# Worksheet: Applications on Sine and Cosine Laws

In this worksheet, we will practice using the cosine and sine law to solve real-world problems.

Q1:

is a triangle, where is the midpoint of , , , and . Find the length of giving the answer to two decimal places.

Q2:

is a triangle, where cm, cm, and . Find the radius of the circumcircle giving the answer to two decimal places.

Q3:

is a triangle where cm, cm and cm. Find the radius of the circumcircle giving the answer to three decimal places.

Q4:

is a triangle where , the ratio between and is , and the area of the circumcircle is cm2. Find the perimeter of triangle giving the answer to the nearest centimeter.

Q5:

is a quadrilateral where , , , and . Find giving the answer to the nearest second and the area of giving the answer to three decimal places.

• A, 88.123 cm2
• B, 37.767 cm2
• C, 134.283 cm2
• D, 67.142 cm2

Q6:

is a quadrilateral where , , and . Find the area of giving the answer to two decimal places.

Q7:

Find the area of given is the point of intersection of and , , , , and . Give the answer to the nearest square centimeter. Q8:

is a parallelogram where is the point of intersection to the diagonals, , and . Find the area of the parallelogram giving the answer to two decimal places.

Q9:

In a parallelogram , and , where . Work out the length of . Give your answer to 2 decimal places.

Q10:

The side length of a regular pentagon is 25.81 cm. Find the length of the diagonal giving the answer to two decimal places.

Q11:

In a triangle , , , and .

If is the altitude from , what is to 2 decimal places?

Calculate the area of the triangle to 2 decimal places.

Q12:

Find the area of the green part of the diagram, given that , , and . Give the answer to the nearest square centimeter. Q13:

is an equilateral triangle inscribed in circle with a radius of 95 cm. Find the area of each shaded segment giving the answer to the nearest square centimeter. Q14:

is a triangle where the ratio between , , and is and the radius of the circumcircle is 6 cm. Find the area of giving the answer to two decimal places.

Q15:

Two planes leave the same airport at the same time. One flies on a bearing of at 500 miles per hour. The other flies on a bearing of at 600 miles per hour. How far apart are the planes after 2 hours?

Q16:

The shown diagram represents a satellite calculating the distances and the angle. Find the distance between the two cities. Give your answer to one decimal place. Note that the diagram is not to scale. Q17:

A 113-foot tower is located on a hill that is inclined to the horizontal, as shown in the figure. A guy wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed. Q18:

A man ran 24 km down a straight road. He then turned and ran a further 9 km along another straight road. Find, to the nearest kilometer, the shortest distance between his start and end points.

Q19:

An airplane flies 220 miles on a bearing of and then changes course and flies 180 miles on a bearing of . How far is the plane from its starting point and on what bearing? Give your answers to one decimal place.

• A172.9 miles,
• B172.9 miles,
• C52.9 miles,
• D362.9 miles,
• E6,141.9 miles,

Q20:

Two boats leave a harbor in different directions. One travels due west at 30 mph, and the other travels on a bearing of at 42 mph. How far apart are they after 90 minutes?

Q21:

Two ships left a port at the same time. One traveled on a bearing of at 18 mph, while the other traveled on a bearing of at 22 mph. Find the distance between the ships after 10 hours. Give your answer to the nearest mile.

Q22:

A plane follows a straight path at a steady speed of 600 miles per hour. After 1 hour  30 minutes, the pilot makes a course correction of , and the flight continues at the same speed, following a straight path, for another 2 hours. How far is the plane from its initial position? Give your answer to the nearest mile.

Q23:

Philadelphia is 140 miles from Washington, DC. Washington, DC, is 442 miles from Boston. Boston is 315 miles from Philadelphia. Find the angles in the triangle whose vertices are at Philadelphia, Washington, DC, and Boston.

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q24:

Los Angeles is 1,744 miles from Chicago, Chicago is 712 miles from New York, and New York is 2,451 miles from Los Angeles. Find the angles in the triangle with its vertices at the three cities.

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Q25:

Two sides of a triangular flower bed measure 40 feet and 65 feet with an angle of between them. Calculate the length of the third side, giving your answer to one decimal place.