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 𝐡 𝐢 is a triangle, where π‘šβˆ π΅=110∘, 𝑏=16 cm, and 𝑐=12 cm. How many possible solutions are there for the other lengths and angles?

  • Ano solutions
  • Bone solution
  • Ctwo solutions

Q2:

𝐴 𝐡 𝐢 is a triangle where π‘Ž=13.8cm, 𝑏=15.9cm and π‘šβˆ π΄=28∘. 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 , π‘š ∠ 𝐡 = 3 2 4 4 β€² 4 5 β€² β€² ∘ , π‘š ∠ 𝐢 = 1 1 9 1 5 β€² 1 5 β€² β€² ∘ or 𝑐=4.86cm, π‘šβˆ π΅=14715β€²15β€²β€²βˆ˜, π‘šβˆ πΆ=444β€²45β€²β€²βˆ˜
  • B 𝑐 = 2 5 . 6 5 c m , π‘š ∠ 𝐡 = 1 1 9 1 5 β€² 1 5 β€² β€² ∘ , π‘š ∠ 𝐢 = 3 2 4 4 β€² 4 5 β€² β€² ∘ or 𝑐=2.43cm, π‘šβˆ π΅=444β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=14715β€²15β€²β€²βˆ˜
  • C 𝑐 = 2 5 . 6 5 c m , π‘š ∠ 𝐡 = 3 2 4 4 β€² 4 5 β€² β€² ∘ , π‘š ∠ 𝐢 = 1 1 9 1 5 β€² 1 5 β€² β€² ∘ or 𝑐=2.43cm, π‘šβˆ π΅=444β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=14715β€²15β€²β€²βˆ˜
  • D 𝑐 = 2 5 . 6 5 c m , π‘š ∠ 𝐡 = 3 2 4 4 β€² 4 5 β€² β€² ∘ , π‘š ∠ 𝐢 = 1 1 9 1 5 β€² 1 5 β€² β€² ∘ or 𝑐=2.43cm, π‘šβˆ π΅=14715β€²15β€²β€²βˆ˜, π‘šβˆ πΆ=444β€²45β€²β€²βˆ˜

Q3:

𝐴 𝐡 𝐢 is a triangle, where π‘šβˆ π΄=40∘, π‘Ž=17cm, and 𝑏=23cm. 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 𝑐 = 1 1 . 1 1 c m , π‘š ∠ 𝐡 = 7 9 3 4 β€² 5 4 β€² β€² ∘ , π‘š ∠ 𝐢 = 6 0 2 5 β€² 6 β€² β€² ∘
  • B 𝑐 = 2 6 . 0 1 c m , π‘š ∠ 𝐡 = 6 0 2 5 β€² 6 β€² β€² ∘ , π‘š ∠ 𝐢 = 7 9 3 4 β€² 5 4 β€² β€² ∘
  • C 𝑐 = 2 6 . 0 1 c m , π‘š ∠ 𝐡 = 6 0 2 5 β€² 6 β€² β€² ∘ , π‘š ∠ 𝐢 = 7 9 3 4 β€² 5 4 β€² β€² ∘ or 𝑐=9.23cm, π‘šβˆ π΅=11934β€²54β€²β€²βˆ˜, π‘šβˆ πΆ=2025β€²6β€²β€²βˆ˜
  • D 𝑐 = 1 1 . 1 1 c m , π‘š ∠ 𝐡 = 7 9 3 4 β€² 5 4 β€² β€² ∘ , π‘š ∠ 𝐢 = 6 0 2 5 β€² 6 β€² β€² ∘ or 𝑐=14.25cm, π‘šβˆ π΅=11934β€²54β€²β€²βˆ˜, π‘šβˆ πΆ=2025β€²6β€²β€²βˆ˜

Q4:

𝐴 𝐡 𝐢 is a triangle, where π‘šβˆ π΄=55∘, 𝐡𝐢=13cm, and 𝐴𝐢=28cm. If the triangle exists, find all the possible values for the other lengths and angles in 𝐴𝐡𝐢, giving the lengths to two decimal places and angles to the nearest degree.

  • A 𝐴 𝐡 = 2 2 . 9 4 cm, π‘š ∠ 𝐡 = 9 0 ∘ , and π‘šβˆ πΆ=35∘
  • B 𝐴 𝐡 = 2 4 . 8 0 cm, π‘š ∠ 𝐡 = 9 0 ∘ , and π‘šβˆ πΆ=35∘
  • CThe triangle does not exist.

