Worksheet: The Sine Ratio

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

Q1:

Find 𝑚𝐵𝐴𝐶 giving the answer to the nearest second.

  • A45355
  • B542744
  • C442455
  • D353216

Q2:

Find sin𝐴, given 𝐴𝐵𝐶 is a right triangle at 𝐶 where 𝐴𝐵=20cm and 𝐵𝐶=16cm.

  • A35
  • B53
  • C54
  • D45

Q3:

Find the measure of 𝐵, given 𝐴𝐵𝐶𝐷 is an isosceles trapezoid where 𝐴𝐵=𝐴𝐷=𝐷𝐶=13cm and 𝐵𝐶=17cm giving the answer to the nearest second.

  • A819
  • B839
  • C829
  • D809

Q4:

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

Q5:

In the given figure, 𝑚𝐵𝐴𝐶=90 and 𝐴𝐷𝐵𝐶. What is𝐴𝐶𝜃sin?

  • A𝐶𝐷
  • B𝐴𝐵
  • C𝐴𝐷
  • D𝐴𝐶
  • E𝐵𝐶

Q6:

Find sin𝐴, given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where the point 𝐷 lies on 𝐵𝐶, 𝐷𝐸 is perpendicular to 𝐴𝐶, 𝐷𝐸=18cm, and 𝐶𝐸=24cm.

  • A35
  • B54
  • C45
  • D43

Q7:

The distance between two villages 𝐴 and 𝐵 is 13 km where 𝐴 is west of 𝐵. Village 𝐶 is located 37 east of north of 𝐴 and 48 north of west of 𝐵. Find the distance between 𝐵 and 𝐶 giving the answer to the nearest kilometer.

Q8:

𝐴, 𝐵, and 𝐶 are three towns that lie in the same horizontal plane. 𝐵 lies 38 km southwest of 𝐴. 𝐴 lies 23 east of north of 𝐶 and 𝐵 lies 7 east of north of 𝐶. Find the length of 𝐴𝐶 giving the answer to the nearest kilometer.

Q9:

In the given figure, 𝑚𝐵𝐴𝐶=90 and 𝐴𝐷𝐵𝐶. What is 𝐵𝐶𝜃sin?

  • A𝐶𝐵
  • B𝐴𝐷
  • C𝐶𝐷
  • D𝐴𝐵
  • E𝐴𝐶

Q10:

𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where 𝑚𝐶=6548 and 𝐴𝐶=12cm. Find the length of 𝐴𝐵 giving the answer to two decimal places.

Q11:

Find the value of sinsin𝑋+𝑌 given 𝑋𝑌𝑍 is a right-angled triangle at 𝑍, 𝑋𝑌=58cm, 𝑌𝑍=40cm and 𝑋𝑍=42cm.

  • A8229
  • B4029
  • C4129
  • D4229

Q12:

In the given figure, 𝐴𝐶=5, 𝐵𝐶=7, and 𝑚𝐵𝐶𝐴=41. Work out the height 𝐴𝐷 of the triangle to 2 decimal places.

Q13:

In the given figure, 𝐴𝐵=6, 𝐵𝐶=11, and 𝑚𝐴𝐵𝐶=61. Work out the height 𝐴𝐷 of the triangle to 2 decimal places.

Q14:

A mountain has a uniform incline of 28. If a person walks 127 meters to the peak of the mountain, how tall is the mountain? Give the answer to the nearest meter.

Q15:

What is the value of sin𝜃 in the given triangle?

  • A513
  • B512
  • C1312
  • D1213
  • E135

Q16:

A circle has a radius of 16 cm. A chord is drawn where the central angle is 80. Find the length of the chord giving the answer to three decimal places.

Q17:

Consider the following right triangle 𝐴𝐵𝐶.

Which of the following statements is true?

