List of top Mathematics Questions on Some Applications of Trigonometry

A kite is flying at a height of \(60 \text{ m}\) above the ground level. Ravi, standing at the roof of the house is holding the string straight and observes the angle of elevation of kite as \(30^{\circ}\). From the bottom of the same building, the angle of elevation of kite is \(45^{\circ}\). Find the length of the string and height of roof from the ground. (Use \(\sqrt{3} = 1.73\))

Simplest form of \(\frac{\sec A}{\sqrt{\sec^2 A - 1}}\) is

A wire is attached from a point A on the ground to the top of a pole BC, making an angle of elevation as \(60^{\circ}\). If \(AB = 5\sqrt{3}\) m, then length of the wire is

For an acute angle \(\theta\), if \(\sin \theta = \frac{1}{9}\), then value of \(\frac{9 \csc \theta + 1}{9 \csc \theta - 1}\) is

Evaluate : \(\frac{\sin^3 60^{\circ} - \tan 30^{\circ}}{\cos^2 45^{\circ}}\)

For acute angles A and B and \(A + 2B\) and \(2A + B\) are acute if \(\tan (A + 2B) = \sqrt{3}\) and \(\sin (2A + B) = \frac{1}{\sqrt{2}}\), then find the measures of angles A and B.}

Prove that :
\(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta\).

If \( \cos A = \frac{4}{5} \), then the value of \( \tan A \) is :

If \( 2 \sin A = 1 \), then the value of \( \tan A + \cot A \) is :

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be \( 45^\circ \). The height (in metres) of the tower is :

Radio towers are built in two sections 'A' and 'B'. Tower is supported by wires from a point 'O' (as shown in figure). Distance between the base of the tower and point 'O' is 6 m. From point 'O', the angle of elevation of the top of the section 'B' is \(30^{\circ}\) and the angle of elevation of the top of section 'A' is \(60^{\circ}\). Based on the above information, answer the following questions :

Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window (W ) of the building, the angle of elevation of top of the tank is 45◦ and angle of depression of its foot is 30◦.

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \sqrt{5\beta}}{2} \] then the value of \( (\alpha + \beta) \) is:

The number of values of \( x \) satisfying \( \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6} \) and \( x<\left[ \frac{-1}{2\sqrt{6}} , \frac{1}{2\sqrt{6}} \right] \) is:

The number of solutions of the equation \( \tan 3x = \cot x \) in \( x \in [0, 2\pi] \) is:

The minimum value of \( (\sin^{-1} x)^2 + (\cos^{-1} x)^2 \) in \( x \in \left[ \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}} \right] \) is \( \frac{a \pi^2}{b} \), then \( a + b \) is:

The value of cosec10° - √3 sec10°

Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.

A ladder 7 m long makes an angle of 30\(^\circ\) with the wall. Find the height of the point on the wall where the ladder touches the wall.

The angle of elevation of the top of a tower at a distance of 10 m from its base is 60\(^\circ\); then the height of the tower is

A kite is at a height of 30 m from the earth and its string makes an angle 60\(^\circ\) with the earth. Then the length of the string is

Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.

If \(\frac {3cos\ 2x+cos^32x}{cos^6x-sin^6x}=x^3-x^2+6\), then find sum of roots.