Sine and Cosine Rules Practice Questions — JAMB UTME

Question 1Question →

In $\Delta ABC$ , side $a = 6\text{ cm}$ , side $b = 10\text{ cm}$ , and the included angle $\angle C = 120^\circ$ . What is the length of side $c$ ?

14\text{ cm}

2\sqrt{19}\text{ cm}

2\sqrt{34}\text{ cm}

16\text{ cm}

Show answer & explanation

Answer: $14\text{ cm}$

Answer

The length of side

c

is

14\text{ cm}

.

According to the Cosine Rule

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

, substituting the given values

a = 6

,

b = 10

, and

\cos 120^\circ = -\frac{1}{2}

yields

c^2 = 36 + 100 - 2(6)(10)\left(-\frac{1}{2}\right) = 136 + 60 = 196

. Taking the positive square root gives

c = 14\text{ cm}

.

Step-by-Step Solution

1

Identify the given values and state the relevant Cosine Rule formula

Given:

a = 6\text{ cm}

,

b = 10\text{ cm}

,

\angle C = 120^\circ

. Formula:

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

.

Since two sides and the included angle (SAS configuration) are known, the Cosine Rule must be used to find the third side.

2

Evaluate

\cos 120^\circ

and substitute all values into the formula

\cos 120^\circ = -\frac{1}{2}

. Thus,

c^2 = 6^2 + 10^2 - 2(6)(10)\left(-\frac{1}{2}\right)

.

Cosine of an obtuse angle in the second quadrant is negative.

3

Simplify the algebraic expression

c^2 = 36 + 100 + 60 = 196

.

Multiplying

-2(60)\left(-\frac{1}{2}\right)

yields

+60

.

4

Take the principal square root to solve for

c

c = \sqrt{196} = 14\text{ cm}

.

Length must be a positive real number.

Key Concept

Cosine Rule for finding an unknown side in SAS triangle configurations

Question 2Question →

In $\Delta ABC$ , side $a = 5\text{ cm}$ , side $b = 5\sqrt{2}\text{ cm}$ , and $\angle A = 30^\circ$ . What are all possible values for $\angle B$ ?

45^\circ

or

135^\circ

45^\circ

only

60^\circ

or

120^\circ

135^\circ

only

Show answer & explanation

Answer: $45^\circ$ or $135^\circ$

Answer

45^\circ

or

135^\circ

Applying the Sine Rule gives

\frac{5}{\sin 30^\circ} = \frac{5\sqrt{2}}{\sin B}

, so

\sin B = \frac{\sqrt{2}}{2}

. The angles whose sine is

\frac{\sqrt{2}}{2}

between

0^\circ

and

180^\circ

are

45^\circ

and

135^\circ

. Checking angle sums:

30^\circ + 45^\circ = 75^\circ < 180^\circ

and

30^\circ + 135^\circ = 165^\circ < 180^\circ

, so both

45^\circ

and

135^\circ

yield valid triangles.

Step-by-Step Solution

1

Set up the Sine Rule formula relating sides

a

,

b

and angles

A

,

B

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule connects two sides and their opposite angles in any non-right triangle.

2

Substitute given values

a = 5

,

b = 5\sqrt{2}

, and

A = 30^\circ

\frac{5}{\sin 30^\circ} = \frac{5\sqrt{2}}{\sin B} \implies \frac{5}{0.5} = \frac{5\sqrt{2}}{\sin B} \implies 10 = \frac{5\sqrt{2}}{\sin B}

Since

\sin 30^\circ = \frac{1}{2}

, simplifying yields the ratio.

3

Solve for

\sin B

\sin B = \frac{5\sqrt{2}}{10} = \frac{\sqrt{2}}{2}

Isolating

\sin B

gives the principal trigonometric ratio value.

4

Determine all valid values for angle

B

in the range

(0^\circ, 180^\circ)

Acute

B_1 = \arcsin\left(\frac{\sqrt{2}}{2}\right) = 45^\circ

; Obtuse

B_2 = 180^\circ - 45^\circ = 135^\circ

. Both are valid since

30^\circ + 135^\circ = 165^\circ < 180^\circ

.

Since

b > a

, the ambiguous case (SSA) produces two valid distinct triangle solutions.

Key Concept

Sine Rule and the Ambiguous Case (SSA)

Estimated Time:1m 0s

Question 3Question →

In $\Delta ABC$ , side lengths are given as $a = 6\text{ cm}$ and $b = 6\sqrt{3}\text{ cm}$ , and $\angle A = 30^\circ$ . If $\angle B$ is an obtuse angle, what is the length of side $c$ ?

