11.7 Non-right-angle Trigonometry

**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.

**Exact triangle.** In triangle $ABC$, $AB = 4$, $AC = 6$, $\angle BAC = 60^\circ$. Give exact answers. (a) Find $BC$. (b) Find the area.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.