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For a triangle with angle $\theta$ and hypotenuse $h$, the height $a$ opposite to $\theta$ can be calculated using the sine function: $a = h \times \sin(\theta)$. This is useful in finding the area of the triangle, which is $\frac{1}{2} \times b \times a$, where $b$ is the base of the triangle.

The length $L$ of a shadow cast by an object of height $h$ at an angle of elevation $\theta$ of the sun can be found using the tangent function: $L = h \times \tan(\theta)$. This is a practical application of trigonometry in real-life scenarios.

For two points at a distance $d$ in space forming an angle $\theta$ with a common point, the distance $x$ along one axis can be found using the cosine function: $x = d \times \cos(\theta)$. This is often used in navigation and astronomy.

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. It is used to solve for unknown sides and angles in a triangle.

The Law of Cosines relates the lengths of a triangle's sides to the cosine of one of its angles: $c^2 = a^2 + b^2 - 2ab\cos(C)$, where $\angle C$ is the angle opposite side $c$. It is useful when solving for unknown sides and angles.

For a cyclic quadrilateral with sides $a, b, c, d$ and opposite angles $A$ and $C$, the area $K$ can be found using Brahmagupta's formula: $K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$, where $s$ is the semiperimeter and $A+C = 180^\circ$. Trigonometry is used to find the necessary angles.

For a circle with an inscribed angle $\theta$ subtending a chord of length $c$, the radius $r$ of the circle can be calculated using the formula: $r = \frac{c}{2\sin(\theta)}$. Trigonometry allows calculation of $r$ without direct measurement.

For a rectangle with length $l$ and width $w$, the diagonal $d$ can be found using the Pythagorean theorem where $d = \sqrt{l^2 + w^2}$. Trigonometry comes into play when angles are involved, calculating $d = l \cos(\theta)$ or $d = w \cos(90^\circ-\theta)$ where $\theta$ is the angle between the diagonal and the length.

For a ramp with vertical height $h$ and horizontal length $l$, the slope angle $\theta$ can be found using the tangent function: $\theta = \arctan(\frac{h}{l})$. This application of trigonometry is important in engineering and design.