6\text{ cm}

12\text{ cm}

6\sqrt{3}\text{ cm}

3\sqrt{3}\text{ cm}

Show answer & explanation

Answer: $6\text{ cm}$

Answer

The length of side

c

is

6\text{ cm}

.

Using the Sine Rule,

\frac{6}{\sin 30^\circ} = \frac{6\sqrt{3}}{\sin B}

, which yields

\sin B = \frac{\sqrt{3}}{2}

. The two possible values for

\angle B

are

60^\circ

(acute) and

120^\circ

(obtuse). The problem explicitly specifies that

\angle B

is obtuse, so

\angle B = 120^\circ

. Subtracting from

180^\circ

gives

\angle C = 30^\circ

. Since

\angle A = \angle C = 30^\circ

, the triangle is isosceles, making side

c

equal to side

a

, which is

6\text{ cm}

.

Step-by-Step Solution

1

Apply the Sine Rule to find

\sin B

\frac{a}{\sin A} = \frac{b}{\sin B} \implies \frac{6}{\sin 30^\circ} = \frac{6\sqrt{3}}{\sin B} \implies 12 = \frac{6\sqrt{3}}{\sin B} \implies \sin B = \frac{\sqrt{3}}{2}

The Sine Rule relates side lengths to the sines of their opposite angles.

2

Determine the value of

\angle B

using the given condition

\angle B = 60^\circ

or

\angle B = 180^\circ - 60^\circ = 120^\circ

. Since

\angle B

is obtuse,

\angle B = 120^\circ

.

The inverse sine function yields two possible angle solutions between

0^\circ

and

180^\circ

(the ambiguous case).

3

Calculate the third angle

\angle C

\angle C = 180^\circ - (\angle A + \angle B) = 180^\circ - (30^\circ + 120^\circ) = 30^\circ

The sum of interior angles in any triangle is

180^\circ

.

4

Find the length of side

c

Since

\angle A = 30^\circ

and

\angle C = 30^\circ

,

\Delta ABC

is isosceles with side

c = a = 6\text{ cm}

.

Sides opposite to equal angles in a triangle are equal in length.

Key Concept

Ambiguous Case of the Sine Rule (SSA Condition)

Alternative Method

Alternatively, apply the Cosine Rule for angle

A

:

a^2 = b^2 + c^2 - 2bc \cos A \implies 6^2 = (6\sqrt{3})^2 + c^2 - 2(6\sqrt{3})c \cos 30^\circ

. Simplifying gives

36 = 108 + c^2 - 18c \implies c^2 - 18c + 72 = 0

. Factoring yields

(c - 6)(c - 12) = 0

, giving

c = 6\text{ cm}

or

c = 12\text{ cm}

. For

c = 12\text{ cm}

,

b^2 + a^2 = 108 + 36 = 144 = c^2

, making

\angle C = 90^\circ

and

\angle B = 60^\circ

(acute). Thus,

c = 6\text{ cm}

corresponds to the obtuse angle

\angle B = 120^\circ

.

Estimated Time:2m 0s

Question 4Question →

In $\Delta PQR$ , side $p = 3\text{ cm}$ , side $q = 8\text{ cm}$ , and the included angle $\angle R = 60^\circ$ . What is the length of side $r$ in cm?

Show answer & explanation

Answer: 7

Answer

The length of side

r

is

7\text{ cm}

.

Using the Cosine Rule

r^2 = p^2 + q^2 - 2pq \cos R

with

p=3

,

q=8

, and

\angle R=60^\circ

yields

r^2 = 9 + 64 - 48(0.5) = 49

, which gives

r = 7\text{ cm}

.

Step-by-Step Solution

1

State the Cosine Rule formula for side

r

r^2 = p^2 + q^2 - 2pq \cos R

The Cosine Rule allows calculating the third side of a triangle when two sides and the included angle are given.

2

Substitute the given values into the formula

r^2 = 3^2 + 8^2 - 2(3)(8) \cos 60^\circ

We are given

p = 3

,

q = 8

, and

\angle R = 60^\circ

.

3

Evaluate the trigonometric expression and simplify

r^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49

Since

\cos 60^\circ = 0.5

, multiplying

2 \times 3 \times 8 \times 0.5

yields

24

.

4

Solve for

r

by taking the positive square root

r = 7\text{ cm}

Length must be a positive value, and

\sqrt{49} = 7

.

Key Concept

Cosine Rule for finding an unknown side given two sides and the included angle (SAS).

Question 5Question →

Two ships, $P$ and $Q$ , leave a port $O$ at the same time. Ship $P$ sails on a bearing of $040^\circ$ at a constant speed of $25\text{ km/h}$ , while Ship $Q$ sails on a bearing of $100^\circ$ at a constant speed of $40\text{ km/h}$ . What is the distance in kilometers between the two ships after $2$ hours?

