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IB Mathematics: Applications and Interpretation SL · 12.10: Geometry & Trigonometry · 12.10.8

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Applications of trigonometry Problem set A

Draw the right-angled triangle or the bearings sketch first. Use your calculator in degree mode and give answers to 3 significant figures unless told otherwise.

Difficulty: first skills applying them stretch

Skills

  1. The cuboid ABCDEFGH has AB = 8 cm, BC = 5 cm and AE = 6 cm.

    8 cm5 cm6 cmABCDEFGH
    1. (a) Find the length of AG.
    2. (b) Find the angle between AG and the base ABCD.
    3. (c) Find the angle between AG and the edge AB.
    4. (d) AG and CE cross at the centre of the cuboid. Find the acute angle between AG and CE.

  2. The cuboid ABCDEFGH has AB = 10 cm, BC = 6 cm and AE = 5 cm.

    10 cm6 cm5 cmABCDEFGH
    1. (a) Find AC, then find the angle ACE.
    2. (b) Find the length of HB.
    3. (c) Find the angle between HB and BD.
    4. (d) Find the area of the triangle BGE.

  3. In the cuboid ABCDEFGH the three face diagonals AC, AF and CF have lengths AC = 10 cm, AF = 8 cm and CF = 9 cm.

    10 cm8 cm9 cmABCDEFGH
    1. (a) Find the length of the diagonal AG. Give your answer to 4 significant figures.
    2. (b) Find the angle between AG and the base ABCD.

  4. EABCD is a right pyramid with a square base ABCD of side 8 cm. Each sloping edge is 11 cm long. M is the midpoint of AB and N is the midpoint of BC.

    11 cmABCEMN
    1. (a) Sketch the triangle ACE and label every length you can.
    2. (b) Find the height of the pyramid.
    3. (c) Find the angle AEC.
    4. (d) Find the angle MEN.

  5. MABCD is a right pyramid. The base ABCD is a rectangle with AB = 9 cm and BC = 6 cm. O is the centre of the base and OM = 7 cm is the height.

    9 cm6 cm7 cmABCMO
    1. (a) Find the angle ACM, which is the angle between the sloping edge CM and the base.
    2. (b) Find the angle ABM.

  6. Two solids.

    ABMO
    1. (a) MABC is a right pyramid. The base ABC is an equilateral triangle of side 6 cm. O is the centre of the base and OM = 5 cm. Find the angle between AM and the base.
    2. (b) A cone has a base of diameter 10 cm and a vertical height of 7 cm. Find the angle between a sloping edge and the base.

In context

  1. A boat leaves a harbour H and sails 5.3 km on a bearing of 072° to a buoy K. It then sails 4.1 km on a bearing of 158° to a wreck W.

    5.3 km4.1 km?HKWNN
    1. (a) Find the distance from W back to H.
    2. (b) Find the bearing on which it must travel from W to return to H. Give the bearing to the nearest degree.

  2. From a coastguard post C, a mast M is 3.6 km away on a bearing of 025° and a wreck W is 2.9 km away on a bearing of 112°.

    3.6 km2.9 kmCMWN
    1. (a) Find the distance from the mast to the wreck.
    2. (b) Find the bearing of the mast from the wreck. Give the bearing to the nearest degree.

  3. A hut H has a pylon P 250 m away on a bearing of 125°. A well W is due east of the hut and due north of the pylon.

    1. (a) Sketch the situation. Mark the north direction at the starting point, the bearing and the right angle.
    2. (b) Find the distance from the well to the pylon.

  4. A ranger walks 6.4 km on a bearing of 048°, then 3.5 km on a bearing of 126°, then 4.1 km on a bearing of 228°. Find how far she is from her start.

    6.4 km3.5 km4.1 km?SPQFN

  5. Find the answer to each part. Draw a sketch with a right-angled triangle first.

    1. (a) Priya's eyes are 1.5 m above the ground. She stands 18 m from a flagpole and sees the top at an angle of elevation of 34°. Find the height of the flagpole.
    2. (b) From a point on level ground 55 m from the foot of a tower, the angle of elevation of the top of the tower is 28°. Find the height of the tower.
    3. (c) The top of a crane 60 m high is seen at an angle of elevation of 25° from a point on level ground. Find the horizontal distance from the point to the crane.
    4. (d) From a balcony 14 m above the ground, the angle of depression of a car is 18°. Find the horizontal distance from the balcony to the car.

  6. Three problems about heights and angles.

    1. (a) A clock tower stands on level ground. From a point on the ground the angle of elevation of the top is 36°. After walking 15 m directly away from the tower, the angle of elevation is 26°. Find the height of the tower.
    2. (b) From the top of a watchtower, the angle of depression of a boat due east of the tower is 31.0°. A second boat is 25 m due south of the first. Its angle of depression from the top is 29.5°. Find the height of the watchtower.
    3. (c) A tree stands at B on level ground. Priya stands at P, 26.0 m from B, and Omar stands at Q, 18.4 m from B. They are 15.2 m apart. Find the angle PBQ.

Stretch

  1. A botanical garden has a glass greenhouse in the shape of a square-based pyramid. The base has sides of 28 m and the height is 17.5 m. Air conditioning is rated at one unit for every 600 m³ of air.

    28 m17.5 mABCEO
    1. (a) How many units are needed? State your assumptions.
    2. (b) On a hot day the glass gains energy at 150 watts for every m² of glass. Find the total power the air conditioning must remove. The base is not glass.
    3. (c) The cleaning rules say scaffolding is needed if any glass has an angle of elevation of more than 48°. Is scaffolding needed?

  2. A house is 9.0 m long and 6.4 m wide. Its roof is a triangular prism with an isosceles triangle as its end. Each side of the roof makes an angle of 38° with the horizontal. A loft conversion can use only the part of the roof space where the roof is at least 1.4 m above the floor.

    6.4 m38°
    1. (a) Find the height of the ridge above the loft floor.
    2. (b) Find the volume of the whole roof space.
    3. (c) Find the percentage of the floor area that can be used.
    4. (d) Find the percentage of the roof space volume that can be used.

jmaths.xyz · AI SL · 12.10.8 Applications of trigonometry: 3D, bearings, elevation