Answer Key
11.7 Non-right-angle Trigonometry
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 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. | 5.74 |
| 2 | In a right-angled triangle, opposite is 5 and hypotenuse is 13. Find the angle, to 3 s.f. | $22.6^\circ$ |
| 3 | In a right-angled triangle, the two shorter sides are 5 and 12. Find the hypotenuse (exact). | 13 |
| 4 | State the exact value of $\sin 60^\circ$. | $\dfrac{\sqrt{3}}{2}$ |
| 5 | A right-angled triangle has legs 6 cm and 8 cm. Find its area. | 24 cm$^2$ |
| 6 | A ship sails on bearing $060^\circ$. State the angle it makes with North (clockwise). | $60^\circ$ |
| 7 | Find the area of a triangle with sides $a = 12$, $b = 9$ and included angle $C = 75^\circ$, to 3 s.f. | 52.2 |
| 8 | In triangle $ABC$, $\angle A = 40^\circ$ and $\angle B = 75^\circ$. Find $\angle C$. | $65^\circ$ |
| 9 | In triangle $ABC$, write the sine rule equation relating $a, b, \angle A, \angle B$. | $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$ |
| 10 | State the cosine rule for $a$ in terms of $b, c, A$. | $a^2 = b^2 + c^2 - 2bc \cos A$ |
| 11 | In triangle $ABC$, $\angle A = 40^\circ$, $\angle B = 70^\circ$, side $a = 8$. Find $b$ to 3 s.f. | 11.7 |
| 12 | In triangle $ABC$, $a = 10$, $b = 13$ and $\angle A = 35^\circ$. Find $\angle B$ (acute case), to 3 s.f. | $48.3^\circ$ |
| 13 | Find side $a$ when $b = 7$, $c = 9$ and $A = 50^\circ$, to 3 s.f. | 7.00 |
| 14 | In a triangle with sides $a = 5$, $b = 6$, $c = 7$, find angle $A$, to 3 s.f. | $44.4^\circ$ |
| 15 | Triangle $ABC$ has $AB = 12$ m, $AC = 9$ m, $\angle BAC = 75^\circ$. Find (a) area, (b) $BC$, to 3 s.f. | (a) 52.2 m$^2$; (b) $BC \approx 13.0$ m |
| 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. | $31.2$ m |
| 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. | 6.43 cm |
| 18 | A ship sails 50 km on bearing $ 60^\circ$ from port. How far north and east is the ship? To 3 s.f. | North 25.0 km; East 43.3 km |
| 19 | In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Find the area exactly. | $6\sqrt{3}$ |
| 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.? | $\approx 299$ m |
| 21 | In triangle $ABC$, $AB = 12$, $AC = 9$, $\angle BAC = 75^\circ$ and $BC \approx 13.0$. Find $\angle ABC$ to 3 s.f. | $42.0^\circ$ |
| 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. | $\angle PAB = 90^\circ$; $PB \approx 94.3$ km |
| 23 | A triangle has sides 7, 9, 11. Find its largest angle to 3 s.f. | $\approx 87.3^\circ$ |
| 24 | Triangle $XYZ$ has $XY = 14$, $\angle X = 50^\circ$, $\angle Z = 70^\circ$. Find $YZ$ to 3 s.f. | $\approx 11.4$ |
| 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. | $\approx 45.6^\circ$ |
| 26 | A box has dimensions $5 \times 4 \times 3$ cm. Find the length of its space diagonal exactly. | $\sqrt{50} = 5\sqrt{2}$ cm |
| 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. | $\approx 25.1^\circ$ |
| 28 | A hiker walks 5 km on bearing $050^\circ$ from $C$ to $V$. State the bearing of $C$ from $V$ (return bearing). | $230^\circ$ |
| 29 | In triangle $ABC$, $a = 8$, $b = 11$, $\angle A = 35^\circ$. Find both possible values of $\angle B$, to 3 s.f. | $B \approx 52.1^\circ$ or $127.9^\circ$ |
| 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. | $\approx 9.32$ km |
| 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. | (a) $85^\circ$; (b) $CL \approx 8.24$ km |
| 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. | $\approx 32.0$ m |
| 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. | $\approx 10.5$ km |
| 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. | Case 1: $B \approx 52.1^\circ$, $C \approx 92.9^\circ$, $c \approx 13.9$. Case 2: $B \approx 127.9^\circ$, $C \approx 17.1^\circ$, $c \approx 4.10$. |
| 35 | In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Find (a) $BC$ exactly, (b) area exactly. | (a) $2\sqrt{7}$; (b) $6\sqrt{3}$ |
| 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. | $\approx 46.7^\circ$ |
| 37 | In triangle $ABC$, $\angle A = 40^\circ$, $\angle B = 75^\circ$, $AB = 12$ cm. Find the area to 3 s.f. | $\approx 49.3$ cm$^2$ |
| 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. | $\approx 49.4$ km |
| 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. | $\approx 232$ m |
| 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. | $\approx 911$ km |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | 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. | 7.71 |