Show answer & explanation

Answer: 70

Answer

The distance between the two ships after 2 hours is 70 km.

The distance traveled by Ship P in 2 hours is

50\text{ km}

and by Ship Q is

80\text{ km}

. The angle between their paths is

100^\circ - 040^\circ = 60^\circ

. Applying the Cosine Rule yields

PQ^2 = 50^2 + 80^2 - 2(50)(80)\cos(60^\circ) = 2500 + 6400 - 4000 = 4900

, giving a distance of

\sqrt{4900} = 70\text{ km}

.

Step-by-Step Solution

1

Calculate the distances traveled by Ship P and Ship Q after 2 hours

OP = 50\text{ km}

and

OQ = 80\text{ km}

Distance equals speed multiplied by time.

2

Find the angle between the lines of travel from port O

\angle POQ = 100^\circ - 40^\circ = 60^\circ

The angle between two bearings from a common origin is the difference between their bearing angles.

3

Use the Cosine Rule to calculate the side length PQ

PQ^2 = 50^2 + 80^2 - 2(50)(80)\cos(60^\circ) = 4900

The Cosine Rule

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

calculates the unknown opposite side given two side lengths and their included angle.

4

Take the square root to find PQ

PQ = 70\text{ km}

Taking the principal square root gives the final linear distance.

Key Concept

Applying the Cosine Rule to solve bearing and distance non-right triangle problems

Estimated Time:2m 0s

Question 6Question →

In triangle $LMN$ , the side lengths are given as $l = 7\text{ cm}$ , $m = 8\text{ cm}$ , and $n = 13\text{ cm}$ . What is the measure of angle $N$ in degrees?

Show answer & explanation

Answer: 120

Answer

The measure of angle

N

is

120^\circ

.

Using the Cosine Rule for angle

N

,

\cos N = \frac{l^2 + m^2 - n^2}{2lm} = \frac{7^2 + 8^2 - 13^2}{2(7)(8)} = \frac{-56}{112} = -0.5

. The angle whose cosine is

-0.5

within the interior angles of a triangle (

0^\circ < N < 180^\circ

) is

120^\circ

.

Step-by-Step Solution

1

Set up the Cosine Rule formula for angle

N

\cos N = \frac{l^2 + m^2 - n^2}{2lm}

The Cosine Rule relates the three sides of any triangle to the cosine of one of its interior angles.

2

Substitute the known side lengths into the formula

\cos N = \frac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8} = \frac{49 + 64 - 169}{112}

Side

n = 13\text{ cm}

is opposite to angle

N

and must be subtracted in the numerator.

3

Simplify the fraction

\cos N = \frac{-56}{112} = -0.5

Evaluating the numerical expression gives a negative value, indicating that angle

N

is obtuse.

4

Calculate the principal inverse cosine angle for the triangle

N = 120^\circ

Since

\cos 60^\circ = 0.5

,

\cos(180^\circ - 60^\circ) = -0.5

, giving

N = 120^\circ

.

Key Concept

Applying the Cosine Rule to find an obtuse angle given three side lengths (SSS)

Question 7Question →

In $\Delta PQR$ , side $p = 4\text{ cm}$ , side $q = 4\sqrt{2}\text{ cm}$ , and $\angle P = 30^\circ$ . If $\angle Q$ is an obtuse angle, what is the measure of $\angle Q$ ?

135^\circ

45^\circ

60^\circ

150^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

The measure of angle

Q

is

135^\circ

.

Applying the Sine Rule gives

\sin Q = \frac{4\sqrt{2} \cdot \sin 30^\circ}{4} = \frac{\sqrt{2}}{2}

. The acute reference angle is

45^\circ

, so the obtuse angle is

180^\circ - 45^\circ = 135^\circ

.

Step-by-Step Solution

1

Apply the Sine Rule to relate sides

p, q

and angles

P, Q

\frac{p}{\sin P} = \frac{q}{\sin Q} \implies \frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin Q}

The Sine Rule connects pairs of opposite sides and angles in any triangle.

2

Solve for

\sin Q

\sin Q = \frac{4\sqrt{2} \cdot \sin 30^\circ}{4} = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}

Substitute

\sin 30^\circ = \frac{1}{2}

and simplify the numerical expression.

3

Determine the obtuse solution for angle

Q

Q = 180^\circ - 45^\circ = 135^\circ

Since

\sin Q = \frac{\sqrt{2}}{2}

, the principal acute angle is

45^\circ

. Because the problem specifies that angle

Q

is obtuse (

90^\circ < Q < 180^\circ

), we take its supplement.

