Determine whether the primary trigonometric ratios, the sine law, or the cosine law should be used first to solve each triangle.

a)

b)

c)

d)

a) Find x to the nearest tenth of a centimetre.

b) Find x using a different method.

a) Find x, to the nearest tenth of a centimetre.

b) Find x using a different method.

While flying at an altitude of 1.5 km, a plane measures angles of depression to opposite ends of a large crater, as shown. Find the width of the crater, to the nearest tenth of a kilometre.

Earth is 149\ 600\ 000 km from the Sun. This distance is equal to 1 A.U. (astronomical unit). Mars is 1.5 A.U. from the Sun. One evening, Mars is seen from Earth to make an angle of 68^\circ with the Sun.

a) Draw a diagram and label the given information.

b) How far apart are Earth and Mars at this point, in kilometres?

c) Do you think the distance between Earth and Mars is always the same? Explain why or why not.

Leia is in a bicycle road race. In the first leg, she rides 12 km from Riverside to Danton. Then, she turns and rides 17 km to Humberville, making a 74^\circ angle from the first leg. The final turn leads back to Riverside.

a) What is the total length of the race, to the nearest kilometre?

b) At what angles are the three towns situated with respect to each other? Round to the nearest degree.

Trey, who is 1.5 m tall, is standing at a distance of 14 m from a building. From his point of view, the bottom and top of the building are separated by 36^\circ, as shown. How tall is the building, to the nearest tenth of a metre?

Ron and Ben are two koala bears frolicking in a meadow. Suddenly, a tasty clump of eucalyptus falls to the ground, catching their attention. Ben glances at Ron, who appears to be 15 m away, then over to the eucalyptus, which appears to be 18 m away. From Ben's point of view, Ron and the eucalyptus are separated by an angle of 45^\circ. Rocco’s top running speed is 1.0 m/s, but Ben can run one and a half times as fast. Can Ben beat Ron to the eucalyptus? State any assumptions you make.

Find the total length of materials required to build the bridge truss shown, to the nearest tenth of a metre.

Describe the steps in your solution and state any assumptions you make.

Lookout Point is accessible from two trails, both of which start from the same altitude and climb upward. Path p travels east to the point and climbs at an average angle of elevation of 20^\circ. Path q travels northeast to the point at an average angle of elevation of 15^\circ. Path p is 2.0 km long. lack and Debbie parked at the base of path p. They hiked a round trip up path p to Lookout Point. then down path q, and then finally straight from the base of path q back to their truck. How far did they hike, to the nearest tenth of a kilometre? State any assumptions you make.

A tetrahedron has edges that are 10 cm in length. Find the height of this tetrahedron. to the nearest tenth of a centimetre.

Doctors Jones and Hwang are astronomers observing the sun from opposite ends of Earth. The radius of Earth is 6400 km.

a) Use this information to verify the distance from Earth to the Sun, which was given in question 5. State any assumptions you make.

b) At approximately what times of day were these observations made by each astronomer? Explain your answer.

Pilots must take wind into account when flying, or the wind will blow them off course and they will not reach the desired destination. Your aircraft cruises at a speed of 100 km/h. There is a strong wind blowing from N60^\circE at a speed of 90 km/h. You need to fly south to home base.

a) Find the direction, \theta, you must aim the plane. to the nearest degree.

b) What will your speed be, over the ground? Round to the nearest unit.

Hanna, Jon. and Robin live in two identical apartment buildings. located 30 in apart. Jon lives two floors higher than Hanna. Robin lives four floors lower than Hanna. There is a 36^\circ angle of separation when Hanna looks from her balcony to those of her two friends.

a) How far apart, vertically, do Jon and Robin live? Round to the nearest tenth of a metre.

b) Explain how you solved this problem and discuss any assumptions you made.

A box is in the shape of a square-based prism. The height of the box is twice the width of the base.

a) Show that the longest thin rod that can be encased in the box has length \sqrt{6}w, where w is the width of the base.

b) Find the angles that such a rod would make with each edge of the box.

A ship travels 100 km at a bearing of N60^\circE and then turns and travels 80 km at a bearing of S2O^\circE before reaching its destination. Suppose the ship travelled directly from its starting point to its destination, following a direct route. What distance and at what bearing would the ship travel? Round to the nearest unit.