Problem of the Week

Problem 8

In the above triangle,

(1) $\begin{equation*} 0 < a \le b \le c \end{equation*}$

and

(2) $\begin{equation*} 0 \le \alpha \le \beta \le \gamma \le 180^\circ \text{.} \end{equation*}$

(i) Show that if $0 \le \theta \le 180^\circ$ , $\sin \theta = \sqrt{1-(\cos\theta)^2}$ .

(ii) Show algebraically using only the cosine rule, inequalities (1) and (2), part (i) and the identity $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$ that $\cos(\alpha+\beta)=-\cos\gamma$ and $\alpha,\beta\le90^\circ$ .

(iii) Deduce that $\alpha+\beta+\gamma=180^\circ$ .

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Problem of the Week

Problem 8

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