题目

Problem

In the triangle $ABC$ ,

(a) Use the sine rule to find the value of $\sin x^\circ$ , giving your answer to 4 decimal places.

(2)

Given also that $AB$ is the longest side of the triangle,

(b) find the value of $x$ , giving your answer to 2 decimal places.

(3)

解答

展开

边 $AB$ 对着角 $C=x^\circ$ ，边 $BC$ 对着角 $A=25^\circ$ 。直接用正弦定理。

展开

Using the sine rule,

\begin{align*} \frac{\sin x^\circ}{21} =&\,\frac{\sin25^\circ}{13}. \end{align*}

Therefore

\begin{align*} \sin x^\circ =&\,\frac{21\sin25^\circ}{13}\\ =&\,0.6827\quad\text{to 4 d.p.} \end{align*}

展开

$\sin x=0.6827$ 会给出一个锐角和一个钝角。因为 $AB$ 是最长边，所以它对着的角 $C=x^\circ$ 必须是最大角，因此选择钝角。

展开

The acute angle with sine $0.6827$ is

\begin{align*} \sin^{-1}(0.6827)=43.05^\circ\ldots \end{align*}

The other possible angle is

\begin{align*} 180^\circ-43.05^\circ\ldots =&\,136.95^\circ\ldots \end{align*}

Since $AB$ is the longest side, angle $ACB$ is the largest angle. Hence

\begin{align*} x=136.95^\circ \end{align*}

to 2 decimal places.