Worksheet: Law of Sines

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

Q1:

𝐴 𝐵 𝐶 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:

𝐿 𝑀 𝑁 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.

  • A 𝑀 𝑁 = 1 3 . 6 c m and 𝐿𝑀=16.4cm
  • B 𝑀 𝑁 = 1 6 . 4 c m and 𝐿𝑀=6.7cm
  • C 𝑀 𝑁 = 1 3 . 6 c m and 𝐿𝑀=6.7cm
  • D 𝑀 𝑁 = 6 . 7 c m and 𝐿𝑀=13.6cm

Q3:

𝑋 𝑌 𝑍 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:

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

  • A 8 ∶ 1 6 ∶ 1 1
  • B 1 1 ∶ 1 6 ∶ 2 2
  • C 8 ∶ 1 1 ∶ 1 6
  • D 2 2 ∶ 1 6 ∶ 1 1
  • E 1 6 ∶ 1 1 ∶ 8

Q5:

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

  • A 2 ∶ √ 6 + √ 2 ∶ 2 √ 2
  • B 1 ∶ √ 6 + √ 2 ∶ √ 2
  • C 2 ∶ √ 6 − √ 2 ∶ 2 √ 2
  • D √ 6 + √ 2 ∶ 2 ∶ 2 √ 2

Q6:

In triangle 𝐴𝐵𝐶, 𝐴𝐶=97m, 𝑚∠𝐵𝐴𝐶=101∘, and 𝑚∠𝐴𝐶𝐵=53∘. Determine the length of 𝐴𝐵 to the nearest meter.

Q7:

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.

Q8:

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

Q9:

𝐴 𝐵 𝐶 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.

  • A 𝑎 = 1 5 c m and 𝑐=25cm
  • B 𝑎 = 3 2 c m and 𝑐=15cm
  • C 𝑎 = 2 5 c m and 𝑐=32cm
  • D 𝑎 = 3 2 c m and 𝑐=25cm

Q10:

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

  • A 9 6 6 0 9 0 s i n s i n ∘ ∘
  • B 9 6 6 0 3 0 s i n s i n ∘ ∘
  • C s i n s i n 6 0 9 6 3 0 ∘ ∘
  • D 9 6 9 0 6 0 s i n s i n ∘ ∘
  • E 9 6 3 0 6 0 s i n s i n ∘ ∘

Q11:

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.

Q12:

𝐴 𝐵 𝐶 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.

  • A 𝑎 = 5 2 c m and 𝑐=39cm
  • B 𝑎 = 7 8 c m and 𝑐=39cm
  • C 𝑎 = 7 8 c m and 𝑐=52cm
  • D 𝑎 = 3 9 c m and 𝑐=78cm

Q13:

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
  • Dtangent rule
  • Eangles sum rule

Q14:

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.

Q15:

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

  • A 1 5 8 5 2 ′ 5 3 ′ ′ ∘
  • B 2 1 7 ′ 7 ′ ′ ∘
  • C 5 3 3 4 ′ 5 9 ′ ′ ∘
  • D 1 1 1 7 ′ 7 ′ ′ ∘

Q16:

In the figure 𝐴𝐶=3.5.

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

Q17:

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.

  • A The actual distance between cities A and B is 12 km, and the actual distance between cities A and C is 7 km.
  • B The actual distance between cities A and B is 12 km, and the actual distance between cities A and C is 21 km.
  • C The actual distance between cities A and B is 36 km, and the actual distance between cities A and C is 21 km.
  • D The actual distance between cities A and B is 9 km, and the actual distance between cities A and C is 16 km.

Q18:

𝐴 𝐵 𝐶 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.

Q19:

𝐴 𝐵 𝐶 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.

Q20:

𝐴 𝐵 𝐶 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.

Q21:

𝐴 𝐵 𝐶 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.

Q22:

In the figure below, 𝐵𝐶=405m, 𝑚∠𝐷𝐴𝐶=21∘ and 𝑚∠𝑌𝐴𝐶=59∘. Find the length of 𝐷𝐶 giving the answer to the nearest metre.

Q23:

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

Q24:

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.

  • A 𝐴 𝐶 = 1 7 . 1 6 m . Width of river =14.37m.
  • B 𝐴 𝐶 = 1 7 . 1 6 m . Width of river =15.01m.
  • C 𝐴 𝐶 = 1 6 . 0 0 m . Width of river =15.01m.
  • D 𝐴 𝐶 = 1 6 . 0 0 m . Width of river =14.37m.

Q25:

𝐴 𝐵 𝐶 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.

  • A 𝑚 ∠ 𝐵 = 2 8 0 ′ 2 ′ ′ ∘
  • B 𝑚 ∠ 𝐵 = 4 2 5 8 ′ 2 8 ′ ′ ∘ or 𝑚∠𝐵=1371′32′′∘
  • C 𝑚 ∠ 𝐵 = 4 2 5 8 ′ 2 8 ′ ′ ∘
  • D 𝑚 ∠ 𝐵 = 4 2 5 8 ′ 2 8 ′ ′ ∘ or 𝑚∠𝐵=15159′58′′∘
  • E 𝑚 ∠ 𝐵 = 2 8 0 ′ 2 ′ ′ ∘ or 𝑚∠𝐵=15159′58′′∘

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