Mathematics

Problem

Two sides of a triangle

ABC

have lengths

AB = 6\,\text{cm}

and

AC = 8\,\text{cm}

. The included angle

\angle A = C

satisfies

\cos 2C = \dfrac{1}{2}

. a. (i) Express

\sin C

solely in terms of

\cos 2C

and hence determine the value of

\sin C

. (Take the principal value of

C

for a triangle.) b. (ii) Calculate the measure of angle

C

in degrees. c. (iii) Using the information above, find the area of triangle

ABC

correct to two decimal places.

Intuitive explanation

This problem combines the Law of Cosines with a double-angle equation. The key insight is that you need to first solve for the actual angle $C$ using the double-angle formula $\cos 2C = \frac{1}{2}$ , then use that angle as the included angle $\angle A$ in the Law of Cosines to find the third side $BC$ . Once you have all three sides, you can find whatever the problem asks (likely the area or another side length).

Intuitive explanation

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