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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:

A bridge is to be constructed over a canyon stretching from point 𝐴 to point 𝐡 as seen in the given figure. A surveyor stands at a point 𝐢 , 30 yards from point 𝐴 , at the edge of the canyon. They measured that π‘š ∠ 𝐢 𝐴 𝐡 = 7 0 ∘ , π‘š ∠ 𝐴 𝐡 𝐢 = 2 8 ∘ . Work out the length of the bridge.

Q2:

𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐴 = 3 0 ∘ and π‘š ∠ 𝐡 = 1 0 5 ∘ . Find the ratio of lengths π‘Ž ∢ 𝑏 ∢ 𝑐 .

  • A 2 ∢ √ 6 βˆ’ √ 2 ∢ 2 √ 2
  • B √ 6 + √ 2 ∢ 2 ∢ 2 √ 2
  • C 1 ∢ √ 6 + √ 2 ∢ √ 2
  • D 2 ∢ √ 6 + √ 2 ∢ 2 √ 2

Q3:

To determine how far a boat is from shore, two radar stations 500 feet apart find the angles out to the boat, as shown in the given figure. Determine the distance of the boat from station 𝐴 and the distance of the boat from shore. Round your answers to the nearest whole foot.

  • A 442 ft, 531 ft
  • B 565 ft, 193 ft
  • C 442 ft, 193 ft
  • D 565 ft, 531 ft
  • E 613 ft, 576 ft

Q4:

The diagram shows an 8-foot solar panel mounted on the roof of a house. The roof is inclined at 2 0 ∘ to the horizontal, and, for maximum yield, the solar panel is placed at 3 8 ∘ to the horizontal. The solar panel is held in position by a vertical support. How long should the support be to hold the solar panel at an inclination of 3 8 ∘ ? Give your answer to one decimal place.

Q5:

In triangle 𝐴 𝐡 𝐢 , 𝐴 𝐢 = 9 7 m , π‘š ∠ 𝐡 𝐴 𝐢 = 1 0 1 ∘ , and π‘š ∠ 𝐴 𝐢 𝐡 = 5 3 ∘ . Determine the length of 𝐴 𝐡 to the nearest meter.

Q6:

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

Q7:

𝐴 𝐡 𝐢 is a triangle, where π‘Ž = 9 , 𝑏 = 6 , and π‘š ∠ 𝐴 = 5 8 . 1 ∘ . Find π‘š ∠ 𝐡 to the nearest tenth of a degree.

Q8:

is an obtuse triangle at where , and . Find lengths and giving the answer to the nearest integer.

  • A and
  • B and
  • C and
  • D and

Q9:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 9 6 and π‘š ∠ 𝐡 = 3 π‘š ∠ 𝐴 = 9 0 ∘ . Find length 𝑐 giving the answer in terms of s i n .

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

Q10:

The diameter of a circle 𝐴 𝐷 is 82 cm. 𝐴 𝐡 and 𝐴 𝐢 are two chords on opposite sides of a circle with lengths 5.1 cm and 48.4 cm respectively. Find the length 𝐡 𝐢 giving the answer to two decimal places.

Q11:

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

  • A π‘Ž = 7 8 c m and 𝑐 = 5 2 c m
  • B π‘Ž = 3 9 c m and 𝑐 = 7 8 c m
  • C π‘Ž = 5 2 c m and 𝑐 = 3 9 c m
  • D π‘Ž = 7 8 c m and 𝑐 = 3 9 c m

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?

  • Acosine rule
  • Bdouble angle rule
  • Ctangent rule
  • Dsine rule
  • Eangles sum rule

Q13:

Two men are standing in front of a minaret 𝐴 𝐷 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.

Q14:

is a triangle where , and . Find giving the answer to the nearest second.

  • A
  • B
  • C
  • D

Q15:

In the figure 𝐴 𝐢 = 3 . 5 .

What is 𝐴 𝐡 ? Give your answer to two decimal places.

