Trigonometric Inequalities

An industrial robot arm has 6 degrees of freedom, each with angular limits. Boston Dynamics Spot: every step = solving a system of trigonometric inequalities in under 1 ms. Too slow - the robot falls. GPS discards satellites below the elevation mask: sin(elevation) > 0.17. An inequality, not an equation. Continuous, per satellite, in real time.

• **Joint constraints in robotics:** Boston Dynamics, industrial ARM manipulators - valid angle range per joint is a trig inequality, evaluated at every trajectory step
• **GPS elevation mask:** satellites below ~10 degrees are discarded - condition sin(elevation) > sin(10 deg), a continuous trigonometric inequality
• **Frustum culling in 3D:** an object is visible when cos(angle_to_camera) >= cos(FOV/2) - inequality checked per object per frame in Unity, Unreal, WebGL
• **Phase unwrapping in DSP:** restoring continuous phase from a wrapped signal - solving |phi| > pi at each signal step

Предварительные знания

• Trigonometric Equations

Inequalities sin x > a and cos x <= a

Every GPS receiver discards satellites below the elevation mask (~10 degrees above the horizon). The condition: $\sin(\text{elevation}) > \sin(10^\circ) \approx 0.17$. A trigonometric inequality. Every navigation chip solves it continuously - for each of the dozens of visible satellites.

The inequality $\sin x > a$ does not describe a point - it describes an **arc** on the unit circle. Geometrically: draw a horizontal line $y = a$ and select the part of the circle that lies **above** that line.

**Template for $\sin x > a$ ($|a| < 1$):** Solve $\sin x = a$ to get $x_0 = \arcsin(a)$ Then: $\sin x > a \iff x \in (x_0 + 2\pi n, \; \pi - x_0 + 2\pi n), \; n \in \mathbb{Z}$ **Template for $\cos x > a$ ($|a| < 1$):** Solve $\cos x = a$ to get $x_0 = \arccos(a)$ Then: $\cos x > a \iff x \in (-x_0 + 2\pi n, \; x_0 + 2\pi n), \; n \in \mathbb{Z}$

For $\cos x \leq a$ the geometry shifts: cosine is the horizontal coordinate of the circle point, so a vertical line $x = a$ is drawn instead. The arc where cosine does not exceed $a$ is the left half of the circle.

Compound Inequalities and Systems

Boston Dynamics Spot: each step solves a system of trigonometric inequalities simultaneously. Hip angle in $[-\pi/2, \pi/2]$, knee angle in $[0, 2\pi/3]$, ankle angle in $[-\pi/4, \pi/4]$. Solved in under 1 ms - otherwise the robot falls. Geometrically: intersection of three arcs on the unit circle.

A compound inequality $a \leq \sin x \leq b$ defines a **band** on the unit circle: the arc bounded above by $y = b$ and below by $y = a$. The solution is the intersection of the individual inequality sets.

A system with $\sin x > a$ and $\cos x > b$ simultaneously - intersection of two arcs on the unit circle, typically landing in a single quadrant. This is exactly what a joint controller computes: angle $\theta$ must satisfy multiple workspace constraints at once.

For compound inequalities, sketch the unit circle by hand and mark the allowed and forbidden arcs. Faster and more reliable than algebraic bookkeeping - especially when three or more constraints are active simultaneously.

Sign Analysis of Trigonometric Expressions

Phase unwrapping in DSP is solving a trigonometric inequality. A wrapped phase $\phi(t) \in (-\pi, \pi]$ jumps by $2\pi$ whenever it crosses the boundary - unwrapping detects when $|\phi| > \pi$ and restores the continuous phase. Sign analysis of $\phi$ at each time step.

When a trigonometric expression involves products or quotients, tracking the sign of each factor solves the inequality. The sign-diagram method: partition the period at zeros and discontinuities, determine the sign on each subinterval.

$\tan x > 0$ means $\sin x / \cos x > 0$. The sign of the fraction follows the signs of numerator and denominator. The points $x = \pi/2 + \pi n$ where $\cos x = 0$ are discontinuities of $\tan$ - they are excluded from the domain, and solution intervals never cross through them.

When solving inequalities involving $\tan x > a$, always state the domain restriction: $x \neq \pi/2 + \pi n$. These are discontinuities - solution intervals stop at them and do not continue through. Omitting the domain is the most common mistake.

Key Ideas

• **Geometric approach:** an inequality describes an arc on the unit circle; the horizontal line $y = a$ splits it into above and below
• **$\sin x > a$** - open interval $(\arcsin a + 2\pi n, \; \pi - \arcsin a + 2\pi n)$; **$\cos x > a$** - $(-\arccos a + 2\pi n, \; \arccos a + 2\pi n)$
• **Compound inequalities and systems:** intersection of solution sets; geometrically, intersection of arcs; typically lands in a single quadrant
• **Sign diagrams** for products and quotients: sign table by quadrant; function discontinuities are hard boundaries of solution intervals

Related Topics

Trigonometric inequalities bridge equation solving and geometric applications:

• Trigonometric Equations — Boundary points of each inequality come from solving the equation
• Law of Sines and Cosines — Triangle existence conditions are trigonometric inequalities
• Trigonometry in Interviews — 3D visibility and ray intersection problems reduce to inequality systems

Вопросы для размышления

• The inequality $\sin x \geq 1$ has solutions - isolated points $x = \pi/2 + 2\pi n$. Does $\sin x \geq 2$ have any solutions? How does the unit circle geometry explain the difference?
• Compare the solution sets for $\sin x = 0.5$ and $\sin x > 0.5$. Why does one give isolated points and the other gives intervals? What changes geometrically?
• GPS uses an elevation mask of 10 degrees. If the mask were raised to 15 degrees, how would the solution set of $\sin(\text{elevation}) > \sin(15^\circ)$ change? Draw this on the unit circle.

Связанные уроки

• trig-04 — Boundary points of every inequality come from solving the equation
• trig-06 — Triangle existence conditions are trigonometric inequalities
• trig-12 — 3D visibility and frustum culling problems reduce to systems of trig inequalities
• calc-03-limits-intro — Boundedness |sin x| <= 1 powers the squeeze theorem for limits
• calc-06-derivative-intro