Pack A — Fluency — 11.7 Non-right-angle Trigonometry

BronzeQuestions 1–10

1.
In a right-angled triangle, the hypotenuse is 10 and one angle is $35^\circ$ . Find the side opposite this angle, to 3 s.f.
2.
In a right-angled triangle, opposite is 5 and hypotenuse is 13. Find the angle, to 3 s.f.
3.
In a right-angled triangle, the two shorter sides are 5 and 12. Find the hypotenuse (exact).
4.
State the exact value of $\sin 60^\circ$ .
5.
A right-angled triangle has legs 6 cm and 8 cm. Find its area.
6.
A ship sails on bearing $060^\circ$ . State the angle it makes with North (clockwise).
7.
Find the area of a triangle with sides $a = 12$ , $b = 9$ and included angle $C = 75^\circ$ , to 3 s.f.
8.
In triangle $ABC$ , $\angle A = 40^\circ$ and $\angle B = 75^\circ$ . Find $\angle C$ .
9.
In triangle $ABC$ , write the sine rule equation relating $a, b, \angle A, \angle B$ .
10.
State the cosine rule for $a$ in terms of $b, c, A$ .

SilverQuestions 11–20

11.
In triangle $ABC$ , $\angle A = 40^\circ$ , $\angle B = 70^\circ$ , side $a = 8$ . Find $b$ to 3 s.f.
12.
In triangle $ABC$ , $a = 10$ , $b = 13$ and $\angle A = 35^\circ$ . Find $\angle B$ (acute case), to 3 s.f.
13.
Find side $a$ when $b = 7$ , $c = 9$ and $A = 50^\circ$ , to 3 s.f.
14.
In a triangle with sides $a = 5$ , $b = 6$ , $c = 7$ , find angle $A$ , to 3 s.f.
15.
Triangle $ABC$ has $AB = 12$ m, $AC = 9$ m, $\angle BAC = 75^\circ$ . Find (a) area, (b) $BC$ , to 3 s.f.
16.
From a point on the ground 50 m from the base of a tower, the angle of elevation of the top is $32^\circ$ . Find the height, to 3 s.f.
17.
A chord subtends an angle of $80^\circ$ at the centre of a circle of radius 5 cm. Find the chord length, to 3 s.f.
18.
A ship sails 50 km on bearing $60^\circ$ from port. How far north and east is the ship? To 3 s.f.
19.
In triangle $ABC$ , $AB = 4$ , $AC = 6$ , $\angle BAC = 60^\circ$ . Find the area exactly.
20.
From the top of a cliff 80 m high, the angle of depression of a boat at sea is $15^\circ$ . How far is the boat from the foot of the cliff, to 3 s.f.?

GoldQuestions 21–30

21.
In triangle $ABC$ , $AB = 12$ , $AC = 9$ , $\angle BAC = 75^\circ$ and $BC \approx 13.0$ . Find $\angle ABC$ to 3 s.f.
22.
From port $P$ , a ship sails 50 km on bearing $060^\circ$ to $A$ , then 80 km on bearing $150^\circ$ to $B$ . Find the angle at $A$ in triangle $PAB$ , and $PB$ to 3 s.f.
23.
A triangle has sides 7, 9, 11. Find its largest angle to 3 s.f.
24.
Triangle $XYZ$ has $XY = 14$ , $\angle X = 50^\circ$ , $\angle Z = 70^\circ$ . Find $YZ$ to 3 s.f.
25.
A triangle has sides 10 and 14 and area 50 m $^2$ . Find the included angle, giving the acute case to 3 s.f.
26.
A box has dimensions $5 \times 4 \times 3$ cm. Find the length of its space diagonal exactly.
27.
A box has dimensions $5 \times 4 \times 3$ . Find the angle the space diagonal makes with the $5 \times 4$ base, to 3 s.f.
28.
A hiker walks 5 km on bearing $050^\circ$ from $C$ to $V$ . State the bearing of $C$ from $V$ (return bearing).
29.
In triangle $ABC$ , $a = 8$ , $b = 11$ , $\angle A = 35^\circ$ . Find both possible values of $\angle B$ , to 3 s.f.
30.
A boat sails 6 km on bearing $080^\circ$ from $X$ to $Y$ , then 8 km on bearing $160^\circ$ from $Y$ to $Z$ . Find $XZ$ to 3 s.f.

PlatinumQuestions 31–40

31.
A hiker walks 5 km on bearing $050^\circ$ from base camp $C$ to viewpoint $V$ . From $V$ they walk 7 km on bearing $145^\circ$ to lake $L$ . Find (a) the interior angle at $V$ , (b) $CL$ to 3 s.f.
32.
A vertical flagpole stands at $F$ on horizontal ground. Surveyor $S_1$ is 80 m due south of $S_2$ . From $S_1$ , $F$ is on bearing $045^\circ$ ; from $S_2$ , $F$ is on bearing $115^\circ$ . The angle of elevation of the top from $S_2$ is $28^\circ$ . Find the height to 3 s.f.
33.
A boat sails 6 km on bearing $080^\circ$ , then 9 km on $150^\circ$ , then 4 km on $250^\circ$ . Find its straight-line distance from the start to 3 s.f.
34.
In triangle $ABC$ , $a = 8$ , $b = 11$ , $\angle A = 35^\circ$ . For each ambiguous-case solution, state $\angle C$ and $c$ , to 3 s.f.
35.
In triangle $ABC$ , $AB = 4$ , $AC = 6$ , $\angle BAC = 60^\circ$ . Find (a) $BC$ exactly, (b) area exactly.
36.
A cuboid has a square base of side 4 and height 6. A diagonal is drawn from one base vertex to the opposite top vertex. Find the angle this diagonal makes with the **base diagonal**, to 3 s.f.
37.
In triangle $ABC$ , $\angle A = 40^\circ$ , $\angle B = 75^\circ$ , $AB = 12$ cm. Find the area to 3 s.f.
38.
A ship sails 30 km on bearing $090^\circ$ from $P$ to $Q$ . From $Q$ it changes course to bearing $030^\circ$ and sails to $R$ . The total straight-line distance $PR$ is 50 km. Find the distance $QR$ , to 3 s.f.
39.
Two stations $A$ and $B$ are 200 m apart on level ground. The angles of elevation of a mountain top $T$ from $A$ and $B$ are $25^\circ$ and $35^\circ$ respectively, with $B$ closer. Find the height of the mountain to 3 s.f., assuming $A$ , $B$ , and the foot of $T$ are collinear.
40.
Two aircraft leave the same airport at the same time. Aircraft A flies at 400 km/h on bearing $060^\circ$ ; aircraft B flies at 350 km/h on bearing $130^\circ$ . Find the distance between them after 2 hours, to 3 s.f.

Pack A — Fluency

Pack A · Fluency