Lesson Worksheet: Law of Sines Mathematics

In this worksheet, we will practice applying the law of sines to find lengths and angle measures in non-right triangles.

Q1:

𝐴𝐵𝐶 is a triangle where sin𝐴=47, sin𝐵=45 and 𝐵𝐶=3.99cm. Find the length of 𝐴𝐵 giving the answer to two decimal places.

Q2:

𝐿𝑀𝑁 is a triangle where 𝑚∠𝐿=5430′∘, 𝑚∠𝑁=2330′∘ and 𝑁𝐿=16.4cm. Find the lengths of 𝑀𝑁 and 𝐿𝑀 giving the answer to one decimal place.

  • A𝑀𝑁=16.4cm and 𝐿𝑀=6.7cm
  • B𝑀𝑁=6.7cm and 𝐿𝑀=13.6cm
  • C𝑀𝑁=13.6cm and 𝐿𝑀=16.4cm
  • D𝑀𝑁=13.6cm and 𝐿𝑀=6.7cm

Q3:

𝑋𝑌𝑍 is a triangle where 𝑌𝑍=8cm, 𝑚∠𝑌=22∘, and 𝑚∠𝑍=23∘. 𝑊 lies on 𝑌𝑍 where 𝑋𝑊⊥𝑌𝑍. Find the length of 𝑋𝑊 giving the answer to two decimal places.

Q4:

𝐴𝐵𝐶 is a triangle where 8𝐴=11𝐵=16𝐶sinsinsin. Find the ratio 𝑎∶𝑏∶𝑐.

  • A16∶11∶8
  • B11∶16∶22
  • C22∶16∶11
  • D8∶16∶11
  • E8∶11∶16

Q5:

𝐴𝐵𝐶 is a triangle where 𝑚∠𝐴=30∘ and 𝑚∠𝐵=105∘. Find the ratio of lengths 𝑎∶𝑏∶𝑐.

  • A1∶√6+√2∶√2
  • B√6+√2∶2∶2√2
  • C2∶√6+√2∶2√2
  • D2∶√6−√2∶2√2

Q6:

For the given figure, 𝐴𝐵=3 and 𝐵𝐶=𝑎. Use the Law of Sines to work out 𝑎. Give your answer to two decimal places.

Q7:

𝐴𝐵𝐶 is a triangle, where 𝑎=9, 𝑏=6, and 𝑚∠𝐴=58.1∘. Find 𝑚∠𝐵 to the nearest tenth of a degree.

Q8:

𝐴𝐵𝐶 is an obtuse triangle at 𝐴, where 𝑏=15cm, tan𝐶=65, and 𝑚∠𝐵=27∘. Find lengths 𝑎 and 𝑐, giving the answer to the nearest integer.

  • A𝑎=15cm, 𝑐=25cm
  • B𝑎=32cm, 𝑐=25cm
  • C𝑎=32cm, 𝑐=15cm
  • D𝑎=25cm, 𝑐=32cm

Q9:

𝐴𝐵𝐶 is a triangle where 𝑎=96 and 𝑚∠𝐵=3𝑚∠𝐴=90∘. Find length 𝑐 giving the answer in terms of sin.

  • Asinsin609630∘∘
  • B966090sinsin∘∘
  • C966030sinsin∘∘
  • D969060sinsin∘∘
  • E963060sinsin∘∘

Q10:

𝐴𝐵𝐶 is a triangle where 2𝐴=3𝐵=4𝐶sinsinsin and the perimeter is 169 cm. Find the values of 𝑎 and 𝑐 giving the answer to the nearest centimeter.

  • A𝑎=39cm and 𝑐=78cm
  • B𝑎=52cm and 𝑐=39cm
  • C𝑎=78cm and 𝑐=52cm
  • D𝑎=78cm and 𝑐=39cm

Q11:

Which rule could be used to find the length of an unknown side of a triangle, given the measures of two angles and the length of one other side?

  • Asine rule
  • Bcosine rule
  • Cdouble angle rule
  • Dangles sum rule
  • Etangent rule

Q12:

𝐴𝐵𝐶 is a triangle where 𝑚∠𝐴=138∘, 𝑎=13cm and 𝑏=7cm. Find 𝑚∠𝐵 giving the answer to the nearest second.

  • A5334′59′′∘
  • B1117′7′′∘
  • C15852′53′′∘
  • D217′7′′∘

Q13:

𝐴𝐵𝐶 is a right triangle at 𝐵. The point 𝐷 lies on 𝐵𝐶, where 𝐶𝐷=17cm, 𝑚∠𝐴𝐷𝐶=46∘, and 𝑚∠𝐶𝐴𝐷=24∘. Find the length of 𝐴𝐵, giving your answer to the nearest centimeter.