Key Concept

Ambiguous case of the Sine Rule (SSA condition)

Estimated Time:1m 30s

Question 8Question →

In $\Delta ABC$ , the side lengths are given as $a = 3\text{ cm}$ , $b = 5\text{ cm}$ , and $c = 7\text{ cm}$ . What is the measure of angle $C$ in degrees?

Show answer & explanation

Answer: 120

Answer

The measure of angle

C

is

120^\circ

.

Using the Cosine Rule

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

, substituting

a=3

,

b=5

, and

c=7

yields

\cos C = \frac{9 + 25 - 49}{30} = -0.5

. Taking the inverse cosine of

-0.5

gives an angle of

120^\circ

.

Step-by-Step Solution

1

Select the appropriate formula for calculating an interior angle given three sides (SSS).

Use the Cosine Rule rearranged for

\cos C

:

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

.

When all three side lengths of a non-right triangle are known, the Cosine Rule is required to find any of its angles.

2

Substitute side lengths

a=3

,

b=5

, and

c=7

into the Cosine Rule equation.

\cos C = \frac{3^2 + 5^2 - 7^2}{2(3)(5)} = \frac{9 + 25 - 49}{30} = -\frac{15}{30} = -0.5.

Simplifying the numerator and denominator determines the exact trigonometric ratio for angle

C

.

3

Find the inverse cosine of

-0.5

in degrees.

C = \arccos(-0.5) = 120^\circ.

A negative cosine value indicates an obtuse angle in the second quadrant (

90^\circ < C < 180^\circ

).

Key Concept

Using the Cosine Rule to determine obtuse angles in SSS triangles

Question 9Question →

In triangle $XYZ$ , side length $x = 3\text{ cm}$ , side length $y = 8\text{ cm}$ , and the included angle $\angle Z = 60^\circ$ . What is the length of side $z$ in cm?

Show answer & explanation

Answer: 7

Answer

The length of side

z

is

7\text{ cm}

.

Using the Cosine Rule

z^2 = x^2 + y^2 - 2xy \cos Z

, substituting

x = 3

,

y = 8

, and

\angle Z = 60^\circ

yields

z^2 = 3^2 + 8^2 - 2(3)(8)(0.5) = 9 + 64 - 24 = 49

. Taking the positive square root gives

z = 7\text{ cm}

.

Step-by-Step Solution

1

Identify known triangle components and select the appropriate rule

Two sides and the included angle (SAS) are given:

x = 3

,

y = 8

,

\angle Z = 60^\circ

, requiring the Cosine Rule.

When given two sides and the included angle (SAS), the Cosine Rule is used to find the third side.

2

Substitute values into the Cosine Rule formula

z^2 = x^2 + y^2 - 2xy \cos Z

z^2 = 3^2 + 8^2 - 2(3)(8)\cos 60^\circ

Direct algebraic substitution of side lengths and angle measure into the Cosine Rule.

3

Evaluate the trigonometric term and simplify the arithmetic expression

z^2 = 9 + 64 - 48(0.5) = 73 - 24 = 49

The exact value of

\cos 60^\circ

is

0.5

.

4

Solve for side length

z

by taking the principal square root

z = \sqrt{49} = 7\text{ cm}

Side length must be positive.

Key Concept

Applying the Cosine Rule to find the third side of a non-right-angled triangle given two sides and an included angle (SAS).

Estimated Time:1m 30s

Question 10Question →

In $\Delta ABC$ , the length of side $a$ is $10\text{ cm}$ , angle $\angle A = 30^\circ$ , and angle $\angle B = 45^\circ$ . What is the length of side $b$ ?

10\sqrt{2}\text{ cm}

5\sqrt{2}\text{ cm}

10\sqrt{3}\text{ cm}

20\text{ cm}

Show answer & explanation

Answer: $10\sqrt{2}\text{ cm}$

Answer

The length of side

b

is

10\sqrt{2}\text{ cm}

.

The option specifying

10\sqrt{2}\text{ cm}

is correct because applying the Sine Rule gives

\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}

, which simplifies directly to

b = 10\sqrt{2}\text{ cm}

.

Step-by-Step Solution

1

State the Sine Rule formula relating sides

a

,

b

and their opposite angles

A

,

B

.

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.

2

Substitute the known values (

a = 10

,

A = 30^\circ

,

B = 45^\circ

) into the formula.

\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}

Isolating the variable

b

requires substituting given numeric angle and side measures.

3

Evaluate exact values of trigonometric functions and solve for

b

.

b = \frac{10 \times \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2}\text{ cm}

Simplifying the algebraic fraction yields the exact length of side

b

.

