Worksheet: Law of Sines

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-angled 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 and 𝑐=25cm
  • B𝑎=32cm and 𝑐=25cm
  • C𝑎=32cm and 𝑐=15cm
  • D𝑎=25cm and 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:

The diameter of a circle 𝐴𝐷 is 82 cm. 𝐴𝐵 and 𝐴𝐶 are two chords on opposite sides of a circle with lengths 5.1 cm and 48.4 cm respectively. Find the length 𝐵𝐶 giving the answer to two decimal places.

Q11:

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

Q12:

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

Q13:

Two men are standing in front of a minaret 𝐴𝐷 at the points 𝐵 and 𝐶 respectively where the distance between them is 25.4 m. Find the height of the minaret giving the answer to one decimal place.

Q14:

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′′∘

Q15:

In the figure 𝐴𝐶=3.5.

What is 𝐴𝐵? Give your answer to two decimal places.

Q16:

The scale of a map is 1∶1.35cmkm. The positions of three towns on a map form a triangle. Towns 𝐵 and 𝐶 are 17 cm apart, and the angles ∠𝐶𝐴𝐵 and ∠𝐴𝐵𝐶 are 83∘ and 65∘ respectively. Find the actual distance between towns 𝐴 and 𝐵 and between towns 𝐴 and 𝐶, giving the answer to the nearest kilometer.

  • AThe actual distance between cities 𝐴 and 𝐵 is 12 km, and the actual distance between cities 𝐴 and 𝐶 is 21 km.
  • BThe actual distance between cities 𝐴 and 𝐵 is 36 km, and the actual distance between cities 𝐴 and 𝐶 is 21 km.
  • CThe actual distance between cities 𝐴 and 𝐵 is 12 km, and the actual distance between cities 𝐴 and 𝐶 is 7 km.
  • DThe actual distance between cities 𝐴 and 𝐵 is 9 km, and the actual distance between cities 𝐴 and 𝐶 is 16 km.

Q17:

𝐴𝐵𝐶 is an equilateral triangle of side length 12 cm that is inscribed in a circle. Find the radius of the circle, giving the answer to two decimal places.

Q18:

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.

Q19:

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.

Q20:

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.

Q21:

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.

Q22:

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

Q23:

𝐴, 𝐵 and 𝐶 represent three coffee shops along a river bank. 𝐴 is on one side and 𝐵 and 𝐶 are on the other. 𝐴 is located where 𝑚∠𝐵𝐶𝐴=61∘ and 𝑚∠𝐶𝐵𝐴=66∘. Find the distance between 𝐴 and 𝐶 and the width of the river giving the answers to two decimal places.

  • A𝐴𝐶=16.00m. Width of river =15.01m.
  • B𝐴𝐶=16.00m. Width of river =14.37m.
  • C𝐴𝐶=17.16m. Width of river =15.01m.
  • D𝐴𝐶=17.16m. Width of river =14.37m.

Q24:

𝐴𝐵𝐶 is a triangle where 𝑎=17.7, 𝑏=25.7 and 𝑚∠𝐴=28∘. Find all possible values of 𝑚∠𝐵 giving the answer to the nearest second.

  • A𝑚∠𝐵=4258′28′′∘ or 𝑚∠𝐵=1371′32′′∘
  • B𝑚∠𝐵=4258′28′′∘
  • C𝑚∠𝐵=280′2′′∘ or 𝑚∠𝐵=15159′58′′∘
  • D𝑚∠𝐵=280′2′′∘
  • E𝑚∠𝐵=4258′28′′∘ or 𝑚∠𝐵=15159′58′′∘

Q25:

𝐴𝐵𝐶 is a triangle, where 𝐵𝐶=13.8 cm, 𝐴𝐶=21.2 cm, and 𝑚∠𝐴=21∘. Find all possible values of length 𝐴𝐵 giving the answer to three decimal places.

  • A𝐴𝐵=22.651cm or 𝐴𝐵=2.859cm
  • B𝐴𝐵=62.625cm or 𝐴𝐵=16.543cm
  • C𝐴𝐵=31.312cm or 𝐴𝐵=8.272cm
  • D𝐴𝐵=15.656cm or 𝐴𝐵=4.136cm

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