| 2 | In a right-angled triangle, opposite is 7 and hypotenuse is 25. Find the angle, to 3 s.f. | $16.3^\circ$ |
| 3 | In a right-angled triangle, the two shorter sides are 8 and 15. Find the hypotenuse (exact). | 17 |
| 4 | State the exact value of $\sin 30^\circ$. | $\dfrac{1}{2}$ |
| 5 | A right-angled triangle has legs 6 cm and 8 cm. Find its area. | 30 cm$^2$ |
| 6 | A ship sails on bearing $060^\circ$. State the angle it makes with North (clockwise). | $120^\circ$ |
| 7 | Find the area of a triangle with sides $a = 10$, $b = 14$ and included angle $C = 50^\circ$, to 3 s.f. | 53.6 |
| 8 | In triangle $ABC$, $\angle A = 40^\circ$ and $\angle B = 75^\circ$. Find $\angle C$. | $70^\circ$ |
| 9 | In triangle $ABC$, write the sine rule equation relating $a, b, \angle A, \angle B$. | $\dfrac{b}{\sin B} = \dfrac{c}{\sin C}$ |
| 10 | State the cosine rule for $a$ in terms of $b, c, A$. | $b^2 = a^2 + c^2 - 2ac \cos B$ |
| 11 | In triangle $ABC$, $\angle A = 35^\circ$, $\angle B = 80^\circ$, side $a = 6$. Find $b$ to 3 s.f. | 10.3 |
| 12 | In triangle $ABC$, $a = 10$, $b = 13$ and $\angle A = 35^\circ$. Find $\angle B$ (acute case), to 3 s.f. | $62.1^\circ$ |
| 13 | Find side $a$ when $b = 6$, $c = 10$ and $A = 75^\circ$, to 3 s.f. | 10.2 |
| 14 | In a triangle with sides $a = 5$, $b = 6$, $c = 7$, find angle $A$, to 3 s.f. | $49.5^\circ$ |
| 15 | Triangle $ABC$ has $AB = 12$ m, $AC = 9$ m, $\angle BAC = 75^\circ$. Find (a) area, (b) $BC$, to 3 s.f. | (a) 74.8; (b) $BC \approx 14.4$ |
| 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. | $37.3$ m |
| 17 | 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. | 15.3 cm |
| 18 | A ship sails 80 km on bearing $ 130^\circ$ from port. How far north and east is the ship? To 3 s.f. | North $-51.4$ km (i.e. 51.4 km south); East 61.3 km |
| 19 | In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Find the area exactly. | $\dfrac{15\sqrt{3}}{4}$ |
| 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.? | $\approx 275$ m |
| 21 | In triangle $ABC$, $AB = 12$, $AC = 9$, $\angle BAC = 75^\circ$ and $BC \approx 13.0$. Find $\angle ABC$ to 3 s.f. | $75.4^\circ$ |
| 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. | $\angle PAB = 85^\circ$; $PB \approx 77.5$ km |
| 23 | A triangle has sides 7, 9, 11. Find its largest angle to 3 s.f. | $\approx 97.9^\circ$ |
| 24 | Triangle $XYZ$ has $XY = 14$, $\angle X = 50^\circ$, $\angle Z = 70^\circ$. Find $YZ$ to 3 s.f. | $\approx 12.0$ |
| 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. | $\approx 47.7^\circ$ |
| 26 | A box has dimensions $5 \times 4 \times 3$ cm. Find the length of its space diagonal exactly. | $\sqrt{77}$ cm |
| 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. | $\approx 27.1^\circ$ |
| 28 | A hiker walks 5 km on bearing $050^\circ$ from $C$ to $V$. State the bearing of $C$ from $V$ (return bearing). | $300^\circ$ |
| 29 | In triangle $ABC$, $a = 8$, $b = 11$, $\angle A = 35^\circ$. Find both possible values of $\angle B$, to 3 s.f. | $B \approx 45.6^\circ$ or $134.4^\circ$ |
| 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. | $\approx 11.5$ km |
| 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. | $\approx 288^\circ$ |
| 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. | $\approx 30.1$ m |
| 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. | $\approx 7.02$ km |
| 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. | Case 1: $B \approx 45.6^\circ$, $C \approx 104.4^\circ$, $c \approx 13.6$. Case 2: $B \approx 134.4^\circ$, $C \approx 15.6^\circ$, $c \approx 3.77$. |
| 35 | In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Find (a) $BC$ exactly, (b) area exactly. | (a) $7$; (b) $10\sqrt{3}$ |
| 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. | $\approx 48.5^\circ$ |
| 37 | In triangle $ABC$, $\angle A = 40^\circ$, $\angle B = 75^\circ$, $AB = 12$ cm. Find the area to 3 s.f. | $\approx 35.3$ cm$^2$ |
| 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. | $\approx 49.0$ km |
| 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. | $\approx 290$ m |
| 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. | $\approx 645$ km |
Problems — Worked Solutions
**Triangle $ABC$ — sine, cosine, area.** $AB = 12$ m, $AC = 9$ m, $\angle BAC = 75^\circ$. (a) Find the area of the triangle, to 3 s.f. (b) Find the length $BC$, to 3 s.f. (c) Find the angle $\angle ABC$, to 3 s.f.