Key Concept

Application of the Sine Rule to find an unknown side length in a non-right-angled triangle.

Estimated Time:45s

Question 11Question →

Two boats depart simultaneously from a port $P$ . Boat $A$ travels along a straight path for $8\text{ km}$ , while Boat $B$ travels along another straight path for $15\text{ km}$ . If the angle between their paths at the port is $60^\circ$ , what is the distance between the two boats in kilometers?

Show answer & explanation

Answer: 13

Answer

The distance between the two boats is 13 km.

The distance between the two boats forms the third side of a triangle where two side lengths (

8\text{ km}

and

15\text{ km}

) and the included angle (

60^\circ

) are known. By the Cosine Rule,

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ) = 64 + 225 - 120 = 169

, so

d = \sqrt{169} = 13\text{ km}

.

Step-by-Step Solution

1

Formulate the geometric model

A triangle with two sides of length

8\text{ km}

and

15\text{ km}

, and an included angle of

60^\circ

.

The paths of the two boats and the distance between them form a triangle with two given side lengths and the included angle.

2

Set up the Cosine Rule formula

d^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ)

The Cosine Rule (

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

) is used when two sides and the included angle are known (SAS configuration).

3

Evaluate the trigonometric term and simplify

d^2 = 64 + 225 - 240(0.5) = 289 - 120 = 169

Since

\cos(60^\circ) = 0.5

, the subtraction term reduces to

120

.

4

Calculate the principal square root

d = 13

Taking the positive square root gives the distance in kilometers.

Key Concept

Applying the Cosine Rule to calculate the unknown side of a triangle given two sides and the included angle (SAS).

Question 12Question →

In $\Delta LMN$ , the length of side $l = 10\text{ cm}$ , side $m = 10\sqrt{3}\text{ cm}$ , and angle $\angle L = 30^\circ$ . Given that angle $\angle M$ is an obtuse angle, what is the measure of angle $\angle M$ ?

120^\circ

60^\circ

150^\circ

45^\circ

Show answer & explanation

Answer: $120^\circ$

Answer

The measure of angle

\angle M

is

120^\circ

.

Using the Sine Rule, we find

\sin M = \frac{10\sqrt{3} \cdot \sin 30^\circ}{10} = \frac{\sqrt{3}}{2}

. The inverse sine gives a reference angle of

60^\circ

. Since the problem explicitly states that angle

\angle M

is obtuse, we select

180^\circ - 60^\circ = 120^\circ

.

Step-by-Step Solution

1

Apply the Sine Rule relating sides

l, m

and their opposite angles

L, M

.

\frac{l}{\sin L} = \frac{m}{\sin M}

The Sine Rule allows us to find an unknown angle given two sides and one non-included opposite angle.

2

Substitute the given values into the formula and solve for

\sin M

.

\sin M = \frac{10\sqrt{3} \cdot \sin 30^\circ}{10} = \sqrt{3} \cdot 0.5 = \frac{\sqrt{3}}{2}

Since

\sin 30^\circ = \frac{1}{2}

, simplifying the fraction gives

\frac{\sqrt{3}}{2}

.

3

Determine the obtuse angle solution for

\sin M = \frac{\sqrt{3}}{2}

.

\angle M = 180^\circ - 60^\circ = 120^\circ

The principal value is

60^\circ

, but because

\angle M

is specified to be obtuse (

90^\circ < \angle M < 180^\circ

), we take the supplementary angle in the second quadrant.

Key Concept

Sine Rule and the Ambiguous Case (SSA Condition)

Estimated Time:1m 30s

Question 13Question →

In $\Delta ABC$ , $\sin A = \frac{3}{5}$ , $\sin B = \frac{4}{5}$ , and the side opposite angle $A$ has length $a = 15\text{ cm}$ . What is the length of side $b$ , in centimeters?

Show answer & explanation

Answer: 20

Answer

The length of side

b

is 20 cm.

Using the Sine Rule

\frac{a}{\sin A} = \frac{b}{\sin B}

, substitute

a = 15

,

\sin A = \frac{3}{5}

, and

\sin B = \frac{4}{5}

. Evaluating

\frac{15}{3/5}

gives

25

. Multiplying

25

by

\frac{4}{5}

yields

20\text{ cm}

.

Step-by-Step Solution

1

Set up the Sine Rule relationship between sides

a

,

b

and their opposite angles

A

,

B

.

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule relates the side lengths of a triangle to the sines of its angles.

2

Substitute

a = 15

,

\sin A = \frac{3}{5}

, and

\sin B = \frac{4}{5}

into the Sine Rule equation.

