Geometry and Trigonometry Practice Questions — JAMB UTME — Page 10

Question 181Question →

In triangle $PQR$ , the side lengths are $p = 8\text{ cm}$ and $q = 15\text{ cm}$ , and the included angle $\angle R = 60^\circ$ . What is the length of side $r$ in centimeters?

Show answer & explanation

Answer: 13

Answer

The length of side

r

is

13\text{ cm}

.

Using the Cosine Rule formula

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

with

p=8

,

q=15

, and

R=60^\circ

, we calculate

r^2 = 64 + 225 - 240(0.5) = 169

. Taking the square root gives

r = 13\text{ cm}

.

Step-by-Step Solution

1

Identify the given values and appropriate trigonometric rule

Givens:

p = 8\text{ cm}

,

q = 15\text{ cm}

,

\angle R = 60^\circ

. Since two sides and the included angle (SAS) are given, use the Cosine Rule:

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

.

The Cosine Rule is required to find the third side when two sides and their included angle are known.

2

Substitute the values into the Cosine Rule formula

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

Replacing variables with their numerical equivalents sets up the algebraic calculation.

3

Calculate the terms and evaluate

r^2

r^2 = 64 + 225 - 240 \times 0.5 = 289 - 120 = 169

Since

\cos 60^\circ = 0.5

, simplify the arithmetic operations.

4

Solve for side length

r

r = \sqrt{169} = 13\text{ cm}

Take the square root of both sides to obtain the length of side

r

.

Key Concept

Applying the Cosine Rule to find an unknown side given two sides and the included angle (SAS)

Question 182Question →

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

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Answer: 14

Answer

The length of side

a

is

14\text{ cm}

.

Applying the Cosine Rule

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

with

b = 10\text{ cm}

,

c = 6\text{ cm}

, and

\angle A = 120^\circ

gives

a^2 = 10^2 + 6^2 - 2(10)(6)(-0.5) = 100 + 36 + 60 = 196

. Taking the square root yields

a = 14\text{ cm}

.

Step-by-Step Solution

1

Identify the given parameters and select the relevant trigonometric rule

Given sides

b = 10\text{ cm}

,

c = 6\text{ cm}

, and included angle

\angle A = 120^\circ

. Use the Cosine Rule:

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

.

The Cosine Rule is required because two sides and the included angle (SAS) are given to find the third side.

2

Substitute the known values into the Cosine Rule equation

a^2 = 10^2 + 6^2 - 2(10)(6) \cos(120^\circ)

This sets up a single equation with the unknown side length

a

.

3

Evaluate the cosine term and simplify the arithmetic expression

a^2 = 100 + 36 - 120(-0.5) = 136 + 60 = 196

Since

120^\circ

is an obtuse angle in the second quadrant,

\cos(120^\circ) = -0.5

, changing the minus sign in the formula to a plus sign.

4

Calculate the principal square root to find

a

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

Side length must be a positive scalar quantity.

Key Concept

Cosine Rule for Side Length in Oblique Triangles

Question 183Question →

A trigonometric function is defined by $y = A \cos(B x) + C$ , where $A > 0$ and $B > 0$ . If the function has a maximum value of $7$ , a minimum value of $-3$ , and a period of $120^\circ$ , what is the value of $A + B + C$ ?

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Answer: 10

Answer

The value of

A + B + C

is 10.

Solving the system

A + C = 7

and

-A + C = -3

gives

C = 2

and

A = 5

. Using the period relationship

120^\circ = 360^\circ / B

yields

B = 3

. Adding these parameters together gives

5 + 3 + 2 = 10

.

Step-by-Step Solution

1

Calculate vertical shift C and amplitude A

C = 2, A = 5

For y = A cos(Bx) + C, Max = A + C and Min = -A + C. Thus C = (Max + Min)/2 = 2 and A = (Max - Min)/2 = 5.

2

Calculate period coefficient B

B = 3

The period T in degrees is given by 360° / B. With T = 120°, B = 360° / 120° = 3.

3

Compute the sum A + B + C

10

Sum the derived parameters: 5 + 3 + 2 = 10.

Key Concept

Deriving amplitude, period coefficient, and vertical shift from trigonometric graph properties

Question 184Question →

What is the smallest positive angle $\theta$ , in degrees, that satisfies the trigonometric equation $\sqrt{3}\tan(3\theta) = 3$ ?

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Answer: 20

Answer

The smallest positive value of

\theta

is

20^\circ

.

Isolating

\tan(3\theta)

gives

\tan(3\theta) = \frac{3}{\sqrt{3}} = \sqrt{3}

. The principal acute angle whose tangent is

\sqrt{3}

is

60^\circ

. Equating

3\theta = 60^\circ

and solving for

\theta

yields

\theta = 20^\circ

.

Step-by-Step Solution

1

Isolate the trigonometric function

\tan(3\theta)

\tan(3\theta) = \sqrt{3}

Dividing both sides by \sqrt{3} isolates the tangent term and rationalizes the fraction to \sqrt{3}.

2

Determine the principal angle for

3\theta

3\theta = 60^\circ

The smallest positive angle whose tangent equals \sqrt{3} is 60^\circ.

3

Solve for

\theta

\theta = 20^\circ

Dividing the principal angle 60^\circ by the coefficient 3 gives the smallest positive angle \theta.

Key Concept

Solving trigonometric equations with multiple angle arguments

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