In \triangle ABC, side b = 10\text{ cm}, side c = 6\text{ cm}, and \angle A = 120^\circ. W… — JAMB UTME

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?

Answer: 14 cm

The length of side

a

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}

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.

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

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.

Calculate the principal square root to find

a

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

Side length must be a positive scalar quantity.

Cosine Rule for Side Length in Oblique Triangles

Question