Trigonometry for Engineering

Trigonometry is a branch of mathematics dealing with the relationships between the sides and angles of triangles. In engineering, trigonometry is extensively used for resolving forces, determining phases in alternating current, and calculating distances.

1. Right Triangles

A right triangle is a triangle in which one angle is exactly $90^\circ$ (a right angle).

• Engineering Meaning: Many physical structures and force vectors can be modeled using right triangles, reducing complex 2D and 3D problems into manageable 1D components.

2. Sine, Cosine, Tangent

The primary trigonometric ratios connecting the angles of a right triangle to its side lengths.

• Sine ($\sin \theta$): Opposite / Hypotenuse
• Cosine ($\cos \theta$): Adjacent / Hypotenuse
• Tangent ($\tan \theta$): Opposite / Adjacent
• Engineering Meaning: Used to resolve any angled force into horizontal (cosine) and vertical (sine) components, making structural or fluid flow analysis much simpler.

3. Inverse Functions

Inverse trigonometric functions (arcsin, arccos, arctan, or $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) are used to find an angle when the side lengths are known.

• Engineering Meaning: Essential for finding the direction (angle) of a resultant force when the horizontal and vertical components are known.

4. Degrees and Radians

Angles can be measured in degrees ($^\circ$) or radians (rad).

• Degrees: A full circle is $360^\circ$. Commonly used in surveying, civil engineering, and general geometry.
• Radians: A full circle is $2\pi$ radians. Radian measure directly links the angle to the arc length ($s = r\theta$).
• Engineering Meaning: Radians are critical in calculus and rotational dynamics (e.g., angular velocity in rad/s). Using degrees in a calculus formula will yield incorrect results.

5. Short Example

A force of $100\text{ N}$ acts at a $30^\circ$ angle to the horizontal.

• Horizontal component: $F_x = 100 \times \cos(30^\circ) = 86.6\text{ N}$
• Vertical component: $F_y = 100 \times \sin(30^\circ) = 50.0\text{ N}$