In \Delta ABC, the side lengths are given as a = 3\text{ cm}, b = 5\text{ cm}, and c = 7\t… — JAMB UTME

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?

Answer: 120 degrees

Answer

The measure of angle

C

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

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.

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

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

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