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Non-right-angled trigonometry

Apply the sine rule, cosine rule and area formula to non-right-angled triangles; solve bearing and radial survey problems.

Measurement · MS-M6

Sub-topics

Sine rule

Apply the sine rule

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

to find unknown sides and angles in non-right-angled triangles; recognise and solve the ambiguous case where two triangles are possible.

Cosine rule

Apply the cosine rule

c^2 = a^2 + b^2 - 2ab\cos C

to find unknown sides in non-right-angled triangles; rearrange to find unknown angles using

\cos C = \dfrac{a^2+b^2-c^2}{2ab}

Area of a triangle

Calculate the area of any triangle using

A = \tfrac{1}{2}ab\sin C

; choose the appropriate formula for a given configuration, including using the cosine rule to find an unknown angle before applying the area formula.

Applications and radial surveys

Solve multi-step problems involving three-figure and compass bearings; find distances and areas using radial (plane-table) surveys; combine the sine rule, cosine rule and area formula in realistic contexts.

Open Math

Non-right-angled trigonometry

Measurement · MS-M6

Name:

Date:

Sine rule

Q1Straightforward

In triangle

ABC

, angle

A = 40°

, angle

B = 70°

, and side

a = 8

cm (the side opposite angle

A

). Use the sine rule to find side

b

(opposite angle

B

). Give your answer to 2 decimal places.

Q2Straightforward

In triangle

PQR

, angle

P = 55°

, angle

Q = 65°

, and side

p = 12

cm (opposite angle

P

). Find side

q

(opposite angle

Q

) using the sine rule. Give your answer to 2 decimal places.

Q3Straightforward

In triangle

ABC

, angle

A = 35°

, angle

C = 80°

, and side

a = 10

cm (opposite angle

A

). Find side

c

(opposite angle

C

). Give your answer to 2 decimal places.

Q4Straightforward

In triangle

XYZ

, angle

X = 50°

, angle

Z = 72°

, and

z = 15

m (opposite angle

Z

). Find side

x

(opposite angle

X

). Give your answer to 2 decimal places.

Q5Moderate

In triangle

ABC

,

a = 10

cm,

b = 14

cm, and angle

A = 32°

(opposite side

a

). Assuming angle

B

is acute, find angle

B

to 1 decimal place.

Q6Moderate

In triangle

PQR

,

p = 8

cm,

r = 11

cm, and angle

P = 40°

(opposite side

p

). Assuming angle

R

is acute, find angle

R

to 1 decimal place.

Q7Moderate

In triangle

ABC

, angle

A = 48°

, angle

B = 76°

, and

b = 25

cm (opposite angle

B

). Find side

c

(opposite angle

C

). Give your answer to 1 decimal place.

Q8Moderate

Two park rangers,

A

and

B

, are stationed

5

km apart. They both spot a fire

F

. Ranger

A

measures the angle

FAB = 52°

and ranger

B

measures angle

FBA = 71°

. Find the distance from ranger

A

to the fire (

AF

) to 2 decimal places.

Q9Moderate

In triangle

ABC

, angle

A = 65°

, angle

C = 42°

, and

b = 18

m (the side opposite angle

B

). Find side

a

(opposite angle

A

) to 2 decimal places.

Q10Challenging

In triangle

ABC

,

a = 12

cm,

b = 15

cm, and angle

A = 38°

(opposite side

a

). Two triangles are possible (the ambiguous case). Find the **larger** of the two possible values of angle

B

, to the nearest degree.

Q11Challenging

A triangular block of land has vertices

A

,

B

and

C

. Side

BC = 320

m, angle

ABC = 68°

, and angle

ACB = 54°

. Find the length of side

AC

to the nearest metre.

Q12Challenging

Points

A

and

B

are on opposite sides of a river. A point

C

is on the same bank as

A

, with

AC = 80

m. From

A

, the angle

CAB = 73°

; from

C

, the angle

ACB = 52°

. Find the width of the river

AB

to the nearest metre.

Cosine rule

Q13Straightforward

In triangle

ABC

,

a = 6

cm,

b = 8

cm and

C = 60°

. Use the cosine rule to find side

c

. Give your answer to 2 decimal places.

Q14Straightforward

In triangle

PQR

,

p = 5

m,

q = 7

m and

R = 120°

. Find side

r

using the cosine rule. Give your answer to 2 decimal places.

Q15Straightforward

In triangle

ABC

,

a = 10

cm,

b = 8

cm and

C = 45°

. Find side

c

using the cosine rule. Give your answer to 2 decimal places.

Q16Straightforward

In triangle

ABC

,

a = 7

cm,

b = 8

cm and

c = 10

cm. Find angle

C

to 1 decimal place.

Q17Moderate

In triangle

PQR

,

p = 5

cm,

q = 6

cm and

r = 9

cm. Find angle

R

to 1 decimal place.

Q18Moderate

In triangle

ABC

,

a = 9

cm,

b = 12

cm and

C = 72°

. Find side

c

to 2 decimal places.

Q19Moderate

Two ships leave port at the same time. Ship

A

sails 15 km on a bearing of 040° and ship

B

sails 20 km on a bearing of 100°. The angle between their paths is 60°. Find the distance between the two ships to 2 decimal places.

