Worksheet: The Sine Ratio

In this worksheet, we will practice finding angles and sides in a right triangle using the sine ratio.

Q1:

Find 𝑚 𝐵 𝐴 𝐶 giving the answer to the nearest second.

  • A 5 4 2 7 4 4
  • B 4 5 3 5 5
  • C 4 4 2 4 5 5
  • D 3 5 3 2 1 6

Q2:

Find s i n 𝐴 , given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐶 where 𝐴 𝐵 = 2 0 c m and 𝐵 𝐶 = 1 6 c m .

  • A 3 5
  • B 4 5
  • C 5 4
  • D 5 3

Q3:

Find the measure of 𝐵 , given 𝐴 𝐵 𝐶 𝐷 is an isosceles trapezium where 𝐴 𝐵 = 𝐴 𝐷 = 𝐷 𝐶 = 1 3 c m and 𝐵 𝐶 = 1 7 c m giving the answer to the nearest second.

  • A 8 3 9
  • B 8 1 9
  • C 8 2 9
  • D 8 0 9

Q4:

Find the value of 𝜃 giving the answer in radians to two decimal places.

Q5:

A builder leans a 5-foot plank of wood against a vertical wall. If one end of the plank is 3 feet from the bottom of the wall, what is the angle between the plank and the wall?

Q6:

In the given figure, 𝐵 𝐴 has been extended to the point 𝐷 such that 𝐵 𝐷 and 𝐷 𝐶 are perpendicular.

Work out the value of s i n 1 1 0 to 4 decimal places.

Work out the value of s i n 7 0 to 4 decimal places.

Work out the length of 𝐷 𝐶 to 2 decimal places.

Work out the area of the triangle 𝐴 𝐵 𝐶 to 2 decimal places.

Q7:

In the given figure, 𝑚 𝐵 𝐴 𝐶 = 9 0 and 𝐴 𝐷 𝐵 𝐶 . What is 𝐴 𝐶 𝜃 s i n ?

  • A 𝐴 𝐵
  • B 𝐵 𝐶
  • C 𝐴 𝐶
  • D 𝐴 𝐷
  • E 𝐶 𝐷

Q8:

In the given figure, 𝐴 𝐶 = 5 , 𝐵 𝐶 = 7 , and 𝑚 𝐵 𝐶 𝐴 = 4 1 .

Calculate the height 𝐴 𝐷 to 2 decimal places.

Calculate the area of the triangle 𝐴 𝐵 𝐶 to 2 decimal places.

Q9:

Find s i n 𝐴 given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where the point 𝐷 lies on 𝐵 𝐶 , 𝐷 𝐸 is perpendicular to 𝐴 𝐶 , 𝐷 𝐸 = 1 8 c m and 𝐶 𝐸 = 2 4 c m .

  • A 4 5
  • B 5 4
  • C 3 5
  • D 4 3

Q10:

A 33 ft ladder leans against a building such that the angle between the ground and the ladder is 8 0 . How high does the ladder reach up the side of the building?

Q11:

The distance between two villages 𝐴 and 𝐵 is 13 km where 𝐴 is west of 𝐵 . Village 𝐶 is located 3 7 east of north of 𝐴 and 4 8 north of west of 𝐵 . Find the distance between 𝐵 and 𝐶 giving the answer to the nearest kilometre.

Q12:

The base of a truck is 2 feet off the ground and a 15-foot ramp is used to load the truck. Find the angle the ramp makes with the horizontal ground.

Q13:

The base of a truck is 2 feet off the ground and a 20-foot ramp is used to load the truck. Find, to two decimal places, the angle the ramp makes with the horizontal ground.

Q14:

𝐴 , 𝐵 , and 𝐶 are three towns that lie in the same horizontal plane. 𝐵 lies 38 km southwest of 𝐴 . 𝐴 lies 2 3 east of north of 𝐶 and 𝐵 lies 7 east of north of 𝐶 . Find the length of 𝐴 𝐶 giving the answer to the nearest kilometre.

Q15:

In the given figure, 𝑚 𝐵 𝐴 𝐶 = 9 0 and 𝐴 𝐷 𝐵 𝐶 . What is 𝐵 𝐶 𝜃 s i n ?