\frac{15}{3/5} = \frac{b}{4/5}

Direct substitution of known values allows us to solve for the unknown side

b

.

3

Simplify the left side of the equation.

15 \times \frac{5}{3} = 25

Dividing 15 by

\frac{3}{5}

is equivalent to multiplying 15 by

\frac{5}{3}

.

4

Multiply both sides by

\frac{4}{5}

to find

b

.

b = 25 \times \frac{4}{5} = 20\text{ cm}

Isolating

b

gives the final length of side

b

.

Key Concept

Sine Rule

Question 14Question →

In $\Delta ABC$ , the lengths of the sides are $a = x\text{ cm}$ , $b = (x + 3)\text{ cm}$ , and $c = (x + 2)\text{ cm}$ . If $\angle C = 60^\circ$ , what is the value of $x$ ?

3

5

7

1

Show answer & explanation

Answer: 5

Answer

5

Applying the Cosine Rule

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

with

a = x

,

b = x + 3

,

c = x + 2

, and

\cos 60^\circ = \frac{1}{2}

gives

(x+2)^2 = x^2 + (x+3)^2 - x(x+3)

. Expanding both sides leads to

x^2 + 4x + 4 = x^2 + 3x + 9

. Subtracting

x^2

and isolating

x

yields

x = 5

.

Step-by-Step Solution

1

Apply the Cosine Rule for side

c

and angle

C

.

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

The Cosine Rule relates all three sides of a triangle to the cosine of an included angle.

2

Substitute

a = x

,

b = x + 3

,

c = x + 2

, and

\cos 60^\circ = \frac{1}{2}

into the formula.

(x + 2)^2 = x^2 + (x + 3)^2 - 2(x)(x + 3)\left(\frac{1}{2}\right)

Inserting the given algebraic side lengths and angle allows solving for

x

.

3

Expand both sides of the equation.

x^2 + 4x + 4 = x^2 + (x^2 + 6x + 9) - (x^2 + 3x)

Cancel the factor of

2

with

\frac{1}{2}

and expand

(x+2)^2

and

(x+3)^2

.

4

Simplify the right-hand side.

x^2 + 4x + 4 = x^2 + 3x + 9

Combine like terms:

(x^2 + x^2 - x^2) + (6x - 3x) + 9 = x^2 + 3x + 9

.

5

Subtract

x^2

from both sides and isolate

x

.

4x - 3x = 9 - 4 \implies x = 5

Subtracting

x^2 + 3x + 4

from both sides gives the linear solution

x = 5

.

Key Concept

Solving for unknown algebraic side lengths using the Cosine Rule.

Question 15Question →

Two straight paths diverge from a junction $J$ at an angle of $120^\circ$ . A person walks $5\text{ km}$ along the first path to point $A$ , and another person walks $16\text{ km}$ along the second path to point $B$ . What is the direct distance between $A$ and $B$ in kilometres?

Show answer & explanation

Answer: 19

Answer

The direct distance between points

A

and

B

is

19\text{ km}

.

Using the Cosine Rule

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

with sides

5\text{ km}

and

16\text{ km}

and included angle

120^\circ

yields

c^2 = 5^2 + 16^2 - 2(5)(16)(-0.5) = 25 + 256 + 80 = 361

. Taking the square root gives

19\text{ km}

.

Step-by-Step Solution

1

Identify given values and setup the Cosine Rule formula

Side

a = 5

, side

b = 16

, and included angle

\theta = 120^\circ

The scenario provides two sides and the included angle (SAS configuration), which requires the Cosine Rule to find the third side.

2

Substitute the values into

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

c^2 = 5^2 + 16^2 - 2(5)(16) \cos(120^\circ)

Populating the formula allows evaluation of the unknown distance squared.

3

Evaluate the trigonometric term and simplify

c^2 = 25 + 256 - 160(-0.5) = 281 + 80 = 361

The cosine of an obtuse angle in the second quadrant (

120^\circ

) is negative:

\cos(120^\circ) = -0.5

.

4

Take the square root to find the distance

c

c = \sqrt{361} = 19\text{ km}

Taking the positive square root gives the physical distance between the two points.

Key Concept

Applying the Cosine Rule to find the length of an unknown side in a non-right triangle given two sides and the included angle (SAS).

Question 16Question →

A surveyor at station $A$ observes two landmarks, $B$ and $C$ . Landmark $B$ is located at a distance of $14\text{ km}$ from $A$ on a bearing of $025^\circ$ . Landmark $C$ is located at a distance of $6\sqrt{2}\text{ km}$ from $A$ on a bearing of $070^\circ$ . What is the direct distance between landmark $B$ and landmark $C$ ?

