Trigonometry: Angles and Wave Basics

Trigonometry connects geometry to measurement. It starts with triangles and angles, then expands into a powerful language for describing rotation, periodic motion, and waves. Engineers use it to resolve forces, physicists use it to model oscillations, and calculus builds on it to analyze changing motion. To understand the larger applications, it helps to build the fundamentals carefully: right-triangle ratios, unit-circle definitions, and the bridge from angles to waves.

Angles: Measuring Rotation

An angle describes how far a ray rotates around a point. Two common units are degrees and radians.

• Degrees divide a full turn into $360^{\circ }$.
• Radians measure an angle by arc length: an angle of $1$ radian subtends an arc equal to the circle’s radius.

Radians matter because they make trigonometric formulas cleaner and match naturally with calculus. A full circle is $2\pi$ radians, so the conversion is:

• ${\theta }_{\text{rad}}={\theta }_{\text{deg}}\cdot \frac{\pi }{180}$
• ${\theta }_{\text{deg}}={\theta }_{\text{rad}}\cdot \frac{180}{\pi }$

Key benchmark angles: $90^{\circ }=\frac{\pi }{2}$, $180^{\circ }=\pi$, $360^{\circ }=2\pi$.

Right-Triangle Trigonometry: Ratios That Define Shape

In a right triangle, with one $90^{\circ }$ angle, the other two angles are acute. Pick one acute angle $\theta$. The sides relative to $\theta$ are:

• Opposite: across from $\theta$
• Adjacent: next to $\theta$ (not the hypotenuse)
• Hypotenuse: longest side, opposite the right angle

The trigonometric ratios are:

• $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$

These ratios depend only on the angle, not the triangle’s size. If you scale the triangle up or down, the ratios stay the same.

Solving for Unknown Sides and Angles

Most practical triangle problems reduce to one of two tasks:

1. Find a side length given an angle and a side.

Example: A ramp makes a $15^{\circ }$ angle with the ground and rises $1.2$ m vertically. The ramp length $L$ satisfies $\sin 15^{\circ }=\frac{1.2}{L}$, so $L=\frac{1.2}{\sin 15^{\circ }}$.

1. Find an angle given side lengths.

If $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$, then $\theta =\arctan \left(\frac{\text{opposite}}{\text{adjacent}}\right)$. In design and navigation, this is how you infer direction from components.

A useful cross-check is the Pythagorean theorem: if you compute a missing side, verify $a^2+b^2=c^2$ where $c$ is the hypotenuse.

Common Triangle Identities

One identity appears constantly:

$${\sin }^2\theta +{\cos }^2\theta =1$$

In a right triangle, it comes from dividing $a^2+b^2=c^2$ by $c^2$. This identity stays true beyond right triangles because it is fundamentally tied to the unit circle.

The Unit Circle: Extending Trig Beyond Acute Angles

Right-triangle definitions only cover angles between $0^{\circ }$ and $90^{\circ }$. The unit circle extends sine and cosine to all real angles.

Take a circle of radius 1 centered at the origin. For an angle $\theta$ measured from the positive $x$-axis, the point on the circle has coordinates:

• $x=\cos \theta$
• $y=\sin \theta$

So $\cos \theta$ and $\sin \theta$ become coordinate functions, not just triangle ratios. This definition automatically includes angles larger than $90^{\circ }$, negative angles, and multiple turns.

Signs by Quadrant

Because the unit circle coordinates can be positive or negative, sine and cosine change sign depending on the quadrant:

• Quadrant I: $\sin >0$, $\cos >0$
• Quadrant II: $\sin >0$, $\cos <0$
• Quadrant III: $\sin <0$, $\cos <0$
• Quadrant IV: $\sin <0$, $\cos >0$

Tangent is $\tan \theta =\frac{\sin \theta }{\cos \theta }$, so it is undefined where $\cos \theta =0$ (at $\theta =\frac{\pi }{2},\frac{3\pi }{2},\dots$).

Reference Angles and Symmetry

A reference angle is the acute angle between the terminal side of $\theta$ and the $x$-axis. It lets you reuse known values like $\sin 30^{\circ }=\frac{1}{2}$ or $\cos 60^{\circ }=\frac{1}{2}$ and then apply the correct sign based on the quadrant.

Unit-circle symmetry also gives helpful relationships:

• $\sin (-\theta )=-\sin \theta$ (odd function)
• $\cos (-\theta )=\cos \theta$ (even function)

From Angles to Waves: Periodic Behavior

A key reason trigonometry is foundational is that sine and cosine naturally describe repeating patterns. A function is periodic if it repeats after a fixed interval $T$:

$$f(t+T)=f(t)$$

For basic trig functions:

• $\sin (\theta )$ and $\cos (\theta )$ have period $2\pi$
• $\tan (\theta )$ has period $\pi$

When the input is time, these become wave models.

The General Sine Wave Form

A common wave model is:

$$y(t)=A\sin (\omega t+\phi )+D$$

Each parameter has a concrete meaning:

• Amplitude $A$: maximum deviation from the midline (strength of the oscillation)
• Angular frequency $\omega$ (rad/s): how fast the wave cycles
• Phase $\phi$: horizontal shift, setting where the cycle starts
• Vertical shift $D$: the midline value

The period is linked to angular frequency:

$$T=\frac{2\pi }{\omega }$$

If you prefer frequency in cycles per second (hertz), $f=\frac{1}{T}$ and $\omega =2\pi f$.

Practical Interpretation

• In AC electricity, voltage is often approximated by a sine wave where amplitude relates to peak voltage and frequency to grid standards.
• In mechanical systems, a vibrating component can be modeled with $A$ as displacement amplitude and $\omega$ tied to the system’s stiffness and mass.

Understanding the parameters helps you read graphs and translate measurements into models. If a wave repeats every 0.5 seconds, then $T=0.5$ and $\omega =\frac{2\pi }{0.5}=4\pi$ rad/s.

Simple Harmonic Motion: Where Trig Meets Physics

Simple harmonic motion (SHM) appears in springs, small-angle pendulums, and many linearized oscillations. In SHM, displacement over time is sinusoidal:

$$x(t)=A\cos (\omega t+\phi )$$

Cosine and sine are interchangeable here; choosing one usually depends on initial conditions. What makes SHM especially important is how derivatives relate back to the original function. When you differentiate a sine or cosine, you get another sine or cosine, which is why these functions solve oscillation equations elegantly.

A core SHM relationship is that acceleration is proportional to negative displacement:

$$a(t)=-{\omega }^{2}x(t)$$

This explains why the motion reverses direction toward equilibrium: the restoring acceleration points back toward the center.

Why These Basics Matter

Right-triangle ratios teach you how angles encode proportions. The unit circle turns those ratios into functions defined for every real angle, enabling modeling of rotation and direction in any quadrant. Once you accept sine and cosine as coordinate functions, waves become an immediate next step, not a separate topic.

Trigonometry is not just a set of formulas to memorize. It is a consistent framework for moving between geometry (angles and triangles), algebra (functions and equations), and real systems (waves, vibration, and circular motion). Mastering angles and wave basics builds a foundation that engineering analysis, physics modeling, and calculus will repeatedly rely on.