Non-right-angled triangles

Apply the sine rule and cosine rule to find unknown sides and angles in non-right-angled triangles, calculate triangle area using

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

, and identify and solve the ambiguous case.

Worked examples

Finding an unknown side using the sine rule

Straightforward

Problem

In triangle

ABC

\angle A = 50°

\angle B = 70°

and

a = 8

cm. Find the length of side

b

to 2 decimal places.

Applying the cosine rule and finding triangle area

Moderate

Problem

In triangle

ABC

a = 7

b = 10

and

\angle C = 60°

. Find side

c

and the area of the triangle.

Solving the ambiguous case

Challenging

Problem

In triangle

ABC

a = 10

b = 12

and

\angle A = 40°

. Determine how many triangles satisfy these conditions and find both possible values of

\angle B

Practise

Q1·Straightforward

In triangle

ABC

\angle A = 50°

\angle B = 70°

and

a = 8

cm. Find the length of side

b

to 2 decimal places.

Q2·Straightforward

In triangle

PQR

\angle P = 35°

\angle Q = 85°

and

p = 12

. Find side

q

to 2 decimal places.

Q3·Straightforward

In triangle

ABC

\angle A = 42°

\angle C = 67°

and

c = 15

. Find side

a

to 2 decimal places.

Q4·Straightforward

In triangle

XYZ

\angle Y = 110°

\angle X = 25°

and

y = 20

. Find side

x

to 2 decimal places.

Q5·Moderate

In triangle

ABC

a = 7

b = 10

and

\angle C = 60°

. Find side

c

to 2 decimal places.

Q6·Moderate

In triangle

PQR

p = 5

q = 8

and

\angle R = 45°

. Find side

r

to 2 decimal places.

Q7·Moderate

Find the area of triangle

ABC

where

a = 6

cm,

b = 9

cm and

\angle C = 55°

. Give your answer to 2 decimal places.

Q8·Moderate

Find the area of triangle

DEF

where

d = 11

e = 7

and

\angle F = 72°

. Give your answer to 2 decimal places.

Q9·Challenging

In triangle

ABC

a = 10

b = 12

and

\angle A = 40°

. How many triangles satisfy these conditions? Enter the number.

Q10·Challenging

In triangle

ABC

a = 8

b = 6

and

\angle A = 30°

. How many triangles satisfy these conditions? Enter the number.

Q11·Challenging

In triangle

ABC

a = 5

b = 10

and

\angle A = 25°

. Find all possible values of

\angle B

, to the nearest degree.

58°

only

58°

122°

25°

155°

No triangle is possible

Q12·Challenging

In triangle

PQR

p = 9

q = 14

and

\angle P = 35°

. How many triangles are possible, and what are the valid values of

\angle Q

to the nearest degree?

One triangle;

\angle Q = 63°

One triangle;

\angle Q = 117°

Two triangles;

\angle Q = 63°

\angle Q = 117°

No triangle is possible

Open Math

Non-right-angled triangles

Functions · MAV-11-04

Name:

Date:

Q1Straightforward

In triangle

ABC

\angle A = 50°

\angle B = 70°

and

a = 8

cm. Find the length of side

b

to 2 decimal places.

Q2Straightforward

In triangle

PQR

\angle P = 35°

\angle Q = 85°

and

p = 12

. Find side

q

to 2 decimal places.

Q3Straightforward

In triangle

ABC

\angle A = 42°

\angle C = 67°

and

c = 15

. Find side

a

to 2 decimal places.

Q4Straightforward

In triangle

XYZ

\angle Y = 110°

\angle X = 25°

and

y = 20

. Find side

x

to 2 decimal places.

Q5Moderate

In triangle

ABC

a = 7

b = 10

and

\angle C = 60°

. Find side

c

to 2 decimal places.

Q6Moderate

In triangle

PQR

p = 5

q = 8

and

\angle R = 45°

. Find side

r

to 2 decimal places.

Q7Moderate

Find the area of triangle

ABC

where

a = 6

cm,

b = 9

cm and

\angle C = 55°

. Give your answer to 2 decimal places.

Q8Moderate

Find the area of triangle

DEF

where

d = 11

e = 7

and

\angle F = 72°

. Give your answer to 2 decimal places.

Q9Challenging

In triangle

ABC

a = 10

b = 12

and

\angle A = 40°

. How many triangles satisfy these conditions? Enter the number.

Q10Challenging

In triangle

ABC

a = 8

b = 6

and

\angle A = 30°

. How many triangles satisfy these conditions? Enter the number.

Q11Challenging

In triangle

ABC

a = 5

b = 10

and

\angle A = 25°

. Find all possible values of

\angle B

, to the nearest degree.

A. $58°$ only
B. $58°$ or $122°$
C. $25°$ or $155°$
D.No triangle is possible

Q12Challenging

In triangle

PQR

p = 9

q = 14

and

\angle P = 35°

. How many triangles are possible, and what are the valid values of

\angle Q

to the nearest degree?

A.One triangle; $\angle Q = 63°$
B.One triangle; $\angle Q = 117°$
C.Two triangles; $\angle Q = 63°$ or $\angle Q = 117°$
D.No triangle is possible

Worked solutions and answers at openmath.au/year-11/advanced/trigonometry-and-measure-of-angles/non-right-angled-triangles