Pythagoras’ theorem and trigonometry：练习题

In a right-angled triangle \(XYZ\), the lengths of the two shorter sides are \(XY = 5\) cm and \(YZ = 12\) cm, where \(\angle XYZ = 90^\circ\). Find the length of the hypotenuse \(XZ\).

In triangle \(PQR\), \(PQ = 7\) cm, \(QR = 9\) cm, and \(\angle PQR = 60^\circ\). Calculate the length of the side \(PR\) correct to two decimal places.

A ship sails from point \(A\) to point \(B\) on a bearing of \(065^\circ\). It then sails from \(B\) to \(C\) on a bearing of \(155^\circ\). Given that \(AB = 8\text{ km}\) and \(BC = 6\text{ km}\), calculate the bearing of \(A\) from \(C\) to the nearest degree.

In triangle \(ABC\), the length of side \(AB = 8\) cm, the length of side \(BC = 5\) cm, and \(\angle ABC = 30^\circ\). Calculate the area of triangle \(ABC\).

A vertical flagpole \( OT \) stands at the centre \( O \) of a horizontal rectangular field \( ABCD \). The length of the field \( AB \) is \( 16\text{ m} \) and the width \( BC \) is \( 12\text{ m} \). If the angle of elevation of the top of the flagpole \( T \) from the corner \( A \) is \( 35^\circ \), calculate the height of the flagpole \( OT \), giving your answer correct to 3 significant figures.

In the diagram, \(ABC\) is a right-angled triangle where \(\angle ABC = 90^\circ\), \(AB = 12\text{ cm}\), and \(BC = 5\text{ cm}\). Find the exact value of \(\sin(\angle BAC)\).

In the diagram, \(D\), \(E\), and \(F\) are three points on horizontal ground. \(DE = 14\text{ m}\), \(EF = 9\text{ m}\), and \(\angle DEF = 62^\circ\).
(a) Calculate the distance \(DF\), giving your answer correct to \(3\) significant figures.
(b) Calculate the area of triangle \(DEF\), giving your answer correct to \(3\) significant figures.

A hiker walks from point \(A\) on a bearing of \(040^\circ\) for \(5 \text{ km}\) to reach point \(B\). From \(B\), he then walks on a bearing of \(130^\circ\) to point \(C\). Given that point \(C\) is directly East of point \(A\), calculate the distance \(BC\).

A vertical flagpole \(TF\) of height \(15\text{ m}\) stands on horizontal ground. A point \(G\) on the ground is \(20\text{ m}\) away from the base \(F\) of the flagpole.
(a) Calculate the distance from \(G\) to the top of the flagpole \(T\).
(b) Calculate the angle of elevation of the top of the flagpole \(T\) from the point \(G\), giving your answer correct to \(1\) decimal place.
(c) Write down the exact value of \(\tan(\angle TGF)\) as a simplified fraction.

In the diagram, \(A\), \(B\), and \(C\) are three points on a horizontal plane. \(B\) is due east of \(A\). The distance \(AB = 14\text{ km}\), \(BC = 9\text{ km}\), and \(\angle ABC = 115^\circ\).
(a) Calculate the distance \(AC\), giving your answer correct to \(3\) significant figures.
(b) Calculate the area of triangle \(ABC\), giving your answer correct to \(1\) decimal place.
(c) Calculate \(\angle BAC\), giving your answer correct to \(1\) decimal place.
(d) Hence, find the bearing of \(C\) from \(A\).