Trigonometry for Engineering

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

1. Right Triangles

A right triangle has one angle equal to $90^\circ$ (a right angle).

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

2. Sine, Cosine, Tangent (SOH CAH TOA)

These are the primary trigonometric ratios relating the angles of a right triangle to the lengths of its sides.

• Sine ($\sin \theta$): Opposite / Hypotenuse
• Cosine ($\cos \theta$): Adjacent / Hypotenuse
• Tangent ($\tan \theta$): Opposite / Adjacent
• Engineering Meaning: Used to split a single angled force into horizontal (cosine) and vertical (sine) components, making it easier to analyze structures or fluid flows.

3. Inverse Functions

Inverse trigonometric functions (arcsin, arccos, arctan or $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) are used to find the 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 $360^\circ$. Used commonly in surveying, civil engineering, and general geometry.
• Radians: A full circle is $2\pi$ radians. Radian measure relates the angle directly to the arc length ($s = r\theta$).
• Engineering Meaning: Radians are crucial 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 $100\text{ N}$ is applied at an angle of $30^\circ$ to the horizontal.

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