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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.7 Non-right-angle Trigonometry

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Triangle ABCABC — sine, cosine, area. AB=12AB = 12 m, AC=9AC = 9 m, ∠BAC=75∘\angle BAC = 75^\circ.

(a) Find the area of the triangle, to 3 s.f.
(b) Find the length BCBC, to 3 s.f.
(c) Find the angle ∠ABC\angle ABC, to 3 s.f.

Working space

2Problem 2 of 12
Exact triangle. In triangle ABCABC, AB=4AB = 4, AC=6AC = 6, ∠BAC=60∘\angle BAC = 60^\circ. Give exact answers.

(a) Find BCBC.
(b) Find the area.

Working space

3Problem 3 of 12
Surveyor flagpole. A vertical flagpole stands at FF on horizontal ground. Surveyor S1S_1 is 80 m due south of S2S_2. From S1S_1, FF is on bearing 045∘045^\circ. From S2S_2, FF is on bearing 115∘115^\circ. The angle of elevation of the top from S2S_2 is 28∘28^\circ.

(a) Find the interior angles of the ground triangle.
(b) Use the sine rule to find S2FS_2 F.
(c) Hence find the height of the flagpole, to 3 s.f.

Working space

4Problem 4 of 12
Hike route. A hiker starts at base camp CC and walks 5 km on bearing 050∘050^\circ to viewpoint VV. From VV they walk 7 km on bearing 145∘145^\circ to lake LL.

(a) Sketch the route and find the interior angle at VV in triangle CVLCVL.
(b) Find CLCL to 3 s.f.
(c) Find the bearing from LL back to CC, to 3 s.f.

Working space

5Problem 5 of 12
Three-leg bearings. A boat sails 6 km on bearing 080∘080^\circ, then 9 km on bearing 150∘150^\circ, then 4 km on bearing 250∘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.

Working space

6Problem 6 of 12
Ambiguous case [EXT]. In triangle ABCABC, a=8a = 8 cm, b=11b = 11 cm, ∠A=35∘\angle A = 35^\circ.

(a) Find both possible values of ∠B\angle B, to 3 s.f.
(b) For each, state ∠C\angle C and cc, to 3 s.f.
(c) Sketch both triangles.

Working space

7Problem 7 of 12
3D trigonometry. A box has dimensions 5×4×35 \times 4 \times 3 cm.

(a) Find the length of the space diagonal exactly.
(b) Find the angle the space diagonal makes with the 5×45 \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.

Working space

8Problem 8 of 12
Mountain height. From a point AA on level ground, the angle of elevation of a mountain top TT is 25∘25^\circ. From a point BB, 200 m closer to the foot of the mountain, the angle of elevation is 35∘35^\circ. Assume AA, BB, and the foot are collinear.

(a) Let the foot-to-BB distance be dd and height hh. Write two equations.
(b) Solve for dd and hh, to 3 s.f.

Working space

9Problem 9 of 12
Aircraft. Two aircraft leave the same airport at the same time. Aircraft AA flies at 400 km/h on bearing 060∘060^\circ; aircraft BB flies at 350 km/h on bearing 130∘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.

Working space

10Problem 10 of 12
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.

Working space

11Problem 11 of 12
Area + missing angle. A triangular plot has sides 12 m and 18 m and area 80 m2^2.

(a) Find the included angle in degrees (acute case), to 3 s.f.
(b) Find the third side, to 3 s.f.

Working space

12Problem 12 of 12
Surveying — distance across a river. Two observation points AA and BB are 150 m apart on the same side of a river. A tree TT on the opposite bank is observed: ∠TAB=75∘\angle TAB = 75^\circ and ∠TBA=60∘\angle TBA = 60^\circ.

(a) State ∠ATB\angle ATB.
(b) Use the sine rule to find ATAT and BTBT, to 3 s.f.
(c) Find the perpendicular distance from the line ABAB to the tree TT, to 3 s.f.

Working space