10\text{ km}

7\sqrt{2}\text{ km}

8\text{ km}

2\sqrt{46}\text{ km}

Show answer & explanation

Answer: $10\text{ km}$

Answer

The direct distance between landmark

B

and landmark

C

is

10\text{ km}

.

The included angle

\angle BAC

between the two bearings is

070^\circ - 025^\circ = 45^\circ

. Applying the Cosine Rule

a^2 = b^2 + c^2 - 2bc \cos A

gives

a^2 = (6\sqrt{2})^2 + 14^2 - 2(6\sqrt{2})(14)\cos 45^\circ = 72 + 196 - 168 = 100

. Taking the square root gives

10\text{ km}

.

Step-by-Step Solution

1

Determine the interior angle

\angle BAC

from the given bearings.

\angle BAC = 070^\circ - 025^\circ = 45^\circ

The difference between two bearings measured clockwise from North from the same point gives the included angle between the lines of sight.

2

Identify the side lengths adjacent to angle

A

in

\Delta ABC

.

c = AB = 14\text{ km}

,

b = AC = 6\sqrt{2}\text{ km}

, and included angle

A = 45^\circ

We have a Side-Angle-Side (SAS) triangle configuration, requiring the Cosine Rule to find the opposite side

a = BC

.

3

Apply the Cosine Rule

a^2 = b^2 + c^2 - 2bc \cos A

.

a^2 = (6\sqrt{2})^2 + 14^2 - 2(6\sqrt{2})(14)\cos 45^\circ

Substituting known values into the Cosine Rule formula.

4

Simplify the terms and solve for

a

.

a^2 = 72 + 196 - 168\sqrt{2}\left(\frac{1}{\sqrt{2}}\right) = 268 - 168 = 100 \implies a = \sqrt{100} = 10\text{ km}

Squaring

6\sqrt{2}

gives

36 \times 2 = 72

,

14^2 = 196

, and simplifying the cosine term yields

168

.

Key Concept

Cosine Rule for SAS non-right triangles in bearing contexts

Estimated Time:2m 0s

Question 17Question →

In triangle $PQR$ , the side length $p = 12\text{ cm}$ , side length $q = 18\text{ cm}$ , and $\sin P = 0.4$ . What is the exact value of $\sin Q$ ?

Show answer & explanation

Answer: 0.6

Answer

The value of

\sin Q

is 0.6.

Applying the Sine Rule

\frac{p}{\sin P} = \frac{q}{\sin Q}

with

p = 12\text{ cm}

,

q = 18\text{ cm}

, and

\sin P = 0.4

gives

\frac{12}{0.4} = \frac{18}{\sin Q}

. Evaluating

\frac{12}{0.4} = 30

leads to

30 = \frac{18}{\sin Q}

, which yields

\sin Q = \frac{18}{30} = 0.6

.

Step-by-Step Solution

1

State the Sine Rule equation for the given triangle sides and angles.

\frac{p}{\sin P} = \frac{q}{\sin Q}

The Sine Rule relates the lengths of the sides of a triangle to the sines of its opposite angles.

2

Substitute the known numerical values into the Sine Rule equation.

\frac{12}{0.4} = \frac{18}{\sin Q}

Substituting

p = 12

,

q = 18

, and

\sin P = 0.4

sets up an equation with a single unknown.

3

Simplify the left side of the equation and solve for

\sin Q

.

\sin Q = \frac{18}{30} = 0.6

Dividing 12 by 0.4 yields 30, so rearranging gives

\sin Q = \frac{18}{30} = 0.6

.

Key Concept

Using the Sine Rule to calculate an unknown sine ratio

Question 18Question →

In $\Delta PQR$ , side $p = 6\text{ cm}$ , side $q = 6\sqrt{2}\text{ cm}$ , and $\angle P = 30^\circ$ . If $\angle Q$ is an acute angle, what is the measure of $\angle Q$ ?

45^\circ

135^\circ

60^\circ

30^\circ

Show answer & explanation

Answer: $45^\circ$

Answer

45^\circ

Using the Sine Rule

\frac{\sin Q}{q} = \frac{\sin P}{p}

, we substitute the given values to find

\sin Q = \frac{6\sqrt{2} \cdot \sin 30^\circ}{6} = \frac{\sqrt{2}}{2}

. Since

\angle Q

is specified as an acute angle,

\angle Q = 45^\circ

.

Step-by-Step Solution

1

Set up the Sine Rule formula relating sides

p, q

and their opposite angles

P, Q

.

\frac{p}{\sin P} = \frac{q}{\sin Q}

The Sine Rule allows finding an unknown angle when two side lengths and one opposite angle are known.

