Worksheet: Ambiguous Case of the Law of Sines

In this worksheet, we will practice using the rule of sines to solve SSA (side-side-angle) ambiguous triangles.

Q1:

𝐴 𝐵 𝐶 is a triangle, where 𝑚 𝐵 = 1 1 0 , 𝑏 = 1 6 cm, and 𝑐 = 1 2 cm. How many possible solutions are there for the other lengths and angles?

  • Ano solutions
  • Btwo solutions
  • Cone solution

Q2:

𝐴 𝐵 𝐶 is a triangle where 𝑎 = 1 3 . 8 c m , 𝑏 = 1 5 . 9 c m and 𝑚 𝐴 = 2 8 . Find all possible values for the other lengths and angles giving lengths to two decimal places and angles to the nearest second.

  • A 𝑐 = 5 1 . 2 9 c m , 𝑚 𝐵 = 3 2 4 4 4 5 , 𝑚 𝐶 = 1 1 9 1 5 1 5 or 𝑐 = 4 . 8 6 c m , 𝑚 𝐵 = 1 4 7 1 5 1 5 , 𝑚 𝐶 = 4 4 4 4 5
  • B 𝑐 = 2 5 . 6 5 c m , 𝑚 𝐵 = 1 1 9 1 5 1 5 , 𝑚 𝐶 = 3 2 4 4 4 5 or 𝑐 = 2 . 4 3 c m , 𝑚 𝐵 = 4 4 4 4 5 , 𝑚 𝐶 = 1 4 7 1 5 1 5
  • C 𝑐 = 2 5 . 6 5 c m , 𝑚 𝐵 = 3 2 4 4 4 5 , 𝑚 𝐶 = 1 1 9 1 5 1 5 or 𝑐 = 2 . 4 3 c m , 𝑚 𝐵 = 4 4 4 4 5 , 𝑚 𝐶 = 1 4 7 1 5 1 5
  • D 𝑐 = 2 5 . 6 5 c m , 𝑚 𝐵 = 3 2 4 4 4 5 , 𝑚 𝐶 = 1 1 9 1 5 1 5 or 𝑐 = 2 . 4 3 c m , 𝑚 𝐵 = 1 4 7 1 5 1 5 , 𝑚 𝐶 = 4 4 4 4 5

Q3:

is a triangle, where , , and . If the triangle exists, find all the possible values for the other lengths and angles giving the lengths to two decimal places and angles to the nearest second.

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

Q4:

is a triangle, where , , and . If the triangle exists, find all the possible values for the other lengths and angles in , giving the lengths to two decimal places and angles to the nearest degree.

  • A cm, , and
  • B cm, , and
  • CThe triangle does not exist.

Q5:

For the given figure, 𝐴 𝐵 = 1 1 , 𝐵 𝐶 = 9 , and 𝑚 𝐵 𝐴 𝐶 = 4 1 . Use the law of sines to work out the size of 𝐴 𝐶 𝐵 . Give your answer to two decimal places.

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

Q6:

𝐴 𝐵 𝐶 is a triangle, where 𝑚 𝐴 = 7 0 , 𝐵 𝐶 = 3 c m , and 𝐴 𝐶 = 3 9 c m . If the triangle exists, find all the possible values for the other lengths and angles in 𝐴 𝐵 𝐶 giving the lengths to two decimal places and angles to the nearest degree.

  • A 𝐴 𝐵 = 𝐴 𝐵 = 3 8 . 8 8 c m , 𝑚 𝐵 = 9 0 , 𝑚 𝐶 = 2 0
  • B 𝐴 𝐵 = 𝐴 𝐵 = 3 6 . 6 5 c m , 𝑚 𝐵 = 9 0 , 𝑚 𝐶 = 2 0
  • CThe triangle does not exist.

Q7:

𝐴 𝐵 𝐶 is a triangle where 𝑚 𝐴 = 4 0 , 𝑎 = 5 c m and 𝑏 = 4 c m . If the triangle exists, find all the possible values for the other lengths and angles in 𝐴 𝐵 𝐶 giving lengths to two decimal places and angles to the nearest second.

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

Q8:

𝐴 𝐵 𝐶 is a triangle, where 𝑎 = 2 8 c m , 𝑏 = 1 7 c m , and 𝑚 𝐶 = 6 0 . Find the missing length rounded to three decimal places and the missing angles rounded to the nearest degree.

  • A 𝑐 = 2 8 . 8 9 6 c m , 𝑚 𝐴 = 7 0 , 𝑚 𝐵 = 5 0
  • B 𝑐 = 3 0 . 8 8 7 c m , 𝑚 𝐴 = 6 4 , 𝑚 𝐵 = 5 6
  • C 𝑐 = 1 5 . 7 6 5 c m , 𝑚 𝐴 = 1 1 7 , 𝑚 𝐵 = 3
  • D 𝑐 = 2 4 . 4 3 4 c m , 𝑚 𝐴 = 8 3 , 𝑚 𝐵 = 3 7

Q9:

𝐴 𝐵 𝐶 is a triangle, where 𝑚 𝐵 = 7 0 , 𝑏 = 3 cm, and 𝑐 = 6 cm. How many possible solutions are there for the other lengths and angles?

  • Aone solution
  • Btwo solutions
  • Cno solutions

Q10:

𝐴 𝐵 𝐶 is a triangle, where 𝑚 𝐵 = 1 3 0 , 𝑏 = 1 7 cm, and 𝑐 = 3 cm. How many possible solutions are there for the other lengths and angles?

  • Ano solutions
  • Btwo solutions
  • Cone solution

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