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 𝐢 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 π‘₯. Give your answer to two decimal places.

Q3:

In the given figure, find π‘₯. Give your answer to two decimal places.

Q4:

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

Q5:

In the given figure, find πœƒ. Give your answer to two decimal places.

Q6:

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

Q7:

𝐴𝐡𝐢 is a triangle, where π‘Ž=13cm, 𝑏=10cm, and cos𝐢=0.2. Find the value of 𝑐, giving your answer to three decimal places.

Q8:

A biker traveled from city 𝐴 to city 𝐡 via city 𝐢 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:

𝐴𝐡𝐢 is a triangle, where 𝐡𝐢=25cm, 𝐴𝐢=13cm, and π‘šβˆ πΆ=142∘. Find the length 𝐴𝐡 giving the answer to three decimal places.

Q10:

𝐴𝐡𝐢 is a triangle where 𝐡𝐢=38cm, π‘šβˆ π΄πΆπ΅=60∘ and the area is 399√3 cm2. Find the other lengths and angles giving lengths to one decimal place and angles to the nearest degree.

  • A𝐴𝐢=42cm, 𝐴𝐡=6.3cm, π‘šβˆ π΅π΄πΆ=553β€²βˆ˜, π‘šβˆ π΄π΅πΆ=253β€²βˆ˜
  • B𝐴𝐢=42cm, 𝐴𝐡=69.3cm, π‘šβˆ π΅π΄πΆ=3457β€²βˆ˜, π‘šβˆ π΄π΅πΆ=253β€²βˆ˜
  • C𝐴𝐢=42cm, 𝐴𝐡=40.1cm, π‘šβˆ π΅π΄πΆ=553β€²βˆ˜, π‘šβˆ π΄π΅πΆ=6457β€²βˆ˜
  • D𝐴𝐢=10.5cm, 𝐴𝐡=34cm, π‘šβˆ π΅π΄πΆ=6457β€²βˆ˜, π‘šβˆ π΄π΅πΆ=553β€²βˆ˜

Q11:

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

  • Ayes
  • Bno

Q12:

𝐴𝐡𝐢 is a triangle, where π‘Ž=27cm, 𝑏=15cm, 𝑐=21cm, and 𝐷∈𝐡𝐢, where 𝐡𝐷=16cm. Find the length of 𝐴𝐷, giving the answer to two decimal places.

Q13:

𝐴𝐡𝐢 is a triangle where π‘Ž=18 cm , 𝑏=10 cm and π‘šβˆ πΆ=76∘. Find the measure of ∠𝐴 giving the answer to the nearest second.

  • A6337β€²3β€²β€²βˆ˜
  • B6621β€²25β€²β€²βˆ˜
  • C725β€²15β€²β€²βˆ˜
  • D3154β€²45β€²β€²βˆ˜

Q14:

𝐴𝐡𝐢 is a triangle where π‘Ž=67cm, 𝑏=49cm and the perimeter is 148 cm. Find the largest angle in 𝐴𝐡𝐢 giving the answer to the nearest second.

  • A4328β€²9β€²β€²βˆ˜
  • B11221β€²48β€²β€²βˆ˜
  • C2641β€²51β€²β€²βˆ˜
  • D10950β€²βˆ˜

Q15:

𝐴𝐡𝐢 is a triangle where π‘Ž=25 cm, 𝑏=20 cm and 𝑐=29 cm. Find π‘šβˆ π΄ giving the answer to the nearest second.

  • A1224β€²31β€²β€²βˆ˜
  • B4240β€²42β€²β€²βˆ˜
  • C5755β€²29β€²β€²βˆ˜
  • D7923β€²50β€²β€²βˆ˜

Q16:

𝐴𝐡𝐢 is a triangle where 115𝐴=15𝐡=118𝐢sinsinsin. Find π‘šβˆ πΆ giving the answer to the nearest second.

  • A10416β€²49β€²β€²βˆ˜
  • B1358β€²56β€²β€²βˆ˜
  • C4627β€²28β€²β€²βˆ˜
  • D11933β€²36β€²β€²βˆ˜

Q17:

Find the value of π‘Ž to 2 decimal places given that the measure of Angle 𝐴 is 64∘, 𝑐 is 10 cm, and 𝑏 is 16 cm.

Q18:

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

  • Aπ‘šβˆ π΄=19∘, π‘šβˆ π΅=74∘, π‘šβˆ πΆ=87∘
  • Bπ‘šβˆ π΄=19∘, π‘šβˆ π΅=87∘, π‘šβˆ πΆ=74∘
  • Cπ‘šβˆ π΄=74∘, π‘šβˆ π΅=19∘, π‘šβˆ πΆ=87∘
  • Dπ‘šβˆ π΄=87∘, π‘šβˆ π΅=19∘, π‘šβˆ πΆ=74∘
  • Eπ‘šβˆ π΄=74∘, π‘šβˆ π΅=87∘, π‘šβˆ πΆ=19∘

Q19:

Find π‘šβˆ π΄ to one decimal place.

Q20:

Complete the expression for triangle 𝐴𝐡𝐢: π‘Ž+π‘βˆ’π‘β‹―=𝐢cos.

  • A2π‘Žπ‘
  • B2𝑏𝑐
  • Cπ‘Žπ‘
  • D2π‘Žπ‘

Q21:

𝐴𝐡𝐢 is a triangle where π‘ƒβˆ’π‘Ž=22cm, π‘ƒβˆ’π‘=32cm, π‘ƒβˆ’π‘=8cm and 2𝑃=π‘Ž+𝑏+𝑐. Find the largest angle in the triangle giving the answer to the nearest second.

  • A9958β€²54β€²β€²βˆ˜
  • B4650β€²49β€²β€²βˆ˜
  • C9458β€²19β€²β€²βˆ˜
  • D3310β€²17β€²β€²βˆ˜

Q22:

π‘‹π‘Œπ‘ is a triangle where the ratio between sin𝑋, sinπ‘Œ and sin𝑍 is 23∢19∢16. Find the smallest angle in π‘‹π‘Œπ‘ giving the answer to the nearest second.

  • A4329β€²52β€²β€²βˆ˜
  • B6329β€²52β€²β€²βˆ˜
  • C5449β€²27β€²β€²βˆ˜
  • D8140β€²41β€²β€²βˆ˜

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:

𝐴𝐡𝐢 is a triangle where π‘ƒβˆ’π‘Ž=20cm, 𝑃+π‘Ž=116cm, 𝑏=41cm and the perimeter is 2𝑃. Find the measure of the smallest angle in 𝐴𝐡𝐢 giving the answer to the nearest second.

  • A517β€²20β€²β€²βˆ˜
  • B6310β€²41β€²β€²βˆ˜
  • C6541β€²59β€²β€²βˆ˜
  • D7142β€²35β€²β€²βˆ˜

Q25:

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

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