Worksheet: Law of Cosines

In this worksheet, we will practice finding the sides and angles in non-right triangles using the law of cosines.

Q1:

𝐸 and 𝐴 are two hot air balloons flying at heights 117 m and 84 m respectively. The angles of depression at a point 𝐶 on the ground from 𝐸 and 𝐴 are 40 and 29 respectively. Find the distance between the balloons giving the answer to the nearest meter.

Q2:

In the given figure, find 𝑥. Give your answer to two decimal places.

Q3:

In the given figure, find 𝑥. Give your answer to two decimal places.

Q4:

In the given figure, using the law of cosines, find 𝜃. Give your answer to two decimal places.

Q5:

In the given figure, find 𝜃. Give your answer to two decimal places.

Q6:

𝐴 , 𝐵 , and 𝐶 are three cities. Find the distance between cities 𝐴 and 𝐵, giving your answer to the nearest kilometer.

Q7:

𝐴 𝐵 𝐶 is a triangle, where 𝑎=13cm, 𝑏=10cm, and cos𝐶=0.2. Find the value of 𝑐, giving your answer to three decimal places.

Q8:

A biker traveled from city 𝐴 to city 𝐵 via city 𝐶 with a uniform speed of 52 km/h. He then returned directly to city 𝐴 with a uniform speed of 89 km/h. Find, in minutes, the total time of his whole journey to two decimal places.

Q9:

𝐴 𝐵 𝐶 is a triangle, where 𝐵𝐶=25cm, 𝐴𝐶=13cm, and 𝑚𝐶=142. Find the length 𝐴𝐵 giving the answer to three decimal places.

Q10:

𝐴 𝐵 𝐶 is a triangle where 𝐵𝐶=38cm, 𝑚𝐴𝐶𝐵=60 and the area is 3993 cm2. Find the other lengths and angles giving lengths to one decimal place and angles to the nearest degree.

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

Q11:

The side lengths of a triangle are 𝑎=5cm, 𝑏=7cm, and 𝑐=10cm. Emma calculated, to one decimal place, the measures of the corresponding angles as 𝐴=27.7, 𝐵=40.5, and 𝐶=111.8. Was she correct?

  • Ayes
  • Bno

Q12:

𝐴 𝐵 𝐶 is a triangle, where 𝑎=27cm, 𝑏=15cm, 𝑐=21cm, and 𝐷𝐵𝐶, where 𝐵𝐷=16cm. Find the length of 𝐴𝐷, giving the answer to two decimal places.

Q13:

𝐴 𝐵 𝐶 is a triangle where 𝑎=18 cm , 𝑏=10 cm and 𝑚𝐶=76. Find the measure of 𝐴 giving the answer to the nearest second.

  • A 6 3 3 7 3
  • B 6 6 2 1 2 5
  • C 7 2 5 1 5
  • D 3 1 5 4 4 5

Q14:

𝐴 𝐵 𝐶 is a triangle where 𝑎=67cm, 𝑏=49cm and the perimeter is 148 cm. Find the largest angle in 𝐴𝐵𝐶 giving the answer to the nearest second.

  • A 4 3 2 8 9
  • B 1 1 2 2 1 4 8
  • C 2 6 4 1 5 1
  • D 1 0 9 5 0

Q15:

𝐴 𝐵 𝐶 is a triangle where 𝑎=25 cm, 𝑏=20 cm and 𝑐=29 cm. Find 𝑚𝐴 giving the answer to the nearest second.

  • A 1 2 2 4 3 1
  • B 4 2 4 0 4 2
  • C 5 7 5 5 2 9
  • D 7 9 2 3 5 0

Q16:

𝐴 𝐵 𝐶 is a triangle where 115𝐴=15𝐵=118𝐶sinsinsin. Find 𝑚𝐶 giving the answer to the nearest second.

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

Q17:

Find the value of 𝑎 to 2 decimal places given that the measure of Angle 𝐴 is 64, 𝑐 is 10 cm, and 𝑏 is 16 cm.

  • A11.84 cm
  • B12.48 cm
  • C215.72 cm
  • D14.69 cm
  • E15.10 cm

Q18:

For the triangle given, find the value of the three angles to the nearest degree.

  • A 𝑚 𝐴 = 1 9 , 𝑚 𝐵 = 7 4 , 𝑚 𝐶 = 8 7
  • B 𝑚 𝐴 = 1 9 , 𝑚 𝐵 = 8 7 , 𝑚 𝐶 = 7 4
  • C 𝑚 𝐴 = 7 4 , 𝑚 𝐵 = 1 9 , 𝑚 𝐶 = 8 7
  • D 𝑚 𝐴 = 8 7 , 𝑚 𝐵 = 1 9 , 𝑚 𝐶 = 7 4
  • E 𝑚 𝐴 = 7 4 , 𝑚 𝐵 = 8 7 , 𝑚 𝐶 = 1 9

Q19:

Find 𝑚𝐴 to one decimal place.

Q20:

Complete the expression for triangle 𝐴𝐵𝐶: 𝑎+𝑏𝑐=𝐶cos.

  • A 2 𝑎 𝑏
  • B 2 𝑏 𝑐
  • C 𝑎 𝑏
  • D 2 𝑎 𝑐

Q21:

𝐴 𝐵 𝐶 is a triangle where 𝑃𝑎=22cm, 𝑃𝑏=32cm, 𝑃𝑐=8cm and 2𝑃=𝑎+𝑏+𝑐. Find the largest angle in the triangle giving the answer to the nearest second.

  • A 9 9 5 8 5 4
  • B 4 6 5 0 4 9
  • C 9 4 5 8 1 9
  • D 3 3 1 0 1 7

Q22:

𝑋 𝑌 𝑍 is a triangle where the ratio between sin𝑋, sin𝑌 and sin𝑍 is 231916. Find the smallest angle in 𝑋𝑌𝑍 giving the answer to the nearest second.

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

Q23:

Which rule could be used to find the length of an unknown side of a triangle, given the other two lengths and the measure of their included angle.

  • Asine rule
  • Bdouble angle rule
  • Ccosine rule
  • Dtangent rule
  • Eangles sum rule

Q24:

𝐴 𝐵 𝐶 is a triangle where 𝑃𝑎=20cm, 𝑃+𝑎=116cm, 𝑏=41cm and the perimeter is 2𝑃. Find the measure of the smallest angle in 𝐴𝐵𝐶 giving the answer to the nearest second.

  • A 5 1 7 2 0
  • B 6 3 1 0 4 1
  • C 6 5 4 1 5 9
  • D 7 1 4 2 3 5

Q25:

Circle 𝑀 has a radius of 8 cm where the length 𝐶𝐵 is 5 cm. Find the angle 𝜃 in radians giving the answer to two decimal places.

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