(a) 52.2 m$^2$. (b) 13.0 m. (c) $42.0^\circ$.
**Exact triangle.** In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Give exact answers. (a) Find $BC$. (b) Find the area.
(a) $BC = 2\sqrt{7}$. (b) Area $= 6\sqrt{3}$.
**Surveyor flagpole.** 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 $28^\circ$. (a) Find the interior angles of the ground triangle. (b) Use the sine rule to find $S_2 F$. (c) Hence find the height of the flagpole, to 3 s.f.
(a) At $S_1$: $45^\circ$; at $S_2$: $65^\circ$; at $F$: $70^\circ$. (b) $S_2 F \approx 60.2$ m. (c) Height $\approx 32.0$ m.
**Hike route.** A hiker starts at base camp $C$ and walks 5 km on bearing $050^\circ$ to viewpoint $V$. From $V$ they walk 7 km on bearing $145^\circ$ to lake $L$. (a) Sketch the route and find the interior angle at $V$ in triangle $CVL$. (b) Find $CL$ to 3 s.f. (c) Find the bearing from $L$ back to $C$, to 3 s.f.
(a) Interior angle at $V$ is $85^\circ$. (b) $CL \approx 8.24$ km. (c) $\approx 288^\circ$.
**Three-leg bearings.** A boat sails 6 km on bearing $080^\circ$, then 9 km on bearing $150^\circ$, then 4 km on bearing $250^\circ$. (a) Compute the east and north components for each leg. (b) Sum the components. (c) Find the straight-line distance from the start, to 3 s.f.
(a) Leg 1: E 5.91, N 1.04; Leg 2: E 4.50, N $-7.79$; Leg 3: E $-3.76$, N $-1.37$. (b) Total E $\approx 6.65$; total N $\approx -8.12$. (c) $\approx 10.5$ km.
**Ambiguous case [EXT].** In triangle $ABC$, $a = 8$ cm, $b = 11$ cm, $\angle A = 35^\circ$. (a) Find both possible values of $\angle B$, to 3 s.f. (b) For each, state $\angle C$ and $c$, to 3 s.f. (c) Sketch both triangles.
(a) $B \approx 52.1^\circ$ or $127.9^\circ$. (b) Case 1: $C \approx 92.9^\circ$, $c \approx 13.9$. Case 2: $C \approx 17.1^\circ$, $c \approx 4.10$.
**3D trigonometry.** A box has dimensions $5 \times 4 \times 3$ cm. (a) Find the length of the space diagonal exactly. (b) Find the angle the space diagonal makes with the $5 \times 4$ base, to 3 s.f. (c) Find the angle the space diagonal makes with the long edge (length 5), to 3 s.f.
(a) $5\sqrt{2}$ cm. (b) $\approx 25.1^\circ$. (c) $\approx 45.0^\circ$.
**Mountain height.** From a point $A$ on level ground, the angle of elevation of a mountain top $T$ is $25^\circ$. From a point $B$, 200 m closer to the foot of the mountain, the angle of elevation is $35^\circ$. Assume $A$, $B$, and the foot are collinear. (a) Let the foot-to-$B$ distance be $d$ and height $h$. Write two equations. (b) Solve for $d$ and $h$, to 3 s.f.
(a) $\tan 35^\circ = h/d$ and $\tan 25^\circ = h/(d + 200)$. (b) $d \approx 399$ m; $h \approx 280$ m.
**Aircraft.** 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 $130^\circ$. (a) Find the distance each has flown after 2 hours. (b) Find the angle between their headings. (c) Find the distance between them after 2 hours, to 3 s.f.
(a) $A: 800$ km; $B: 700$ km. (b) $70^\circ$. (c) $\approx 864$ km.
**Largest angle.** A triangle has sides 7, 9, and 11. (a) Identify the largest angle and find it, to 3 s.f. (b) Find the area of the triangle, to 3 s.f. (c) Determine whether the triangle is acute, right, or obtuse.
(a) Largest angle is opposite side 11; $\approx 85.9^\circ$. (b) $\approx 30.6$ unit$^2$. (c) Acute (all angles $< 90^\circ$).
**Area + missing angle.** A triangular plot has sides 12 m and 18 m and area 80 m$^2$. (a) Find the included angle in degrees (acute case), to 3 s.f. (b) Find the third side, to 3 s.f.
(a) $\approx 47.7^\circ$. (b) $\approx 13.4$ m.
**Surveying — distance across a river.** Two observation points $A$ and $B$ are 150 m apart on the same side of a river. A tree $T$ on the opposite bank is observed: $\angle TAB = 75^\circ$ and $\angle TBA = 60^\circ$. (a) State $\angle ATB$. (b) Use the sine rule to find $AT$ and $BT$, to 3 s.f. (c) Find the perpendicular distance from the line $AB$ to the tree $T$, to 3 s.f.
(a) $45^\circ$. (b) $AT \approx 184$ m; $BT \approx 205$ m. (c) $\approx 178$ m.