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

Q2:

A house is 66 metres tall. The angle of elevation from the base of the house to the top of a tower is 3 9 ∘ , and the angle of elevation from the top of the house to the top of the tower is 2 4 ∘ . Find the height of the tower giving the answer to the nearest metre.

Q3:

𝐴 𝐵 𝐶 is a triangle where s i n 𝐴 = 4 7 , s i n 𝐵 = 4 5 and 𝐵 𝐶 = 3 . 9 9 c m . Find the length of 𝐴 𝐵 giving the answer to two decimal places.

Q4:

An aeroplane needs to head due north, but there is a wind blowing from the southwest at 60 km/h. The plane flies with an airspeed of 550 km/h. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

  • A 8 5 . 9 0 3 ∘
  • B 4 . 0 9 6 ∘
  • C 3 . 8 5 7 ∘
  • D 4 . 4 2 4 ∘
  • E 4 . 8 2 3 ∘

Q5:

Ramy, Shady, and Engy stand at three points, , , and respectively. Suppose that , , and Ramy is exactly 12 feet away from Shady.

Find the distance between Shady and Engy, to two decimal places.

  • A 9.38 feet
  • B 14.51 feet
  • C 9.12 feet
  • D 9.93 feet
  • E 5.61 feet

Find the distance between Ramy and Engy, to two decimal places.

  • A 9.12 feet
  • B 7.27 feet
  • C 3.73 feet
  • D 15.79 feet
  • E 5.48 feet

Q6:

𝐿 𝑀 𝑁 is a triangle where 𝑚 ∠ 𝐿 = 5 4 3 0 ′ ∘ , 𝑚 ∠ 𝑁 = 2 3 3 0 ′ ∘ 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

Q7:

𝑋 𝑌 𝑍 is a triangle where 𝑌 𝑍 = 8 c m , 𝑚 ∠ 𝑌 = 2 2 ∘ and 𝑚 ∠ 𝑍 = 2 3 ∘ . 𝑊 lies on 𝑌 𝑍 where 𝑋 𝑊 ⊥ 𝑌 𝑍 . Find the length of 𝑋 𝑍 giving the answer to two decimal places.

Q8:

𝐴 𝐵 𝐶 is a triangle where 8 𝐴 = 1 1 𝐵 = 1 6 𝐶 s i n s i n s i n . Find the ratio 𝑎 ∶ 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

Q9:

𝐴 𝐵 𝐶 is a triangle where 𝑚 ∠ 𝐴 = 3 0 ∘ and 𝑚 ∠ 𝐵 = 1 0 5 ∘ . Find the ratio of lengths 𝑎 ∶ 𝑏 ∶ 𝑐 .

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

Q10:

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

Q11:

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.

Q12:

In triangle 𝐴 𝐵 𝐶 , 𝐴 𝐶 = 9 7 m , 𝑚 ∠ 𝐵 𝐴 𝐶 = 1 0 1 ∘ , and 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 3 ∘ . Determine the length of 𝐴 𝐵 to the nearest meter.

Q13:

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

Q14:

𝐴 𝐵 𝐶 is a triangle, where 𝑎 = 9 , 𝑏 = 6 , and 𝑚 ∠ 𝐴 = 5 8 . 1 ∘ . Find 𝑚 ∠ 𝐵 to the nearest tenth of a degree.

Q15:

𝐴 𝐵 𝐶 is an obtuse-angled triangle at 𝐴 where 𝑏 = 1 5 c m , t a n 𝐶 = 6 5 and 𝑚 ∠ 𝐵 = 2 7 ∘ . Find lengths 𝑎 and 𝑐 giving the answer to the nearest integer.

  • A 𝑎 = 3 2 c m and 𝑐 = 1 5 c m
  • B 𝑎 = 2 5 c m and 𝑐 = 3 2 c m
  • C 𝑎 = 1 5 c m and 𝑐 = 2 5 c m
  • D 𝑎 = 3 2 c m and 𝑐 = 2 5 c m

Q16:

𝐴 𝐵 𝐶 is a triangle where 𝑎 = 9 6 and 𝑚 ∠ 𝐵 = 3 𝑚 ∠ 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 ∘ ∘

Q17:

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

Q18:

𝐴 𝐵 𝐶 is a triangle where 2 𝐴 = 3 𝐵 = 4 𝐶 s i n s i n s i n and the perimeter is 169 cm. Find the values of 𝑎 and 𝑐 giving the answer to the nearest centimetre.

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

Q19:

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

Q20:

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

Q21:

𝐴 𝐵 𝐶 is a triangle where 𝑚 ∠ 𝐴 = 1 3 8 ∘ , 𝑎 = 1 3 c m and 𝑏 = 7 c m . Find 𝑚 ∠ 𝐵 giving the answer to the nearest second.

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

Q22:

In the figure 𝐴 𝐶 = 3 . 5 .

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

Q23:

Cities A, B, and C are located such that city A is due west 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.

Q24:

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 kilometre.

  • 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

Q25:

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.

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