Q20Moderate

A triangular garden has sides 10 m, 12 m and 16 m. Find the largest angle of the garden to 1 decimal place.

Q21Moderate

In triangle

XYZ

,

x = 15

cm,

y = 12

cm and

z = 20

cm. Find angle

X

to 1 decimal place.

Q22Challenging

Two roads leave an intersection. Road A runs 12 km to town

X

and road B runs 18 km to town

Y

. The angle between the roads at the intersection is 85°. Find the straight-line distance from town

X

to town

Y

to 2 decimal places.

Q23Challenging

From town

P

, town

Q

is 25 km away on a bearing of 048° and town

R

is 38 km away on a bearing of 130°. Find the distance

QR

to 2 decimal places.

Q24Challenging

A triangular block of land has sides 32 m, 45 m and 60 m. Find the angle opposite the 60 m side to 1 decimal place.

Area of a triangle

Q25Straightforward

In triangle

ABC

,

a = 8

cm,

b = 6

cm and

C = 40°

. Find the area of the triangle to 2 decimal places.

Q26Straightforward

In triangle

PQR

,

PQ = 12

cm,

PR = 9

cm and

\angle QPR = 55°

. Find the area of the triangle to 2 decimal places.

Q27Straightforward

In triangle

ABC

,

a = 5

m,

b = 7

m and

C = 120°

. Find the area to 2 decimal places.

Q28Straightforward

The area of triangle

ABC

is 30 cm²,

a = 12

cm and

b = 10

cm. Find the included angle

C

to the nearest degree.

Q29Moderate

A triangular garden has two sides of 15 m and 20 m with an included angle of 68°. Find the area of the garden to 2 decimal places.

Q30Moderate

In triangle

ABC

,

a = 8

cm,

b = 11

cm and

C = 50°

. Find the area to 2 decimal places.

Q31Moderate

Two sides of a triangle are 9 cm and 14 cm. The area of the triangle is 45 cm². Find the **obtuse** angle between the two sides, to 1 decimal place.

Q32Moderate

A triangular plot of land has sides

AB = 80

m and

AC = 65

m, with

\angle BAC = 75°

. Find the area of the plot to the nearest square metre.

Q33Moderate

The area of triangle

ABC

is 84 cm², angle

A = 35°

, and side

b = 20

cm. Find side

c

to 2 decimal places.

Q34Challenging

A triangular plot has sides 45 m and 60 m with an included angle of 108°. Find the area to the nearest square metre.

Q35Challenging

A triangular block of land has sides 25 m, 35 m and 28 m. Find the area of the block to the nearest square metre.

Q36Challenging

In triangle

ABC

, sides

a = 10

cm,

b = 12

cm and

c = 14

cm. Find the area of the triangle to 2 decimal places.

Applications and radial surveys

Q37Straightforward

A ship sails 40 km on a bearing of 050°. How far north of its starting point is it? Give your answer to 2 decimal places.

Q38Straightforward

A car travels 30 km on a bearing of 130°. How far east of its starting point is it? Give your answer to 2 decimal places.

Q39Straightforward

In a radial survey from point

O

, point

A

is measured on a bearing of 020° and point

B

is measured on a bearing of 085°. What is the angle

\angle AOB

at point

O

?

Q40Straightforward

In a radial survey from centre point

O

, boundary point

A

is at bearing 035° and boundary point

B

is at bearing 125°. What is the angle

\angle AOB

?

Q41Moderate

From port

P

, ship

A

sails 20 km on a bearing of 040° and ship

B

sails 25 km on a bearing of 100°. Find the distance between the two ships to 2 decimal places.

Q42Moderate

In a radial survey from point

O

, two boundary markers are recorded:

OA = 35

m on a bearing of 010° and

OB = 28

m on a bearing of 080°. Find the straight-line distance

AB

to 2 decimal places.

Q43Moderate

In a radial survey from point

O

, two boundary points are measured:

OA = 42

m at bearing 010° and

OB = 35

m at bearing 080°. Find the area of triangle

OAB

to the nearest square metre.

Q44Moderate

A helicopter flies 30 km from base

P

on a bearing of 040° to point

A

. It then flies 40 km on a bearing of 100° to point

B

. Find the straight-line distance from base

P

to point

B

, to 2 decimal places.

Q45Challenging

A radial survey from point

O

records three boundary points:

OA = 42

m at bearing 010°,

OB = 38

m at bearing 070°, and

OC = 45

m at bearing 140°. Find the total area of the region

ABC

(the quadrilateral formed by

O

and the three boundary points) to the nearest square metre.

Q46Challenging

From point

A

, the bearing to

B

is 058° and

AB = 150

m. From

A

, the bearing to

C

is 114°. If

AC = 120

m, find the distance

BC

to 2 decimal places.

Q47Challenging

A boat sails from port

P

on a bearing of 040° for 15 km to point

A

. It then turns and sails on a bearing of 110° for 20 km to point

B

. Find the distance

PB

to 2 decimal places.

Q48Challenging

From lighthouse

L

, ship

A

is 12 km away on a bearing of 035° and ship

B

is 15 km away on a bearing of 110°. Find the distance between the two ships to 2 decimal places.

Worked solutions and answers at openmath.au/year-12/standard-2/non-right-angled-trigonometry