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In this lesson, we will learn how to find the sides and angles in non-right triangles using the law of cosines.

Q1:

πΈ and π΄ are two hot air balloons flying at heights 117 m and 84 m respectively. The angles of depression at a point πΆ on the ground from πΈ and π΄ are 4 0 β and 2 9 β respectively. Find the distance between to the balloons giving the answer to the nearest metre.

Q2:

In the given figure, find π₯ . Give your answer to two decimal places.

Q3:

Q4:

In the given figure, using the law of cosines, find π . Give your answer to two decimal places.

Q5:

In the given figure, find π . Give your answer to two decimal places.

Q6:

π΄ , π΅ , and πΆ are three cities. Find the distance between cities π΄ and π΅ , giving your answer to the nearest kilometre.

Q7:

A plane travels 800 meters along the runway before taking off at an angle of 1 0 β . It travels a further 1 0 0 0 meters at this angle as seen in the figure. Work out the distance of the plane from its starting point. Give your answer to 2 decimal places.

Q8:

π΄ π΅ πΆ is a triangle, where π = 1 3 c m , π = 1 0 c m , and c o s πΆ = 0 . 2 . Find the value of π , giving your answer to three decimal places.

Q9:

A biker travelled from city π΄ to city π΅ via city πΆ with a uniform speed of 52 km/h. He then returned directly to city π΄ with a uniform speed of 89 km/h. Find, in minutes, the total time of his whole journey to two decimal places.

Q10:

π΄ π΅ πΆ is a triangle, where π΅ πΆ = 2 5 c m , π΄ πΆ = 1 3 c m , and π β πΆ = 1 4 2 β . Find the length π΄ π΅ giving the answer to three decimal places.

Q11:

π΄ π΅ πΆ is a triangle where π΅ πΆ = 3 8 c m , π β π΄ πΆ π΅ = 6 0 β and the area is 3 9 9 β 3 cm^{2}. Find the other lengths and angles giving lengths to one decimal place and angles to the nearest degree.

Q12:

The side lengths of a triangle are , , and . Sarah calculated, to one decimal place, the sizes of the corresponding angles as , , and . Was she correct?

Q13:

π΄ π΅ πΆ is a triangle, where π = 2 7 c m , π = 1 5 c m , π = 2 1 c m , and π· β π΅ πΆ , where π΅ π· = 1 6 c m . Find the length of π΄ π· , giving the answer to two decimal places.

Q14:

π΄ π΅ πΆ is a triangle where π = 1 8 cm , π = 1 0 cm and π β πΆ = 7 6 β . Find the measure of β π΄ giving the answer to the nearest second.

Q15:

π΄ π΅ πΆ is a triangle where π = 6 7 c m , π = 4 9 c m and the perimeter is 148 cm. Find the largest angle in π΄ π΅ πΆ giving the answer to the nearest second.

Q16:

π΄ π΅ πΆ is a triangle where π = 2 5 cm, π = 2 0 cm and π = 2 9 cm. Find π β π΄ giving the answer to the nearest second.

Q17:

π΄ π΅ πΆ is a triangle where 1 1 5 π΄ = 1 5 π΅ = 1 1 8 πΆ s i n s i n s i n . Find π β πΆ giving the answer to the nearest second.

Q18:

Find the value of π to 2 decimal places given that the measure of Angle π΄ is 6 4 β , π is 10 cm, and π is 16 cm.

Q19:

For the triangle given, find the value of the three angles to the nearest degree.

Q20:

Find π β π΄ to one decimal place.

Q21:

Complete the expression for triangle π΄ π΅ πΆ : π + π β π β― = πΆ 2 2 2 c o s .

Q22:

π΄ π΅ πΆ is a triangle where π β π = 2 2 c m , π β π = 3 2 c m , π β π = 8 c m and 2 π = π + π + π . Find the largest angle in the triangle giving the answer to the nearest second.

Q23:

π π π is a triangle where the ratio between s i n π , s i n π and s i n π is 2 3 βΆ 1 9 βΆ 1 6 . Find the smallest angle in π π π giving the answer to the nearest second.

Q24:

Which rule could be used to find the length of an unknown side of a triangle, given the other two lengths and the measure of their included angle.

Q25:

π΄ π΅ πΆ is a triangle where π β π = 2 0 c m , π + π = 1 1 6 c m , π = 4 1 c m and the perimeter is 2 π . Find the measure of the smallest angle in π΄ π΅ πΆ giving the answer to the nearest second.

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