Trigonometry

In the circle with radius $r=1$, the coordinate of point $P$ is $(\cos \alpha, \sin \alpha)$. If $\alpha = 90^\circ$, what is $\sin \alpha$?

If $\tan \theta = 1$, find the value of $\sin \theta \cdot \cos \theta$ based on the geometric representation.

Looking at the triangle, identify the value of $cosec A$ if $BC = a$, $AC = b$ and $AB = c$.

In the rectangle $ABCD$, diagonal $AC = 10$ and $BC = 6$. Find $\sin(\angle BAC)$.

In the coordinate plane, a line passes through the origin and point $(3, 4)$. What is the $\cos \theta$ of the angle $\theta$ it makes with the positive x-axis?

Two poles of heights $6$ m and $11$ m stand on a plane ground. If the distance between their feet is $12$ m, find the sine of the angle $\theta$ the line joining their tops makes with the horizontal.

If $\sec \theta = \frac{17}{8}$, find the value of $\frac{1 - \sin \theta}{\cos \theta}$ from the diagram.

A rectangle $ABCD$ has side $AB = 20\text{ cm}$ and diagonal $AC = 25\text{ cm}$. If $\angle CAD = \phi$, find the value of $\cos \phi$.

If $5\tan \theta = 4$, evaluate the expression $\frac{5\sin \theta - 3\cos \theta}{5\sin \theta + 2\cos \theta}$.

In a right triangle $ABC$, right-angled at $C$, if $\tan A = \frac{1}{\sqrt{3}}$, calculate the value of $\sin A \cos B + \cos A \sin B$.

Value of $(\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ)$ is:

The value of $2\cos^2 60^\circ - 1$ is:

Calculate: $\cos^2 45^\circ - \sin^2 45^\circ$.

What is the value of $\csc 30^\circ + \sec 60^\circ$?

In the given right-angled triangle $ABC$, the angle $\angle ABC = 90^\circ$ and $\angle ACB = 30^\circ$. If the length of the hypotenuse $AC$ is $12$ units, find the value of the expression: $2 \sin^2(45^\circ) + \cos(60^\circ) \times \frac{BC}{AC}$.

Calculate the value of $\frac{4}{\cot^2 30^\circ} + \frac{1}{\sin^2 60^\circ} - \cos^2 45^\circ$.

If $2\theta$ is an acute angle and $2\sin 2\theta = \sqrt{3}$, find $\theta$.

Two pillars of equal height stand on either side of a road $100$ m wide. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are $30^{\circ}$ and $60^{\circ}$. Find the height of each pillar.

In $\triangle PQR$, $\angle Q = 90^{\circ}$ and $\angle R = 45^{\circ}$. If the perimeter of the triangle is $10 + 10\sqrt{2}$ cm, find the length of the hypotenuse $PR$.

Find the value of $\theta$ ($0^{\circ} \leq \theta \leq 90^{\circ}$) if $\frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4$.

If $\cos \theta = \sin 20^{\circ}$, find $\theta$.

Simplify $\cos(90^{\circ} - \theta) \cdot \sec(90^{\circ} - \theta)$.

If $A + B = 90^{\circ}$, which of the following is true?

Evaluate: $\text{cosec}^{2} 57^{\circ} - \tan^{2} 33^{\circ}$.

In the given right-angled triangle $ABC$, the angle at $B$ is $90^\circ$. If the angle at $A$ is $\theta$ and the angle at $C$ is $(90^\circ - \theta)$, find the value of the expression: $\frac{\sin \theta}{\cos(90^\circ - \theta)} + \frac{\tan(90^\circ - \theta)}{\cot \theta}$

Find $A$ if $\sin (A + 10^{\circ}) = \cos (A - 10^{\circ})$.

The value of $\tan 48^{\circ} \cdot \tan 23^{\circ} \cdot \tan 42^{\circ} \cdot \tan 67^{\circ}$ is:

A mechanical linkage consists of two bars. The torque $T$ is given by $T = 5 \left( \frac{\cos 40^\circ}{\sin 50^\circ} \right) - 2 \left( \frac{\sin 25^\circ}{\cos 65^\circ} \right)$. Calculate the magnitude of $T$.

A surveyor measures the distance between two points $A$ and $B$. In his calculation, he uses the expression $K = \sin(90^\circ - \theta) \sec \theta + \tan \theta \tan(90^\circ - \theta)$. What is the constant value of $K$?

A gardener is creating a flower bed in the shape of a sector. If the central angle is $70^\circ$, find the value of $\frac{\sin 70^\circ}{\cos 20^\circ} + \frac{\cos 70^\circ}{\sin 20^\circ}$.

Find the angle of elevation of the sun when the shadow of a 10 m high pole is $10\sqrt{3}$ m.

A string of a kite is 100 m long and it makes an angle of $30^\circ$ with the horizontal. Find the height of the kite.

The angle of elevation of the top of a tower from a point 100 m from its foot is $30^\circ$. The height of the tower is:

If the length of the shadow of a tower is equal to its height, the angle of elevation of the sun is:

A man on the deck of a ship, 10 m above water level, observes the angle of elevation of the top of a cliff as $60^\circ$ and the angle of depression of the base of the cliff as $30^\circ$. Find the height of the cliff.

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower.

A round balloon of radius $r$ subtends an angle $\alpha$ at the eye of an observer while the angle of elevation of its center is $\beta$. Find the height of the center of the balloon.

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagpole of height $h$. At a point on the plane, the angle of elevation of the bottom and the top of the flagpole are $\alpha$ and $\beta$ respectively. If the height of the tower is $H$, which of the following expressions correctly represents $H$ in terms of $h$, $\alpha$, and $\beta$?

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 (Take $\sqrt{3} = 1.732$).

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