2 c m

Q15:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝑚 𝐴 = 0 . 9 4 2 r a d and 𝐴 𝐵 = 1 9 c m . Find 𝑚 𝐶 in radians and the lengths 𝐵 𝐶 and 𝐴 𝐶 giving all answers to three decimal places.

  • A 𝑚 𝐶 = 2 . 2 0 0 r a d , 𝐵 𝐶 = 2 6 . 1 2 5 c m , 𝐴 𝐶 = 2 3 . 4 9 3 c m
  • B 𝑚 𝐶 = 0 . 6 2 9 r a d , 𝐵 𝐶 = 1 1 . 1 7 5 c m , 𝐴 𝐶 = 3 2 . 3 0 3 c m
  • C 𝑚 𝐶 = 2 . 2 0 0 r a d , 𝐵 𝐶 = 3 2 . 3 0 3 c m , 𝐴 𝐶 = 2 6 . 1 2 5 c m
  • D 𝑚 𝐶 = 0 . 6 2 9 r a d , 𝐵 𝐶 = 2 6 . 1 2 5 c m , 𝐴 𝐶 = 3 2 . 3 0 3 c m

Q16:

𝐴 𝐵 𝐶 is an isosceles triangle where 𝐴 𝐵 = 𝐴 𝐶 = 4 7 c m and 𝐵 𝐶 = 1 0 c m . Find the angles in the triangle giving the answer to the nearest second.

  • A 𝑚 𝐴 = 1 6 7 5 1 1 8 , 𝑚 𝐵 = 6 4 2 1 , 𝑚 𝐶 = 6 4 2 1
  • B 𝑚 𝐴 = 1 6 7 4 7 1 0 , 𝑚 𝐵 = 6 6 2 5 , 𝑚 𝐶 = 6 6 2 5
  • C 𝑚 𝐴 = 1 2 8 4 2 , 𝑚 𝐵 = 8 3 5 5 3 9 , 𝑚 𝐶 = 8 3 5 5 3 9
  • D 𝑚 𝐴 = 1 2 1 2 5 0 , 𝑚 𝐵 = 8 3 5 3 3 5 , 𝑚 𝐶 = 8 3 5 3 3 5

Q17:

is a trapezium, where , , , , and . Find the lengths of and giving the answer to the nearest centimetre.

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q18:

𝐴 𝐵 𝐶 𝐷 is a trapezium where 𝐴 𝐷 𝐵 𝐶 , 𝐵 𝐶 = 2 0 c m , 𝑚 𝐴 𝐵 𝐶 = 6 1 , 𝐴 𝐷 = 1 1 c m and 𝑚 𝐵 𝐴 𝐶 = 5 9 . Find the area of the trapezium giving the answer to the nearest square centimetre.

Q19:

𝐴 𝐵 𝐶 𝐷 is a trapezium, where 𝐴 𝐷 𝐵 𝐶 , 𝐴 𝐷 = 2 0 c m , 𝑚 𝐵 = 5 5 , 𝑚 𝐷 = 8 0 , and 𝑚 𝐴 𝐶 𝐵 = 5 2 . Find the area of the trapezium giving the answer to the nearest square centimetre.

Q20:

𝐴 𝐵 𝐶 𝐷 is a parallelogram where 𝑀 is the point of intersection to the diagonals, 𝐴 𝐶 = 2 1 . 1 c m , 𝑚 𝐴 𝑀 𝐷 = 8 0 5 4 and 𝑚 𝐶 𝐴 𝐵 = 5 3 5 4 . Find the area of the parallelogram giving the answer to two decimal places.

Q21:

𝐴 𝐵 𝐶 is a triangle where 𝐴 𝐵 = 7 c m , 𝑚 𝐵 = 6 0 and the area of the triangle is 9 1 3 cm2. Find the perimeter of 𝐴 𝐵 𝐶 giving the answer to two decimal places.

Q22:

In a parallelogram 𝐴 𝐵 𝐶 𝐷 , 𝐵 𝐶 = 8 i n c h e s and 𝐴 𝐵 = 5 i n c h e s , where 𝑚 𝐴 𝐵 𝐶 = 1 3 4 . Work out the length of 𝐴 𝐶 . Give your answer to 2 decimal places.

Q23:

The side length of a regular pentagon 𝐴 𝐵 𝐶 𝐷 𝐸 is 25.81 cm. Find the length of the diagonal 𝐴 𝐶 giving the answer to two decimal places.

Q24:

In a triangle 𝐴 𝐵 𝐶 , 𝐴 𝐵 = 3 , 𝐴 𝐶 = 8 , and 𝑚 𝐵 𝐴 𝐶 = 5 2 .

If 𝐵 𝐷 is the altitude from 𝐵 , what is 𝐵 𝐷 to 2 decimal places?

Calculate the area of the triangle 𝐴 𝐵 𝐶 to 2 decimal places.

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