  • A 𝐶 𝐵
  • B 𝐴 𝐷
  • C 𝐴 𝐶
  • D 𝐶 𝐷
  • E 𝐴 𝐵

Q16:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝑚 𝐶 = 6 5 4 8 and 𝐴 𝐶 = 1 2 c m . Find the length of 𝐴 𝐵 giving the answer to two decimal places.

Q17:

Find the value of s i n s i n 𝑋 + 𝑌 given 𝑋 𝑌 𝑍 is a right-angled triangle at 𝑍 , 𝑋 𝑌 = 5 8 c m , 𝑌 𝑍 = 4 0 c m and 𝑋 𝑍 = 4 2 c m .

  • A 4 1 2 9
  • B 4 0 2 9
  • C 8 2 2 9
  • D 4 2 2 9

Q18:

In the given figure, 𝐴 𝐵 = 6 , 𝐵 𝐶 = 1 1 , and 𝑚 𝐴 𝐵 𝐶 = 6 1 .

Calculate the height 𝐴 𝐷 to 2 decimal places.

Calculate the area of the triangle 𝐴 𝐵 𝐶 to 2 decimal places.

Q19:

In the given figure, 𝐴 𝐶 = 5 , 𝐵 𝐶 = 7 , and 𝑚 𝐵 𝐶 𝐴 = 4 1 . Work out the height 𝐴 𝐷 of the triangle to 2 decimal places.

Q20:

In the given figure, 𝐴 𝐵 = 6 , 𝐵 𝐶 = 1 1 , and 𝑚 𝐴 𝐵 𝐶 = 6 1 . Work out the height 𝐴 𝐷 of the triangle to 2 decimal places.

Q21:

A mountain has a uniform incline of 2 8 . If a person walks 127 meters to the peak of the mountain, how tall is the mountain? Give the answer to the nearest meter.

  • A 60 m
  • B 239 m
  • C 271 m
  • D 68 m
  • E 112 m

Q22:

What is the value of s i n 𝜃 in the given triangle?

  • A 1 3 5
  • B 1 3 1 2
  • C 5 1 3
  • D 5 1 2
  • E 1 2 1 3

Q23:

Find the area of the triangle 𝐴 𝐵 𝐶 given 𝐴 𝐵 = 𝐴 𝐶 = 2 1 c m and 𝐵 𝐶 = 2 0 c m . Give the answer to two decimal places.

  • A 210 cm2
  • B 184.66 cm2
  • C 100 cm2
  • D 369.32 cm2

Q24:

A circle has a radius of 16 cm. A chord is drawn where the central angle is 8 0 . Find the length of the chord giving the answer to three decimal places.

Q25:

Consider the following right triangle 𝐴 𝐵 𝐶 .

Which of the following statements is true?

  • A s i n c o s 𝛼 = 𝑐 𝑎 , 𝛼 = 𝑐 𝑏
  • B s i n c o s 𝛼 = 𝑎 𝑏 , 𝛼 = 𝑎 𝑐
  • C s i n c o s 𝛼 = 𝑎 𝑐 , 𝛼 = 𝑏 𝑐
  • D s i n c o s 𝛼 = 𝑏 𝑐 , 𝛼 = 𝑎 𝑐
  • E s i n c o s 𝛼 = 𝑏 𝑎 , 𝛼 = 𝑐 𝑎

Find the value of ( 𝛼 ) + ( 𝛼 ) s i n c o s (usually written as: s i n 𝛼 + 𝑐 𝑜 𝑠 𝛼 ).

  • A 𝑎 𝑏 + 𝑎 𝑐 = 𝑎 𝑐 + 𝑏 𝑏 𝑐
  • B 𝑐 𝑎 + 𝑐 𝑏 = 𝑏 𝑐 + 𝑎 𝑐 𝑎 𝑏
  • C 𝑎 𝑏 + 𝑐 𝑏 = 𝑎 + 𝑐 𝑏
  • D 𝑎 𝑐 + 𝑏 𝑐 = 𝑎 + 𝑏 𝑐 = 𝑐 𝑐 = 1
  • E 𝑏 𝑎 + 𝑐 𝑎 = 𝑏 + 𝑐 𝑎 = 2 𝑐 𝑎 𝑎

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