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IB Mathematics: Applications and Interpretation SL · 12.10: Geometry & Trigonometry · 12.10.4

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Rules of trigonometry: mixed practice Problem set A

Draw or label a triangle first. Choose the rule, show the substitution, and keep full accuracy until the last line. Answers to 3 significant figures; angles in degrees.

Difficulty: first skills applying them stretch

Skills

  1. The line \(L_1\) has equation \(3x+4y=24\). The line \(L_2\) has equation \(x-2y=8\).

    1. (a) Find the coordinates of the points where \(L_1\) meets the axes.
    2. (b) Find the acute angle between \(L_1\) and the x-axis.
    3. (c) Show that \(L_1\) and \(L_2\) meet on the x-axis.
    4. (d) Find the acute angle between \(L_1\) and \(L_2\).

  2. In the diagram, angle ABC is a right angle and D lies on BC. BD = 6 cm, angle BAD = 28° and angle DAC = 22°. Find the length x = DC.

    6 cmx28°, 22°ABDC

  3. In the diagram, angle ABC is a right angle and D lies on BC. BD = 7 cm, DC = 3 cm and angle ACB = 24°. Find the size of angle BAD, marked θ.

    7 cm3 cmθ24°ABDC

  4. In triangle ABC, AB = 6.1 cm, AC = 8.4 cm and angle BAC = 57°. Find the length of BC.

    6.1 cm8.4 cmx57°ABC

  5. In triangle ABC, AB = 8.3 cm, BC = 9.6 cm and CA = 7.1 cm.

    8.3 cm9.6 cm7.1 cmθBCA
    1. (a) Find the size of angle ABC, marked θ.
    2. (b) Find the area of the triangle.

In context

  1. Two coastguard stations, A and B, are 9.6 km apart on a straight, level coast. A ship S is at sea. From A the ship is seen at an angle of 52° to the coast line AB, and from B at an angle of 71° to BA.

    9.6 kmyd52°71°ABS
    1. (a) Find the distance y = AS from station A to the ship.
    2. (b) Find the shortest distance d from the ship to the coast.

  2. A triangular garden ABC has AB = 12.4 m, AC = 9.7 m and angle BAC = 64°. A fence runs along BC.

    12.4 m9.7 mx64°θABC
    1. (a) Find the length x of the fence BC.
    2. (b) Find the size of angle ABC, marked θ.

  3. A triangular sail ABC has area 14.5 m². The edge AB is 6.2 m long and the angle at A is 62°. The edge AC has length x and the edge BC has length y.

    6.2 mxy62°ABC
    1. (a) Find x.
    2. (b) Find y.

Stretch

  1. In triangle ABC, D lies on BC. AB = 11 cm, AC = 14 cm, AD = 8 cm and angle ADB = 65°. Find the length of BC.

    11 cm14 cm8 cmx65°BCDA

  2. The area of triangle PQR is 28 cm². PQ = 7.4 cm, PR = 9.6 cm and angle QPR, marked θ, is acute.

    7.4 cm9.6 cmxθPQR
    1. (a) Find the size of θ.
    2. (b) Find the length of QR.

  3. In the diagram, D lies on BC. AB = 9 cm, angle ABD = 52°, angle ADB = 71° and angle ACD = 30°. Find the value of x = AC.

    9 cmx52°30°71°BDCA

  4. A flagpole BT stands on the edge of a vertical cliff. The foot of the cliff is O, with OB = h metres and BT = d metres. From a point P on level ground, the angle of elevation of the bottom of the flagpole, B, is 32° and the angle of elevation of the top, T, is 36°.

    hd32°, 36°PBT
    1. (a) Show that \(h=\dfrac{d\tan 32^\circ}{\tan 36^\circ-\tan 32^\circ}\).
    2. (b) The flagpole is 6 m long. Find the height of the cliff.

  5. A vertical floodlight mast OT of height h metres stands between two points P and Q on level ground. P, O and Q are in a straight line and PQ = 75 m. The angle of elevation of the top of the mast T from P is 27° and from Q is 48°.

    75 mh27°48°PQT
    1. (a) Write down PO and OQ in terms of h.
    2. (b) Hence find the value of h.

jmaths.xyz · AI SL · 12.10.4 Mixed practice: rules of trigonometry