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:

𝐴𝐡𝐢 is a triangle where sin𝐴=47, sin𝐡=45 and 𝐡𝐢=3.99cm. Find the length of 𝐴𝐡 giving the answer to two decimal places.

Q2:

𝐿𝑀𝑁 is a triangle where π‘šβˆ πΏ=5430β€²βˆ˜, π‘šβˆ π‘=2330β€²βˆ˜ 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:

π‘‹π‘Œπ‘ is a triangle where π‘Œπ‘=8cm, π‘šβˆ π‘Œ=22∘, and π‘šβˆ π‘=23∘. π‘Š lies on π‘Œπ‘ where π‘‹π‘ŠβŠ₯π‘Œπ‘. Find the length of π‘‹π‘Š giving the answer to two decimal places.

Q4:

𝐴𝐡𝐢 is a triangle where 8𝐴=11𝐡=16𝐢sinsinsin. Find the ratio π‘ŽβˆΆπ‘βˆΆπ‘.

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

Q5:

𝐴𝐡𝐢 is a triangle where π‘šβˆ π΄=30∘ and π‘šβˆ π΅=105∘. Find the ratio of lengths π‘ŽβˆΆπ‘βˆΆπ‘.

  • A1∢√6+√2∢√2
  • B√6+√2∢2∢2√2
  • C2∢√6+√2∢2√2
  • D2∢√6βˆ’βˆš2∢2√2

Q6:

In triangle 𝐴𝐡𝐢, 𝐴𝐢=97m, π‘šβˆ π΅π΄πΆ=101∘, and π‘šβˆ π΄πΆπ΅=53∘. Determine the length of 𝐴𝐡 to the nearest meter.

Q7:

For the given figure, 𝐴𝐡=3 and 𝐡𝐢=π‘Ž. Use the Law of Sines to work out π‘Ž. Give your answer to two decimal places.

Q8:

𝐴𝐡𝐢 is a triangle, where π‘Ž=9, 𝑏=6, and π‘šβˆ π΄=58.1∘. Find π‘šβˆ π΅ to the nearest tenth of a degree.

Q9:

𝐴𝐡𝐢 is an obtuse-angled triangle at 𝐴 where 𝑏=15cm, tan𝐢=65 and π‘šβˆ π΅=27∘. Find lengths π‘Ž and 𝑐 giving the answer to the nearest integer.

  • Aπ‘Ž=15cm and 𝑐=25cm
  • Bπ‘Ž=32cm and 𝑐=25cm
  • Cπ‘Ž=32cm and 𝑐=15cm
  • Dπ‘Ž=25cm and 𝑐=32cm

Q10:

𝐴𝐡𝐢 is a triangle where π‘Ž=96 and π‘šβˆ π΅=3π‘šβˆ π΄=90∘. Find length 𝑐 giving the answer in terms of sin.

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

Q11:

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

Q12:

𝐴𝐡𝐢 is a triangle where 2𝐴=3𝐡=4𝐢sinsinsin and the perimeter is 169 cm. Find the values of π‘Ž and 𝑐 giving the answer to the nearest centimeter.

  • Aπ‘Ž=39cm and 𝑐=78cm
  • Bπ‘Ž=52cm and 𝑐=39cm
  • Cπ‘Ž=78cm and 𝑐=52cm
  • Dπ‘Ž=78cm and 𝑐=39cm

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
  • Dangles sum rule
  • Etangent rule

Q14:

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

Q15:

𝐴𝐡𝐢 is a triangle where π‘šβˆ π΄=138∘, π‘Ž=13cm and 𝑏=7cm. Find π‘šβˆ π΅ giving the answer to the nearest second.

  • A5334β€²59β€²β€²βˆ˜
  • B1117β€²7β€²β€²βˆ˜
  • C15852β€²53β€²β€²βˆ˜
  • D217β€²7β€²β€²βˆ˜

Q16:

In the figure 𝐴𝐢=3.5.

What is 𝐴𝐡? 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 ∠𝐢𝐴𝐡 and ∠𝐴𝐡𝐢 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.

Q18:

𝐴𝐡𝐢 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:

𝐴𝐡𝐢 is a right triangle at 𝐡. The point 𝐷 lies on οƒͺ𝐡𝐢, where 𝐢𝐷=17cm, π‘šβˆ π΄π·πΆ=46∘, and π‘šβˆ πΆπ΄π·=24∘. Find the length of 𝐴𝐡, giving your answer to the nearest centimeter.

Q20:

𝐴𝐡𝐢 is a triangle with a perimeter of 49 cm where the ratio between π‘šβˆ π΄, π‘šβˆ π΅ and π‘šβˆ πΆ is 9∢5∢4. Find the length of the smallest side giving the answer to two decimal places.

Q21:

𝐴𝐡𝐢 is a triangle, where π‘šβˆ π΄=4611β€²17β€²β€²βˆ˜, π‘šβˆ π΅=274β€²46β€²β€²βˆ˜, and length π‘Ž=21.4cm. Find the length of the shortest side of 𝐴𝐡𝐢 giving the answer to one decimal place.

Q22:

In the given figure, π΅πΆπ·π‘Œ is a rectangle and 𝐡 is a point on the straight line 𝐴𝐢. 𝐡𝐢=405m, π‘šβˆ π·π΄πΆ=21∘, and π‘šβˆ π‘Œπ΄πΆ=59∘. Find the length of 𝐷𝐢 giving the answer to the nearest meter.

Q23:

𝐴𝐡𝐢 is a triangle where π‘šβˆ π΄=π‘šβˆ π΅=33∘ and 𝑐=36cm. Find length 𝑏 giving the answer to two decimal places.

Q24:

𝐴, 𝐡 and 𝐢 represent three coffee shops along a river bank. 𝐴 is on one side and 𝐡 and 𝐢 are on the other. 𝐴 is located where π‘šβˆ π΅πΆπ΄=61∘ and π‘šβˆ πΆπ΅π΄=66∘. Find the distance between 𝐴 and 𝐢 and the width of the river giving the answers to two decimal places.

  • A𝐴𝐢=16.00m. Width of river =15.01m.
  • B𝐴𝐢=16.00m. Width of river =14.37m.
  • C𝐴𝐢=17.16m. Width of river =15.01m.
  • D𝐴𝐢=17.16m. Width of river =14.37m.

Q25:

𝐴𝐡𝐢 is a triangle where π‘Ž=17.7, 𝑏=25.7 and π‘šβˆ π΄=28∘. Find all possible values of π‘šβˆ π΅ giving the answer to the nearest second.

  • Aπ‘šβˆ π΅=4258β€²28β€²β€²βˆ˜ or π‘šβˆ π΅=1371β€²32β€²β€²βˆ˜
  • Bπ‘šβˆ π΅=4258β€²28β€²β€²βˆ˜
  • Cπ‘šβˆ π΅=280β€²2β€²β€²βˆ˜ or π‘šβˆ π΅=15159β€²58β€²β€²βˆ˜
  • Dπ‘šβˆ π΅=280β€²2β€²β€²βˆ˜
  • Eπ‘šβˆ π΅=4258β€²28β€²β€²βˆ˜ or π‘šβˆ π΅=15159β€²58β€²β€²βˆ˜

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