Problem solving — Surveying & Navigation

1Problem 1 of 36

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.

Working space

2Problem 2 of 36

Exact triangle. In triangle

ABC

,

AB = 4

,

AC = 6

,

\angle BAC = 60^\circ

. Give exact answers.

(a) Find

BC

.
(b) Find the area.

Working space

3Problem 3 of 36

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.

Working space

4Problem 4 of 36

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.

Working space

5Problem 5 of 36

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.

Working space

6Problem 6 of 36

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.

Working space

7Problem 7 of 36

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.

Working space

8Problem 8 of 36

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.

Working space

9Problem 9 of 36

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.

Working space

10Problem 10 of 36

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 36

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.

Working space

12Problem 12 of 36

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.

Working space

13Problem 13 of 36

Exact values [EXT]. State the exact value of:

(a)

\sin 60^\circ

(b)

\cos 30^\circ

(c)

\tan 45^\circ

(d)

\sin\dfrac{\pi}{6}

Working space

14Problem 14 of 36

Quadrant II [EXT]. Find the exact value of

\cos\!\left(\dfrac{5\pi}{6}\right)

. Show the reference angle, the quadrant, and the sign.

Working space

15Problem 15 of 36

Fan-blade sector [EXT]. A circular fan blade of radius 9 cm sweeps out a sector with central angle

\dfrac{4\pi}{9}

rad.

(a) Find the length of the arc swept.
(b) Find the area of the swept sector.
(c) Find the perimeter of the swept sector.

Working space

16Problem 16 of 36

Trig equations exactly [EXT]. Solve each in the given range.

(a)

2\cos x = -1

for

0 \leq x \leq 2\pi

(b)

\sin x = \dfrac{\sqrt{2}}{2}

for

0^\circ \leq x \leq 360^\circ

(c)

\tan x = -\sqrt{3}

for

0 \leq x \leq 2\pi

Working space

17Problem 17 of 36

Pythagorean identity [EXT]. Given

\sin\theta = \dfrac{3}{5}

and

\theta

is in Quadrant II, find:

(a)

\cos\theta

(b)

\tan\theta

(c)

\sin\theta \cos\theta

Working space

18Problem 18 of 36

Segment area [EXT]. A chord of a circle of radius 10 cm subtends an angle of

\dfrac{2\pi}{3}

rad at the centre.

(a) Find the area of the larger of the two regions (i.e. the major sector).
(b) Find the area of the triangle formed by the chord and the two radii.
(c) Hence find the area of the minor segment, to 3 s.f.

Working space

19Problem 19 of 36

Identity proof [EXT]. Prove the identity

\dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}

, stating any necessary restrictions on

\theta

.

Working space

20Problem 20 of 36

Quadratic in $\sin$ [EXT]. Solve

2\sin^2 x + \sin x - 1 = 0

for

0 \leq x \leq 2\pi

.

Working space

21Problem 21 of 36

Sum formula [EXT]. Use

\frac{7\pi}{12} = \frac{\pi}{4} + \frac{\pi}{3}

and the sum formula to find:

(a)

\sin\!\left(\dfrac{7\pi}{12}\right)

(b)

\cos\!\left(\dfrac{7\pi}{12}\right)

Working space

22Problem 22 of 36

Sector optimisation [EXT]. A sector has perimeter 24 cm.

(a) Let the radius be

r

cm. Express the central angle in radians in terms of

r

.
(b) Express the area in terms of

r

.
(c) Find the radius that maximises the area, and state the maximum area.

Working space

23Problem 23 of 36

Annular sector [EXT]. An annular ring has inner radius 4 cm, outer radius 7 cm.

(a) Find the area of the full annulus.
(b) An annular sector of angle

\frac{\pi}{3}

rad is cut from the ring. Find its area, exact and to 3 s.f.
(c) Find the perimeter of the annular sector (including both arcs and the two radial edges), to 3 s.f.

Working space

24Problem 24 of 36

Mini-investigation — equation that almost has a closed form [EXT]. Consider the equation

\sin x = \dfrac{x}{2}

for

x \geq 0

.

(a) Verify by inspection that

x = 0

is a solution.
(b) Sketch

y = \sin x

and

y = x/2

on the same axes for

-\pi \leq x \leq \pi

. How many other solutions can you identify?
(c) Use a calculator / GDC to find the positive non-zero solution to 3 s.f.
(d) Comment on why this equation has no closed-form algebraic solution.

Working space

25Problem 25 of 36

Tide model. The depth

d

(m) of water in a harbour follows

d(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 5

, where

t

is hours after midnight.

(a) State the amplitude, period, and mean depth.
(b) Find the maximum and minimum depths.
(c) Find the first time after midnight at which the depth is exactly 7 m, to 3 s.f.