2

Substitute the given values

p = 6

,

q = 6\sqrt{2}

, and

P = 30^\circ

into the formula.

\frac{6}{\sin 30^\circ} = \frac{6\sqrt{2}}{\sin Q}

Populating known quantities permits solving for

\sin Q

.

3

Solve for

\sin Q

.

\sin Q = \frac{6\sqrt{2} \cdot \sin 30^\circ}{6} = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}

Simplifying the algebraic fraction yields the exact sine ratio for angle

Q

.

4

Determine the acute angle

Q

corresponding to

\sin Q = \frac{\sqrt{2}}{2}

.

\angle Q = 45^\circ

The principal acute angle with sine equal to

\frac{\sqrt{2}}{2}

is

45^\circ

.

Key Concept

Applying the Sine Rule to calculate an unknown acute angle in a non-right-angled triangle

Question 19Question →

In $\Delta ABC$ , side $a = 4\text{ cm}$ , side $b = 4\sqrt{2}\text{ cm}$ , and $\angle A = 30^\circ$ . If $\angle B$ is an obtuse angle, what is the measure of $\angle B$ ?

45^\circ

135^\circ

105^\circ

150^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

135^\circ

Applying the Sine Rule gives

\frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin B}

, which simplifies to

\sin B = \frac{\sqrt{2}}{2}

. The inverse sine operation yields an acute angle of

45^\circ

and an obtuse angle of

180^\circ - 45^\circ = 135^\circ

. Since the stem specifies that angle B is obtuse, the correct value is

135^\circ

.

Step-by-Step Solution

1

Apply the Sine Rule formula relating sides

a

,

b

and their opposite angles

A

,

B

.

\frac{a}{\sin A} = \frac{b}{\sin B}

The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.

2

Substitute the known values into the equation:

a = 4

,

b = 4\sqrt{2}

, and

A = 30^\circ

.

\frac{4}{\sin 30^\circ} = \frac{4\sqrt{2}}{\sin B} \implies \frac{4}{0.5} = \frac{4\sqrt{2}}{\sin B} \implies 8 = \frac{4\sqrt{2}}{\sin B}

\sin 30^\circ = 0.5

.

3

Solve for

\sin B

.

\sin B = \frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2}

Rearranging the equation yields the value for

\sin B

.

4

Find the obtuse angle whose sine is

\frac{\sqrt{2}}{2}

.

\angle B = 180^\circ - 45^\circ = 135^\circ

Sine is positive in both the first and second quadrants. The acute reference angle is

\arcsin\left(\frac{\sqrt{2}}{2}\right) = 45^\circ

, so the supplementary obtuse angle is

180^\circ - 45^\circ = 135^\circ

.

Key Concept

Sine Rule and the Ambiguous Case

Question 20Question →

In $\Delta ABC$ , side $a = 10\text{ cm}$ , side $b = 16\text{ cm}$ , and $\sin A = \frac{3}{8}$ . If $\angle B$ is an obtuse angle, what is the exact value of $\cos B$ ?

-\frac{4}{5}

\frac{4}{5}

\frac{3}{5}

-\frac{3}{5}

Show answer & explanation

Answer: $-\frac{4}{5}$

Answer

-\frac{4}{5}

Applying the Sine Rule gives

\sin B = \frac{b \sin A}{a} = \frac{16 \times \frac{3}{8}}{10} = \frac{3}{5}

. Since

\angle B

is an obtuse angle, it lies in the second quadrant where cosine is negative. Using

\cos B = -\sqrt{1 - \sin^2 B}

, we get

\cos B = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\frac{4}{5}

.

Step-by-Step Solution

1

Apply the Sine Rule to calculate

\sin B

.

\sin B = \frac{b \sin A}{a} = \frac{16 \times \frac{3}{8}}{10} = \frac{6}{10} = \frac{3}{5}

The Sine Rule states that

\frac{a}{\sin A} = \frac{b}{\sin B}

.

2

Determine the sign of

\cos B

based on the given angle type.

Since

\angle B

is an obtuse angle (

90^\circ < B < 180^\circ

),

\cos B < 0

.

Cosine is negative in the second quadrant.

3

Calculate

\cos B

using the Pythagorean trigonometric identity.

\cos B = -\sqrt{1 - \sin^2 B} = -\sqrt{1 - \left(\frac{3}{5}\right)^2} = -\sqrt{\frac{16}{25}} = -\frac{4}{5}

Substitute

\sin B = \frac{3}{5}

into

\cos B = -\sqrt{1 - \sin^2 B}

.

Key Concept

Sine Rule and Trigonometric Ratios of Obtuse Angles

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