Trigonometric Ratios: Practice Questions

In a right-angled triangle, if an acute angle \(\theta\) increases from \(10^\circ\) to \(80^\circ\), which of the following statements about its trigonometric ratios is correct?

In a right-angled triangle, if \(\tan \theta = \frac{7}{24}\) and \(\theta\) is an acute angle, find the value of \(\frac{1}{\cos \theta - \sin \theta}\).

A ship is at point \(A\). It sails \(40\) km on a bearing of \(030^\circ\) to point \(B\), and then sails \(30\) km on a bearing of \(120^\circ\) to point \(C\). Find the bearing of \(C\) from \(A\), correct to the nearest degree.

Find the value of \(\cos 60^\circ + \tan 45^\circ - \sin 30^\circ\).

Given that \(\theta\) is an acute angle and \(\tan \theta = \frac{\sqrt{7}}{3}\), find the value of \(\cos^2 \theta - \sin^2 \theta\).

In a right-angled triangle, if an acute angle is \(\theta\), simplify the trigonometric expression \(\cos \theta \cdot \tan \theta\).

In a right-angled triangle \(ABC\), the right angle is at \(B\). If \(\tan A = 2\) and the length of the side \(AB = 5\text{ cm}\), find the exact length of the hypotenuse \(AC\). Leave your answer in surd form.

In the figure, \(PQRS\) is a rectangle. \(T\) is a point on \(RS\) such that \(PT\) is perpendicular to the diagonal \(QS\). If \(PQ = 4\text{ cm}\) and \(QR = 3\text{ cm}\), find the length of \(ST\).

In the figure, P is a point on the horizontal ground. A vertical building AB stands such that its base B is on the ground. From point P, the angle of elevation of the top A is \( 28^\circ \). A flagpole AC of height \( 5 \text{ m} \) is fixed vertically on the top of the building. From the same point P, the angle of elevation of the top of the flagpole C is \( 35^\circ \).

(a) Let the height of the building AB be \( h \text{ m} \) and the horizontal distance PB be \( x \text{ m} \). Express \( x \) in terms of \( h \) using \( \triangle ABP \).
(b) Express \( x \) in terms of \( h \) using \( \triangle CBP \).
(c) Calculate the height of the building \( h \), correct to 2 decimal places.
(d) Find the shortest distance from point P to the top of the flagpole C, correct to 1 decimal place.

In the figure, ABC is a triangle on a horizontal ground where \(\angle ABC = 90^\circ\). A vertical transmission tower CD of height \(h\) m stands at point C. From point A, the angle of elevation of the top of the tower D is \(20^\circ\). From point B, the angle of elevation of D is \(40^\circ\). It is given that the distance between A and B is \(100\) m.

(a) Express the horizontal distances BC and AC in terms of \(h\).
(b) By considering the right-angled triangle ABC, form an equation in terms of \(h\).
(c) Calculate the height of the tower \(h\), correct to 2 decimal places.
(d) Find the angle of elevation of D from point M, where M is the midpoint of AB. Correct your answer to 1 decimal place.