Working space

26Problem 26 of 36

Ferris wheel. A Ferris wheel of radius 15 m has its centre 18 m above ground. It rotates anti-clockwise with period 4 minutes. A capsule starts at the lowest point at

t = 0

.

(a) Explain why

h(t) = 18 - 15\cos\!\left(\dfrac{\pi t}{2}\right)

models its height.
(b) State the max and min heights.
(c) Find the first time the capsule is 25 m above ground, to 3 s.f.
(d) State, without further calculation, the total time per revolution that the capsule is above 25 m.

Working space

27Problem 27 of 36

Periodic features. For

y = A\sin(Bx) + C

with

A = 3

,

B = \dfrac{\pi}{6}

,

C = 4

:

(a) State the amplitude.
(b) State the period.
(c) State the maximum and minimum values of

y

.
(d) Sketch

y

for

0 \leq x \leq 12

, marking the

y

-intercept and the first maximum.

Working space

28Problem 28 of 36

Daylight in a city. Daylight varies from 9 h on 21 December (

t = 0

, where

t

is in months) to 15 h on 21 June (

t = 6

).

(a) State the amplitude and the mean.
(b) Write a model

L(t) = -A\cos\!\left(\dfrac{\pi t}{6}\right) + D

.
(c) Predict the daylight on 21 March (

t = 3

).
(d) Find

t

where the model first gives

L = 13

h, to 3 s.f.

Working space

29Problem 29 of 36

Pendulum. A pendulum swings horizontally and its position from rest follows

x(t) = 10\cos(\pi t)

, where

x

is cm and

t

is seconds.

(a) State the amplitude and period.
(b) Find

x(0)

,

x(0.5)

,

x(1)

.
(c) Find the times in

[0, 2]

s at which

x = 0

.
(d) Find the maximum speed (in cm/s) by symbolic differentiation,

v(t) = x'(t)

.

Working space

30Problem 30 of 36

Convert sin ↔ cos. Rewrite each function using a single sine OR cosine and a phase shift.

(a)

y = \sin x

as a cosine
(b)

y = \cos x

as a sine
(c)

y = -\cos x

as a cosine with a phase shift

Working space

31Problem 31 of 36

Fitting from a table. Data observed:

|

t

(s) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
|

y

| 5 | 9 | 5 | 1 | 5 | 9 |

(a) State the period.
(b) State the amplitude and mean.
(c) Write a model

y(t) = A\sin(Bt) + C

given that

y(0)

is at the mean and rising.

Working space

32Problem 32 of 36

Daylight + threshold. Using

L(t) = -3\cos(\pi t / 6) + 12

(h,

t

in months from 21 Dec):

(a) For how many months per year is

L > 13

h?
(b) State the start and end times of this interval, to 3 s.f.
(c) Sketch

L

for

0 \leq t \leq 12

marking

L = 13

on the

y

-axis.

Working space

33Problem 33 of 36

Periodic vs non-periodic. Decide whether each function is periodic. Justify, and (where periodic) state the period.

(a)

y = \sin x + \cos x

(b)

y = \sin x + x

(c)

y = \sin(2x) + \sin(3x)

[EXT]
(d)

y = \sin x \cdot \cos x

Working space

34Problem 34 of 36

Modelling — sound wave. A sound wave is modelled by

P(t) = 0.5\sin(2\pi \cdot 440 \cdot t)

, where

P

is pressure in Pa and

t

in seconds.

(a) State the amplitude and frequency.
(b) Find the period.
(c) Find

P(0)

,

P(1/1760)

.
(d) Comment on what the frequency 440 Hz represents musically.

Working space

35Problem 35 of 36

Inverse modelling. A tidal model gives

d(t) = 4\sin\!\left(\dfrac{\pi t}{6}\right) + 7

where

t

is hours since the previous high tide (so

d

peaks at

t = 3

).

(a) State the amplitude, period, and average depth.
(b) Find the depth at

t = 0

.
(c) Find the first

t > 0

at which the depth is again 7 m.
(d) Find the duration of one "low-tide window" defined as

d < 5

.

Working space

36Problem 36 of 36

Investigation — fit & predict. Population of foxes in a region is observed monthly:

|

t

| 1 | 4 | 7 | 10 |
|---|---|---|---|---|
|

P

| 410 | 240 | 410 | 580 |

(a) Argue that a sinusoidal model is reasonable.
(b) Estimate amplitude, mean, and period from the data.
(c) Fit a model

P(t) = A\sin(B(t - C)) + D

.
(d) Use the model to predict the population at

t = 12

.

Working space

Problem-solving Pack

Problem-solving Pack