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:

𝐴𝐡𝐢 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:

𝐴𝐡𝐢 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.

  • A𝑐=51.29cm, π‘šβˆ π΅=3244β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=11915β€²15β€²β€²βˆ˜ or 𝑐=4.86cm, π‘šβˆ π΅=14715β€²15β€²β€²βˆ˜, π‘šβˆ πΆ=444β€²45β€²β€²βˆ˜
  • B𝑐=25.65cm, π‘šβˆ π΅=11915β€²15β€²β€²βˆ˜, π‘šβˆ πΆ=3244β€²45β€²β€²βˆ˜ or 𝑐=2.43cm, π‘šβˆ π΅=444β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=14715β€²15β€²β€²βˆ˜
  • C𝑐=25.65cm, π‘šβˆ π΅=3244β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=11915β€²15β€²β€²βˆ˜ or 𝑐=2.43cm, π‘šβˆ π΅=444β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=14715β€²15β€²β€²βˆ˜
  • D𝑐=25.65cm, π‘šβˆ π΅=3244β€²45β€²β€²βˆ˜, π‘šβˆ πΆ=11915β€²15β€²β€²βˆ˜ or 𝑐=2.43cm, π‘šβˆ π΅=14715β€²15β€²β€²βˆ˜, π‘šβˆ πΆ=444β€²45β€²β€²βˆ˜

Q3:

𝐴𝐡𝐢 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.

  • A𝑐=11.11cm, π‘šβˆ π΅=7934β€²54β€²β€²βˆ˜, π‘šβˆ πΆ=6025β€²6β€²β€²βˆ˜
  • B𝑐=26.01cm, π‘šβˆ π΅=6025β€²6β€²β€²βˆ˜, π‘šβˆ πΆ=7934β€²54β€²β€²βˆ˜
  • C𝑐=26.01cm, π‘šβˆ π΅=6025β€²6β€²β€²βˆ˜, π‘šβˆ πΆ=7934β€²54β€²β€²βˆ˜ or 𝑐=9.23cm, π‘šβˆ π΅=11934β€²54β€²β€²βˆ˜, π‘šβˆ πΆ=2025β€²6β€²β€²βˆ˜
  • D𝑐=11.11cm, π‘šβˆ π΅=7934β€²54β€²β€²βˆ˜, π‘šβˆ πΆ=6025β€²6β€²β€²βˆ˜ or 𝑐=14.25cm, π‘šβˆ π΅=11934β€²54β€²β€²βˆ˜, π‘šβˆ πΆ=2025β€²6β€²β€²βˆ˜

Q4:

𝐴𝐡𝐢 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.

  • A𝐴𝐡=22.94 cm, π‘šβˆ π΅=90∘, and π‘šβˆ πΆ=35∘
  • B𝐴𝐡=24.80 cm, π‘šβˆ π΅=90∘, 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:

𝐴𝐡𝐢 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.

  • A𝐴𝐡=𝐴𝐡=38.88cm, π‘šβˆ π΅=90∘, π‘šβˆ πΆ=20∘
  • BThe triangle does not exist.
  • C𝐴𝐡=𝐴𝐡=36.65cm, π‘šβˆ π΅=90∘, π‘šβˆ πΆ=20∘

Q7:

𝐴𝐡𝐢 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.

  • A𝑐=7.35cm, π‘šβˆ π΅=3056β€²46β€²β€²βˆ˜, π‘šβˆ πΆ=1093β€²14β€²β€²βˆ˜
  • B𝑐=3.40cm, π‘šβˆ π΅=3056β€²46β€²β€²βˆ˜, π‘šβˆ πΆ=1093β€²14β€²β€²βˆ˜ or 𝑐=2.77cm, π‘šβˆ π΅=1493β€²14β€²β€²βˆ˜, π‘šβˆ πΆ=93β€²14β€²β€²βˆ˜
  • C𝑐=3.40cm, π‘šβˆ π΅=3056β€²46β€²β€²βˆ˜, π‘šβˆ πΆ=1093β€²14β€²β€²βˆ˜
  • D𝑐=7.35cm, π‘šβˆ π΅=1093β€²14β€²β€²βˆ˜, π‘šβˆ πΆ=3056β€²46β€²β€²βˆ˜
  • ENo triangle exists.

Q8:

𝐴𝐡𝐢 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.

  • A𝑐=24.434cm, π‘šβˆ π΄=83∘, π‘šβˆ π΅=37∘
  • B𝑐=15.765cm, π‘šβˆ π΄=117∘, π‘šβˆ π΅=3∘
  • C𝑐=28.896cm, π‘šβˆ π΄=70∘, π‘šβˆ π΅=50∘
  • D𝑐=30.887cm, π‘šβˆ π΄=64∘, π‘šβˆ π΅=56∘

Q9:

𝐴𝐡𝐢 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:

𝐴𝐡𝐢 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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