Q16:

Cities A, B, and C are located such that city A is east of city B, city C is on a bearing of 3 5 ∘ from city B, and city C is 100 miles from city A and 70 miles from city B. Find the distance between cities A and B giving your answer to one decimal place.

Q17:

The scale of a map is 1 ∢ 1 . 3 5 c m k m . The position of three towns on a map form a triangle. Towns B and C are 17 cm apart, and the angles of towns A and B are 8 3 ∘ and 6 5 ∘ 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 city A and B is 12 km and the actual distance between city A and C is 7 km
  • B The actual distance between city A and B is 36 km and the actual distance between city A and C is 21 km
  • C The actual distance between city A and B is 9 km and the actual distance between city A and C is 16 km
  • D The actual distance between city A and B is 12 km and the actual distance between city A and C is 21 km

Q18:

𝐴 𝐡 𝐢 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:

is a right triangle at . The point lies on , where , , and . Find the length of , giving your answer to the nearest centimeter.

  • A 37 cm
  • B 16 cm
  • C 5 cm
  • D 28 cm
  • E 10 cm

Q20:

James, Anthony, and Jennifer stand at three points, 𝐴 , 𝐡 , and 𝐢 respectively. Suppose that π‘š ∠ 𝐴 𝐡 𝐢 = 4 8 ∘ , π‘š ∠ 𝐡 𝐴 𝐢 = 5 4 ∘ , and James is exactly 12 feet away from Anthony.

Find the distance between Anthony and Jennifer, to two decimal places.

Find the distance between James and Jennifer, to two decimal places.

Q21:

𝐴 𝐡 𝐢 is a triangle with a perimeter of 49 cm where the ratio between π‘š ∠ 𝐴 , π‘š ∠ 𝐡 and π‘š ∠ 𝐢 is 9 ∢ 5 ∢ 4 . Find the length of the smallest side giving the answer to two decimal places.

Q22:

𝐴 𝐡 𝐢 is a triangle, where π‘š ∠ 𝐴 = 4 6 1 1 β€² 1 7 β€² β€² ∘ , π‘š ∠ 𝐡 = 2 7 4 β€² 4 6 β€² β€² ∘ , and length π‘Ž = 2 1 . 4 c m . Find the length of the shortest side of 𝐴 𝐡 𝐢 giving the answer to one decimal place.

Q23:

𝑋 π‘Œ 𝑍 is a triangle where π‘Œ 𝑍 = 8 c m , π‘š ∠ π‘Œ = 2 2 ∘ and π‘š ∠ 𝑍 = 2 3 ∘ . π‘Š lies on π‘Œ 𝑍 where 𝑋 π‘Š βŠ₯ π‘Œ 𝑍 . Find the length of 𝑋 𝑍 giving the answer to two decimal places.

Q24:

𝐴 𝐡 𝐢 is a triangle where 8 𝐴 = 1 1 𝐡 = 1 6 𝐢 s i n s i n s i n . Find the ratio π‘Ž ∢ 𝑏 ∢ 𝑐 .

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

Q25:

𝐿 𝑀 𝑁 is a triangle where π‘š ∠ 𝐿 = 5 4 3 0 β€² ∘ , π‘š ∠ 𝑁 = 2 3 3 0 β€² ∘ and 𝑁 𝐿 = 1 6 . 4 c m . Find the lengths of 𝑀 𝑁 and 𝐿 𝑀 giving the answer to one decimal place.

  • A 𝑀 𝑁 = 1 6 . 4 c m and 𝐿 𝑀 = 6 . 7 c m
  • B 𝑀 𝑁 = 6 . 7 c m and 𝐿 𝑀 = 1 3 . 6 c m
  • C 𝑀 𝑁 = 1 3 . 6 c m and 𝐿 𝑀 = 1 6 . 4 c m
  • D 𝑀 𝑁 = 1 3 . 6 c m and 𝐿 𝑀 = 6 . 7 c m

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