Worksheet: Applications of the Sine Rule

In this worksheet, we will practice applying the sine rule to find a missing side length or a missing angle in a triangle.

Q1:

For the given figure, , , and . Work out the length of . Give your answer to two decimal places.

Q2:

For the given figure, , , and . Work out the length of . Give your answer to two decimal places.

Q3:

Find all the missing angles and lengths of the given triangle, giving the lengths to three decimal places.

• A , ,
• B , ,
• C , ,
• D , ,

Q4:

A lighthouse with a height of 89 metres is constructed on a cliff. From a boat in the sea, the angle of elevation to the top of the lighthouse is , and the angle of elevation to the base of the lighthouse is . Find the height of the cliff from sea level giving the answer to the nearest metre.

Q5:

For the given figure, , , and . Work out the size of . Give your answer to two decimal places.

• A
• B
• C
• D
• E

Q6:

In the given figure, , , and .

Use the law of sines to work out the size of . Give your answer to two decimal places.

Work out the length of . Give your answer to two decimal places.

Q7:

For the given figure, , , and . Work out the size of . Give your answer to two decimal places.

• A
• B
• C
• D
• E

Q8:

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

• A ,
• B ,
• C ,
• D ,

Q9:

Two boys are standing on a riverbank. represents the first boy’s location and represents the location of the second boy. A boat is sailing on the river and stops at a point . Find the distance between the second boy and the boat giving the answer to the nearest metre.

Q10:

The angle of depression of a car parked on the ground from the top of a hill is . A view point is 14 metres vertically below the top of the hill, and the angle of depression to the car is . Find the height of the hill giving the answer to the nearest metre.

Q11:

A man observed a stationary car from the top of a tower where the angle of depression is . When the man descended 9 metres, the angle of depression became . Find the distance between the car and the tower giving the answer to the nearest metre.

Q12:

A tower is 161 metres tall and is constructed on a rock. From a point on the ground on the same horizontal level as the base of the rock, the angles of elevation to the base of the tower and to the top of the tower are and respectively. Find the height of the rock giving the answer to the nearest metre.

Q13:

From point on a riverbank, a man looked at a house located on the other side of the river at point and found the direction to be north of east. He walked 147 metres east parallel to the river and arrived at point where point was north of east. Given the two riverbanks are parallel and points , , and are on the same horizontal level, find the width of the river giving the answer to the nearest metre.

Q14:

A communications tower is located at the top of a steep hill, as shown in the given figure. The angle of inclination of the hill is . A guy wire is to be attached to the top of the tower and to the ground 165 metres downhill from the base of the tower. The angle formed by the guy wire and the hill is . Find the length of the cable required for the guy wire to the nearest whole meter.

Q15:

In the figure, , , , and .

Q16:

Find the length of .

Q17:

Find the value of .

Q18:

Find all the missing angles and lengths of the given triangle, giving the lengths to three decimal places.

• A , ,
• B , ,
• C , ,
• D , ,

Q19:

The positions of three towns are shown in the diagram. Towns and are 89 km apart. Find the distance between towns and and the distance between towns and . Give your answers to the nearest kilometre.

• A ,
• B ,
• C ,
• D ,