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Worksheet: Law of Cosines

Q1:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž = 1 3 c m , 𝑏 = 1 0 c m , and c o s 𝐢 = 0 . 2 . Find the value of 𝑐 , giving your answer to three decimal places.

Q2:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž = 2 8 c m , 𝑏 = 2 6 c m , and c o s 𝐢 = βˆ’ 0 . 2 . Find the value of 𝑐 , giving your answer to three decimal places.

Q3:

Find all the angles in the triangle giving the answers to the nearest second.

  • A π‘š ∠ 𝐴 = 9 6 3 2 β€² 5 8 β€² β€² ∘ , π‘š ∠ 𝐡 = 3 6 4 β€² ∘ , π‘š ∠ 𝐢 = 4 7 2 3 β€² 2 β€² β€² ∘
  • B π‘š ∠ 𝐴 = 3 6 4 β€² ∘ , π‘š ∠ 𝐡 = 4 7 2 3 β€² 2 β€² β€² ∘ , π‘š ∠ 𝐢 = 9 6 3 2 β€² 5 8 β€² β€² ∘
  • C π‘š ∠ 𝐴 = 3 6 4 β€² ∘ , π‘š ∠ 𝐡 = 9 6 3 2 β€² 5 8 β€² β€² ∘ , π‘š ∠ 𝐢 = 4 7 2 3 β€² 2 β€² β€² ∘
  • D π‘š ∠ 𝐴 = 9 6 3 2 β€² 5 8 β€² β€² ∘ , π‘š ∠ 𝐡 = 4 7 2 3 β€² 2 β€² β€² ∘ , π‘š ∠ 𝐢 = 3 6 4 β€² ∘

Q4:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 2 2 c m , 𝑏 = 1 9 c m , and 𝑐 = 2 0 c m . Find the measure of the largest angle in 𝐴 𝐡 𝐢 , giving the answer to the nearest second.

  • A 5 7 5 0 β€² 2 1 β€² β€² ∘
  • B 5 3 3 2 β€² 9 β€² β€² ∘
  • C 1 1 1 2 2 β€² 3 1 β€² β€² ∘
  • D 6 8 3 7 β€² 2 9 β€² β€² ∘

Q5:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 7 c m , 𝑏 = 1 3 c m , and 𝑐 = 1 4 c m . Find the measure of the largest angle in 𝐴 𝐡 𝐢 , giving the answer to the nearest second.

  • A 6 7 1 1 β€² 6 β€² β€² ∘
  • B 2 9 4 5 β€² 2 8 β€² β€² ∘
  • C 9 6 5 6 β€² 3 4 β€² β€² ∘
  • D 8 3 3 β€² 2 6 β€² β€² ∘

Q6:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 2 𝑏 and π‘š ∠ 𝐢 = 3 8 ∘ . Find π‘š ∠ 𝐡 giving the answer to the nearest second.

  • A π‘š ∠ 𝐡 = 1 2 2 7 β€² 9 β€² β€² ∘
  • B π‘š ∠ 𝐡 = 2 9 3 9 β€² ∘
  • C π‘š ∠ 𝐡 = 1 1 5 4 β€² 1 4 β€² β€² ∘
  • D π‘š ∠ 𝐡 = 2 6 5 5 β€² 4 6 β€² β€² ∘

Q7:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 1 2 cm, 𝑏 = 2 0 cm and 𝑐 = 2 6 cm. Find the smallest angle in 𝐴 𝐡 𝐢 giving the answer to the nearest second.

  • A 4 7 4 1 β€² 4 2 β€² β€² ∘
  • B 1 0 5 5 7 β€² 4 3 β€² β€² ∘
  • C 1 5 3 3 9 β€² 2 5 β€² β€² ∘
  • D 2 6 2 0 β€² 3 5 β€² β€² ∘
  • E 3 3 3 3 9 β€² 2 5 β€² β€² ∘

Q8:

The lengths of two sides of a triangle are ο€» √ 2 6 + 3  and ο€» √ 2 6 βˆ’ 3  and the angle between them is 6 0 ∘ . Find the length of the third side giving the answer in an exact form.

  • A 53 cm
  • B √ 6 2 cm
  • C √ 8 7 cm
  • D √ 5 3 cm

Q9:

A biker travelled 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.

Q10:

Find all the other angles and lengths of the triangle giving lengths to two decimal places and angles to the nearest second.

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