  • Asincos𝛼=𝑏𝑐,𝛼=𝑎𝑐
  • Bsincos𝛼=𝑎𝑐,𝛼=𝑏𝑐
  • Csincos𝛼=𝑏𝑎,𝛼=𝑐𝑎
  • Dsincos𝛼=𝑎𝑏,𝛼=𝑎𝑐
  • Esincos𝛼=𝑐𝑎,𝛼=𝑐𝑏

Find the value of (𝛼)+(𝛼)sincos (usually written as: sin𝛼+𝑐𝑜𝑠𝛼).

  • A𝑐𝑎+𝑐𝑏=𝑏𝑐+𝑎𝑐𝑎𝑏
  • B𝑎𝑐+𝑏𝑐=𝑎+𝑏𝑐=𝑐𝑐=1
  • C𝑎𝑏+𝑐𝑏=𝑎+𝑐𝑏
  • D𝑎𝑏+𝑎𝑐=𝑎𝑐+𝑏𝑏𝑐
  • E𝑏𝑎+𝑐𝑎=𝑏+𝑐𝑎=2𝑐𝑎𝑎

Q18:

In the given figure, 𝐵𝐴 has been extended to the point 𝐷 such that 𝐵𝐷 and 𝐷𝐶 are perpendicular. Work out the length of 𝐷𝐶 to two decimal places.

Q19:

What is the value of sin𝜃 in the given triangle?

  • A34
  • B35
  • C53
  • D43
  • E45

Q20:

Find 𝑚𝐴 given 𝐴𝐵𝐶𝐷 is an isosceles trapezoid where 𝐴𝐵=𝐴𝐷=𝐷𝐶=15cm and 𝐵𝐶=33cm. Give the answer to the nearest second.

  • A1275212
  • B1265212
  • C1285212
  • D1255212

Q21:

A ladder 3 meters long leans against a wall. Find the vertical height of the ladder, given that the angle between the ladder and the ground is 45.

  • A332 m
  • B32 m
  • C322 m
  • D3 m

Q22:

𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where 𝑚𝐴=1.138rad and 𝐵𝐶=7cm. Find the lengths 𝐴𝐶, 𝐵𝐶, and the angle 𝐶 in radian, giving the answers to three decimal places.

  • A𝐶=0.433rad, 𝐴𝐵=3.234cm, 𝐴𝐶=7.711cm
  • B𝐶=0.567rad, 𝐴𝐵=16.690cm, 𝐴𝐶=16.690cm
  • C𝐶=1.433rad, 𝐴𝐵=3.234cm, 𝐴𝐶=16.690cm
  • D𝐶=0.567rad, 𝐴𝐵=7.711cm, 𝐴𝐶=3.234cm

Q23:

𝐴𝐵𝐶𝐷 is an isosceles trapezoid where 𝐴𝐷𝐵𝐶, 𝐴𝐵=𝐶𝐷=19cm, 𝐴𝐷=16cm, and 𝐵𝐶=40cm. Find 𝑚𝐴 and 𝑚𝐵 giving the answer to the nearest second.

  • A𝑚𝐴=50500, 𝑚𝐵=129100
  • B𝑚𝐴=140500, 𝑚𝐵=321632
  • C𝑚𝐴=129100, 𝑚𝐵=50500
  • D𝑚𝐴=1221632, 𝑚𝐵=50500

Q24:

For the given figure, 𝐴𝐵=8.5, 𝐴𝐶=9.3, and 𝑚𝐴𝐵𝐶=71. Work out the measure of 𝐴𝐶𝐵. Give your answer to two decimal places.

Q25:

The positions of three towns are shown in the diagram. Towns 𝐴 and 𝐵 are 130 km apart. Find the distance between towns 𝐴 and 𝐶 and the distance between towns 𝐵 and 𝐶. Give your answers to the nearest kilometer.

  • A𝐴𝐶=110km, 𝐵𝐶=93km
  • B𝐴𝐶=83km, 𝐵𝐶=96km
  • C𝐴𝐶=110km, 𝐵𝐶=96km
  • D𝐴𝐶=110km, 𝐵𝐶=130km

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