Q5:

For the given figure, 𝐴𝐡=11, 𝐡𝐢=9, and π‘šβˆ π΅π΄πΆ=41∘. Use the law of sines to work out the measure of ∠𝐴𝐢𝐡. Give your answer to two decimal places.

Q6:

𝐴 𝐡 𝐢 is a triangle, where π‘šβˆ π΄=70∘, 𝐡𝐢=3cm, and 𝐴𝐢=39cm. If the triangle exists, find all the possible values for the other lengths and angles in △𝐴𝐡𝐢 giving the lengths to two decimal places and angles to the nearest degree.

  • A 𝐴 𝐡 = 𝐴 𝐡 = 3 8 . 8 8 c m , π‘š ∠ 𝐡 = 9 0 ∘ , π‘š ∠ 𝐢 = 2 0 ∘
  • BThe triangle does not exist.
  • C 𝐴 𝐡 = 𝐴 𝐡 = 3 6 . 6 5 c m , π‘š ∠ 𝐡 = 9 0 ∘ , π‘š ∠ 𝐢 = 2 0 ∘

Q7:

𝐴 𝐡 𝐢 is a triangle where π‘šβˆ π΄=40∘, π‘Ž=5cm and 𝑏=4cm. If the triangle exists, find all the possible values for the other lengths and angles in 𝐴𝐡𝐢 giving lengths to two decimal places and angles to the nearest second.

  • A 𝑐 = 7 . 3 5 c m , π‘š ∠ 𝐡 = 3 0 5 6 β€² 4 6 β€² β€² ∘ , π‘š ∠ 𝐢 = 1 0 9 3 β€² 1 4 β€² β€² ∘
  • B 𝑐 = 3 . 4 0 c m , π‘š ∠ 𝐡 = 3 0 5 6 β€² 4 6 β€² β€² ∘ , π‘š ∠ 𝐢 = 1 0 9 3 β€² 1 4 β€² β€² ∘ or 𝑐=2.77cm, π‘šβˆ π΅=1493β€²14β€²β€²βˆ˜, π‘šβˆ πΆ=93β€²14β€²β€²βˆ˜
  • C 𝑐 = 3 . 4 0 c m , π‘š ∠ 𝐡 = 3 0 5 6 β€² 4 6 β€² β€² ∘ , π‘š ∠ 𝐢 = 1 0 9 3 β€² 1 4 β€² β€² ∘
  • D 𝑐 = 7 . 3 5 c m , π‘š ∠ 𝐡 = 1 0 9 3 β€² 1 4 β€² β€² ∘ , π‘š ∠ 𝐢 = 3 0 5 6 β€² 4 6 β€² β€² ∘
  • ENo triangle exists.

Q8:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž=28cm, 𝑏=17cm, and π‘šβˆ πΆ=60∘. Find the missing length rounded to three decimal places and the missing angles rounded to the nearest degree.

  • A 𝑐 = 2 4 . 4 3 4 c m , π‘š ∠ 𝐴 = 8 3 ∘ , π‘š ∠ 𝐡 = 3 7 ∘
  • B 𝑐 = 1 5 . 7 6 5 c m , π‘š ∠ 𝐴 = 1 1 7 ∘ , π‘š ∠ 𝐡 = 3 ∘
  • C 𝑐 = 2 8 . 8 9 6 c m , π‘š ∠ 𝐴 = 7 0 ∘ , π‘š ∠ 𝐡 = 5 0 ∘
  • D 𝑐 = 3 0 . 8 8 7 c m , π‘š ∠ 𝐴 = 6 4 ∘ , π‘š ∠ 𝐡 = 5 6 ∘

Q9:

𝐴 𝐡 𝐢 is a triangle, where π‘šβˆ π΅=70∘, 𝑏=3 cm, and 𝑐=6 cm. How many possible solutions are there for the other lengths and angles?

  • Aone solution
  • Bno solutions
  • Ctwo solutions

Q10:

𝐴 𝐡 𝐢 is a triangle, where π‘šβˆ π΅=130∘, 𝑏=17 cm, and 𝑐=3 cm. How many possible solutions are there for the other lengths and angles?

  • Ano solutions
  • Bone solution
  • Ctwo solutions

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