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Worksheet: Law of Cosines

Q1:

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

Q2:

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 meter.

  • A 201 m
  • B 291 m
  • C 182 m
  • D 293 m
  • E 307 m

Q3:

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

Q4:

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

  • A π‘š ∠ 𝐴 = 8 7 ∘ , π‘š ∠ 𝐡 = 1 9 ∘ , π‘š ∠ 𝐢 = 7 4 ∘
  • B π‘š ∠ 𝐴 = 7 4 ∘ , π‘š ∠ 𝐡 = 1 9 ∘ , π‘š ∠ 𝐢 = 8 7 ∘
  • C π‘š ∠ 𝐴 = 1 9 ∘ , π‘š ∠ 𝐡 = 8 7 ∘ , π‘š ∠ 𝐢 = 7 4 ∘
  • D π‘š ∠ 𝐴 = 7 4 ∘ , π‘š ∠ 𝐡 = 8 7 ∘ , π‘š ∠ 𝐢 = 1 9 ∘
  • E π‘š ∠ 𝐴 = 1 9 ∘ , π‘š ∠ 𝐡 = 7 4 ∘ , π‘š ∠ 𝐢 = 8 7 ∘

Q5:

Find π‘š ∠ 𝐴 to one decimal place.

Q6:

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

Q7:

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 = 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 ∘
  • 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 = 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 ∘

Q8:

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.

Q9:

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

  • A 4 6 5 0 β€² 4 9 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 9 9 5 8 β€² 5 4 β€² β€² ∘

Q10:

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 8 1 4 0 β€² 4 1 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 4 3 2 9 β€² 5 2 β€² β€² ∘

Q11:

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.

  • Atangent rule
  • Bsine rule
  • Cdouble angle rule
  • Dcosine rule
  • Eangles sum rule

Q12:

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 6 5 4 1 β€² 5 9 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 5 1 7 β€² 2 0 β€² β€² ∘

Q13:

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

Q14:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝐴 𝐢 = 1 5 c m and 𝐷 𝐡 = 2 3 c m . 𝑀 is the point of intersection of the diagonals where π‘š ∠ 𝐴 𝑀 𝐡 = 4 1 ∘ . Find the length of 𝐴 𝐷 giving the answer to two decimal places.

Q15:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral where 𝐴 𝐡 = 1 0 c m , π‘š ∠ 𝐴 𝐷 𝐡 = 7 4 ∘ , π‘š ∠ 𝐴 𝐡 𝐷 = 3 2 ∘ , 𝐡 𝐢 = 1 6 c m and 𝐷 𝐢 = 1 8 c m . Find π‘š ∠ 𝐢 𝐡 𝐷 giving the answer to the nearest second.

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

Q16:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž = 1 4 c m , 𝑏 = 9 c m , and π‘š ∠ 𝐢 = 7 0 ∘ . Find the perimeter of the triangle, giving the answer to the nearest centimeter.

  • A 42 cm
  • B 38 cm
  • C 39 cm
  • D 37 cm

Q17:

Find the length of π‘Ž to one decimal place.

Q18:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where π‘š ∠ 𝐴 = 6 0 ∘ , the perimeter is 78 cm, the length of the small diagonal is 21 cm and 𝐴 𝐡 < 𝐴 𝐷 . Find the measure of ∠ 𝐴 𝐷 𝐡 giving the answer to the nearest second.

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

Q19:

𝐴 𝐡 𝐢 is a triangle where 𝑃 + π‘Ž = 3 5 c m , 𝑃 βˆ’ π‘Ž = 1 c m where 𝑃 is half the perimeter of the triangle and 𝑏 = 𝑐 + 9 . Find the other lengths and angles of the triangle giving lengths to the nearest centimeter and angles to the nearest minute.

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

Q20:

The side lengths of a triangle are π‘Ž = 5 c m , 𝑏 = 7 c m , and 𝑐 = 1 0 c m . Emma calculated, to one decimal place, the measures of the corresponding angles as 𝐴 = 2 7 . 7 ∘ , 𝐡 = 4 0 . 5 ∘ , and 𝐢 = 1 1 1 . 8 ∘ . Was she correct?

  • Ayes
  • Bno

Q21:

𝐴 𝐡 𝐢 𝐷 is a cyclic quadrilateral where 𝐴 𝐡 = 1 9 c m , 𝐡 𝐢 = 5 c m , 𝐢 𝐷 = 1 9 c m and 𝐷 𝐴 = 8 c m . Find the length of 𝐴 𝐢 giving the answer to one decimal place.

Q22:

Complete the expression for triangle 𝐴 𝐡 𝐢 : 𝑏 = 𝑐 + βˆ’ 2 Γ— 2 2 .

  • A 𝑐 2 , 𝑐 𝑏 𝐡 c o s
  • B 𝑏 2 , 𝑏 π‘Ž 𝐢 c o s
  • C 𝑐 2 , 𝑐 π‘Ž
  • D π‘Ž 2 , 𝑐 π‘Ž 𝐡 c o s

Q23:

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 2 6 4 1 β€² 5 1 β€² 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 1 0 9 5 0 β€² ∘

Q24:

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 7 9 2 3 β€² 5 0 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 5 7 5 5 β€² 2 9 β€² β€² ∘

Q25:

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 6 6 2 1 β€² 2 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 7 2 5 β€² 1 5 β€² β€² ∘