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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 B

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The line \(L_1\) has equation \(5x+2y=30\). The line \(L_2\) has equation \(2x-y=12\).

    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 = 8 cm, angle BAD = 31° and angle DAC = 19°. Find the length x = DC.

    8 cmx31°, 19°ABDC

  3. In the diagram, angle ABC is a right angle and D lies on BC. BD = 4 cm, DC = 5 cm and angle ACB = 31°. Find the size of angle BAD, marked θ.

    4 cm5 cmθ31°ABDC

  4. In triangle ABC, AB = 9.6 cm, AC = 7.3 cm and angle BAC = 49°. Find the length of BC.

    9.6 cm7.3 cmx49°ABC

  5. In triangle ABC, AB = 6.7 cm, BC = 9.4 cm and CA = 7.9 cm.

    6.7 cm9.4 cm7.9 cmθBCA
    1. (a) Find the size of angle ABC, marked θ.
    2. (b) Find the area of the triangle.

In context

  1. A kite K is flying above level ground. Two cameras, A and B, are 145 m apart on the ground. The ground line AB and the kite are in the same vertical plane. From A the kite is seen at an angle of 63° to the ground line AB, and from B at an angle of 48° to BA.

    145 myd63°48°ABK
    1. (a) Find the distance y = AK from camera A to the kite.
    2. (b) Find the height d of the kite above the ground.

  2. A triangular orienteering course has checkpoints P, Q and R. PQ = 6.8 km, PR = 5.2 km and angle QPR = 71°. Runners go directly from Q to R.

    6.8 km5.2 kmx71°θPQR
    1. (a) Find the length x of the leg QR.
    2. (b) Find the size of angle PQR, marked θ.

  3. A triangular solar panel ABC has area 3.2 m². The edge AB is 2.4 m long and the angle at A is 58°. The edge AC has length x and the edge BC has length y.

    2.4 mxy58°ABC
    1. (a) Find x.
    2. (b) Find y.

Stretch

  1. In triangle ABC, D lies on BC. AB = 9 cm, AC = 12 cm, AD = 7 cm and angle ADB = 72°. Find the length of BC.

    9 cm12 cm7 cmx72°BCDA

  2. The area of triangle PQR is 22 cm². PQ = 8.2 cm, PR = 6.5 cm and angle QPR, marked θ, is acute.

    8.2 cm6.5 cmxθPQR
    1. (a) Find the size of θ.
    2. (b) Find the length of QR.

  3. In the diagram, D lies on BC. AB = 7.5 cm, angle ABD = 47°, angle ADB = 76° and angle ACD = 34°. Find the value of x = AC.

    7.5 cmx47°34°76°BDCA

  4. An aerial BT is fixed to the roof of a tall building. The foot of the building is O, with OB = h metres and BT = d metres. From a point P on level ground, the angle of elevation of the base of the aerial, B, is 28° and the angle of elevation of the top, T, is 31°.

    hd28°, 31°PBT
    1. (a) Show that \(h=\dfrac{d\tan 28^\circ}{\tan 31^\circ-\tan 28^\circ}\).
    2. (b) The aerial is 4 m long. Find the height of the building.

  5. A vertical tree OT of height h metres stands between two walkers at P and Q on level ground. P, O and Q are in a straight line and PQ = 90 m. The angle of elevation of the top of the tree T from P is 35° and from Q is 56°.

    90 mh35°56°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