Trigonometry

Determine the value of $\sec^2 30^\circ - \tan^2 30^\circ$.

Find the value of $5\cos^2 \theta + 5\sin^2 \theta$.

Simplify the expression: $(1 + \cot^2 \theta) \sin^2 \theta$.

Which of the following is equivalent to $\tan \theta \cdot \cos \theta$?

A right-angled triangle $ABC$ is shown in the coordinate plane. Points $A$ and $B$ lie on the $x$ and $y$ axes respectively, such that $OA = 3$ units and $OB = 4$ units. If $\angle OAB = \theta$, evaluate the expression $(\sin \theta + \cos \theta)^2 + (\sin \theta - \cos \theta)^2$.

If $\tan \theta + \sin \theta = m$ and $\tan \theta - \sin \theta = n$, then find the value of $m^2 - n^2$.

Evaluate $\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} + \sin \theta \cos \theta$.

A civil engineer uses the identity $\frac{1 + \cot A + \tan A}{\text{cosec} A - \sec A}$ to find the slope of a drainage pipe. What is the simplified expression for this slope?

In a mechanical linkage, the displacement is found to be $\sec^4 \theta - \sec^2 \theta$. What is the equivalent expression in terms of $\tan \theta$?

The efficiency of a solar panel tilt is represented by $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta}$. Which of the following is the correct simplified identity?

A vertical pole and a vertical tower are on the same level ground. From the top of the pole 20 m high, the angle of elevation of the top of the tower is $60^\circ$ and the angle of depression of the foot of the tower is $30^\circ$. Find the height of the tower.

From the top of a 60 m high building, the angle of depression of an object on the ground is $30^\circ$. How far is the object from the base of the building?

A kite is flying at a height of 75 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is $60^\circ$. Find the length of the string, assuming there is no slack.

An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is $45^\circ$. What is the total height of the chimney?

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $30^\circ$ and $45^\circ$ respectively. If the bridge is at a height of 3 m from the banks, find the width of the river.

A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle of $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the original height of the tree.

The angle of elevation of the top of a building from the foot of a tower is $30^\circ$ and the angle of elevation of the top of the tower from the foot of the building is $60^\circ$. If the tower is 50 m high, find the height of the building.

A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from $30^\circ$ to $60^\circ$ as he walks towards the building. Find the distance he walked.

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $45^\circ$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

The angle of elevation of the top of a tower from two points at distances $a$ and $b$ from the base and in the same straight line with it are complementary. Find the height of the tower.