11.7 Non-right-angle Trigonometry

In a right-angled triangle, the hypotenuse is 12 and one angle is $ 40^\circ$. Find the side opposite this angle, to 3 s.f.

In a right-angled triangle, opposite is 7 and hypotenuse is 25. Find the angle, to 3 s.f.

In a right-angled triangle, the two shorter sides are 8 and 15. Find the hypotenuse (exact).

State the exact value of $\sin 30^\circ$.

A right-angled triangle has legs 6 cm and 8 cm. Find its area.

A ship sails on bearing $060^\circ$. State the angle it makes with North (clockwise).

Find the area of a triangle with sides $a = 10$, $b = 14$ and included angle $C = 50^\circ$, to 3 s.f.

In triangle $ABC$, $\angle A = 40^\circ$ and $\angle B = 75^\circ$. Find $\angle C$.

In triangle $ABC$, write the sine rule equation relating $a, b, \angle A, \angle B$.

State the cosine rule for $a$ in terms of $b, c, A$.

In triangle $ABC$, $\angle A = 35^\circ$, $\angle B = 80^\circ$, side $a = 6$. Find $b$ to 3 s.f.

In triangle $ABC$, $a = 10$, $b = 13$ and $\angle A = 35^\circ$. Find $\angle B$ (acute case), to 3 s.f.

Find side $a$ when $b = 6$, $c = 10$ and $A = 75^\circ$, to 3 s.f.

In a triangle with sides $a = 5$, $b = 6$, $c = 7$, find angle $A$, to 3 s.f.

Triangle $ABC$ has $AB = 12$ m, $AC = 9$ m, $\angle BAC = 75^\circ$. Find (a) area, (b) $BC$, to 3 s.f.

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.

A chord subtends an angle of $ 100^\circ$ at the centre of a circle of radius 10 cm. Find the chord length, to 3 s.f.

A ship sails 80 km on bearing $ 130^\circ$ from port. How far north and east is the ship? To 3 s.f.

In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Find the area exactly.

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.?

In triangle $ABC$, $AB = 12$, $AC = 9$, $\angle BAC = 75^\circ$ and $BC \approx 13.0$. Find $\angle ABC$ to 3 s.f.

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.

A triangle has sides 7, 9, 11. Find its largest angle to 3 s.f.

Triangle $XYZ$ has $XY = 14$, $\angle X = 50^\circ$, $\angle Z = 70^\circ$. Find $YZ$ to 3 s.f.

A triangle has sides 10 and 14 and area 50 m$^2$. Find the included angle, giving the acute case to 3 s.f.

A box has dimensions $5 \times 4 \times 3$ cm. Find the length of its space diagonal exactly.

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.

A hiker walks 5 km on bearing $050^\circ$ from $C$ to $V$. State the bearing of $C$ from $V$ (return bearing).

In triangle $ABC$, $a = 8$, $b = 11$, $\angle A = 35^\circ$. Find both possible values of $\angle B$, to 3 s.f.

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.

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.

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.

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.

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.

In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Find (a) $BC$ exactly, (b) area exactly.

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.

In triangle $ABC$, $\angle A = 40^\circ$, $\angle B = 75^\circ$, $AB = 12$ cm. Find the area to 3 s.f.

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.

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.

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.