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Lesson: Law of Cosines

Video

13:49

Sample Question Videos

Worksheet • 25 Questions • 4 Videos

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 4 0 ∘ and 2 9 ∘ respectively. Find the distance between to the balloons giving the answer to the nearest metre.

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 kilometre.

Q7:

A plane travels 800 meters along the runway before taking off at an angle of 1 0 ∘ . It travels a further 1 0 0 0 meters at this angle as seen in the figure. Work out the distance of the plane from its starting point. Give your answer to 2 decimal places.

Q8:

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.

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:

𝐴 𝐡 𝐢 is a triangle, where 𝐡 𝐢 = 2 5 c m , 𝐴 𝐢 = 1 3 c m , and π‘š ∠ 𝐢 = 1 4 2 ∘ . Find the length 𝐴 𝐡 giving the answer to three decimal places.

Q11:

𝐴 𝐡 𝐢 is a triangle where 𝐡 𝐢 = 3 8 c m , π‘š ∠ 𝐴 𝐢 𝐡 = 6 0 ∘ and the area is 3 9 9 √ 3 cm2. Find the other lengths and angles giving lengths to one decimal place and angles to the nearest degree.

  • A 𝐴 𝐢 = 4 2 c m , 𝐴 𝐡 = 4 0 . 1 c m , π‘š ∠ 𝐡 𝐴 𝐢 = 5 5 3 β€² ∘ , π‘š ∠ 𝐴 𝐡 𝐢 = 6 4 5 7 β€² ∘
  • B 𝐴 𝐢 = 1 0 . 5 c m , 𝐴 𝐡 = 3 4 c m , π‘š ∠ 𝐡 𝐴 𝐢 = 6 4 5 7 β€² ∘ , π‘š ∠ 𝐴 𝐡 𝐢 = 5 5 3 β€² ∘
  • C 𝐴 𝐢 = 4 2 c m , 𝐴 𝐡 = 6 . 3 c m , π‘š ∠ 𝐡 𝐴 𝐢 = 5 5 3 β€² ∘ , π‘š ∠ 𝐴 𝐡 𝐢 = 2 5 3 β€² ∘
  • D 𝐴 𝐢 = 4 2 c m , 𝐴 𝐡 = 6 9 . 3 c m , π‘š ∠ 𝐡 𝐴 𝐢 = 3 4 5 7 β€² ∘ , π‘š ∠ 𝐴 𝐡 𝐢 = 2 5 3 β€² ∘

Q12:

The side lengths of a triangle are , , and . Sarah calculated, to one decimal place, the sizes of the corresponding angles as , , and . Was she correct?

  • Ano
  • Byes

Q13:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž = 2 7 c m , 𝑏 = 1 5 c m , 𝑐 = 2 1 c m , and 𝐷 ∈ 𝐡 𝐢 , where 𝐡 𝐷 = 1 6 c m . Find the length of 𝐴 𝐷 , giving the answer to two decimal places.

Q14:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 1 8 cm , 𝑏 = 1 0 cm and π‘š ∠ 𝐢 = 7 6 ∘ . Find the measure of ∠ 𝐴 giving the answer to the nearest second.

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

Q15:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 6 7 c m , 𝑏 = 4 9 c m and the perimeter is 148 cm. Find the largest angle in 𝐴 𝐡 𝐢 giving the answer to the nearest second.

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

Q16:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 2 5 cm, 𝑏 = 2 0 cm and 𝑐 = 2 9 cm. Find π‘š ∠ 𝐴 giving the answer to the nearest second.

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

Q17:

𝐴 𝐡 𝐢 is a triangle where 1 1 5 𝐴 = 1 5 𝐡 = 1 1 8 𝐢 s i n s i n s i n . Find π‘š ∠ 𝐢 giving the answer to the nearest second.

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

Q18:

Find the value of π‘Ž to 2 decimal places given that the measure of Angle 𝐴 is 6 4 ∘ , 𝑐 is 10 cm, and 𝑏 is 16 cm.

  • A 14.69 cm
  • B 15.10 cm
  • C 11.84 cm
  • D 215.72 cm
  • E 12.48 cm

Q19:

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

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

Q20:

Find π‘š ∠ 𝐴 to one decimal place.

Q21:

Complete the expression for triangle 𝐴 𝐡 𝐢 : π‘Ž + 𝑏 βˆ’ 𝑐 β‹― = 𝐢 2 2 2 c o s .

  • A 2 π‘Ž 𝑏
  • B 2 𝑏 𝑐
  • C π‘Ž 𝑏
  • D 2 π‘Ž 𝑐

Q22:

𝐴 𝐡 𝐢 is a triangle where 𝑃 βˆ’ π‘Ž = 2 2 c m , 𝑃 βˆ’ 𝑏 = 3 2 c m , 𝑃 βˆ’ 𝑐 = 8 c m and 2 𝑃 = π‘Ž + 𝑏 + 𝑐 . Find the largest angle in the triangle giving the answer to the nearest second.

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

Q23:

𝑋 π‘Œ 𝑍 is a triangle where the ratio between s i n 𝑋 , s i n π‘Œ and s i n 𝑍 is 2 3 ∢ 1 9 ∢ 1 6 . Find the smallest angle in 𝑋 π‘Œ 𝑍 giving the answer to the nearest second.

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

Q24:

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.

  • Acosine rule
  • Bangles sum rule
  • Csine rule
  • Ddouble angle rule
  • Etangent rule

Q25:

𝐴 𝐡 𝐢 is a triangle where 𝑃 βˆ’ π‘Ž = 2 0 c m , 𝑃 + π‘Ž = 1 1 6 c m , 𝑏 = 4 1 c m and the perimeter is 2 𝑃 . Find the measure of the smallest angle in 𝐴 𝐡 𝐢 giving the answer to the nearest second.

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