Q14:

𝐴𝐵𝐶 is a triangle with a perimeter of 49 cm where the ratio between 𝑚∠𝐴, 𝑚∠𝐵 and 𝑚∠𝐶 is 9∶5∶4. Find the length of the smallest side giving the answer to two decimal places.

Q15:

𝐴𝐵𝐶 is a triangle, where 𝑚∠𝐴=4611′17′′∘, 𝑚∠𝐵=274′46′′∘, and length 𝑎=21.4cm. Find the length of the shortest side of 𝐴𝐵𝐶 giving the answer to one decimal place.

Q16:

In the given figure, 𝐵𝐶𝐷𝑌 is a rectangle and 𝐵 is a point on the straight line 𝐴𝐶. 𝐵𝐶=405m, 𝑚∠𝐷𝐴𝐶=21∘, and 𝑚∠𝑌𝐴𝐶=59∘. Find the length of 𝐷𝐶 giving the answer to the nearest meter.

Q17:

𝐴𝐵𝐶 is a triangle where 𝑚∠𝐴=𝑚∠𝐵=33∘ and 𝑐=36cm. Find length 𝑏 giving the answer to two decimal places.

Q18:

In triangle 𝐴𝐵𝐶, 𝑚∠𝐴=36.4∘, 𝑚∠𝐵=55.3∘, and 𝑎=4. Find the length of 𝑏 to the nearest tenth.

Q19:

𝐴𝐵𝐶 is a triangle, where 𝑐−𝑏=4cm and the ratio between sin:sin:sin𝐴𝐵𝐶 is 579::. Find 𝑎 and 𝑏, giving the answer to the nearest centimeter.

  • A𝑎=10cm, 𝑏=14cm
  • B𝑎=10cm, 𝑏=18cm
  • C𝑎=14cm, 𝑏=10cm
  • D𝑎=18cm, 𝑏=14cm

Q20:

𝐴𝐵𝐶 is a triangle where 𝑏=7cm, 𝑐=14cm and 𝑚∠𝐶=7744′41′′∘. What type(s) could ∠𝐵 be?

  • AObtuse
  • BAcute
  • CAcute or obtuse

Q21:

𝐴𝐵𝐶 is a triangle, where 𝑚∠𝐴=6618′15′′∘, 𝑚∠𝐵=776′7′′∘, and length 𝑎=136.4cm. Find the length of the longest side of 𝐴𝐵𝐶 giving the answer to one decimal place.

Q22:

𝐴𝐵𝐶 is a triangle where 𝑐=16cm and 5𝑚∠𝐵=2𝑚∠𝐴=100∘. Find 𝑎 giving the answer to two decimal places.

Q23:

𝐴𝐵𝐶 is a triangle where 𝑚∠𝐴=76∘, 𝑚∠𝐵=50∘ and 𝑎+𝑏=√13+5cm. Find the values of 𝑎 and 𝑏 giving the answer to two decimal places.

  • A𝑎=4.01cm and 𝑏=3.80cm
  • B𝑎=3.80cm and 𝑏=4.81cm
  • C𝑎=4.81cm and 𝑏=3.80cm
  • D𝑎=4.81cm and 𝑏=4.01cm

Q24:

𝐴𝐵𝐶 is a triangle, where 𝐴𝐵=21.837cm, 𝑚∠𝐴=54∘, and 𝑚∠𝐵=56∘. 𝐷 lies on 𝐴𝐵, where 𝐶𝐷⟂𝐴𝐵. Find the length of 𝐶𝐷, giving your answer to three decimal places.

Q25:

𝐴𝐵𝐶 is a triangle where 𝑐=22cm, 𝑚∠𝐴=50∘ and 𝑚∠𝐵=48∘. Point 𝐷 lies on 𝐵𝐶 where 𝐴𝐷⟂𝐵𝐶. Find 𝑚∠𝐶 to the nearest degree and lengths 𝑎, 𝑏 and 𝐴𝐷 giving the answers to two decimal places.

  • A𝑚∠𝐶=98∘, 𝑎=17.02cm, 𝑏=16.51cm, 𝐴𝐷=16.35cm
  • B𝑚∠𝐶=82∘, 𝑎=16.51cm, 𝑏=17.02cm, 𝐴𝐷=16.35cm
  • C𝑚∠𝐶=82∘, 𝑎=17.02cm, 𝑏=16.51cm, 𝐴𝐷=14.72cm
  • D𝑚∠𝐶=82∘, 𝑎=17.02cm, 𝑏=16.51cm, 𝐴𝐷=16.35cm

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