Sine inequalities
Read the y-coordinate on the unit circle.
Repeat final arcs with +2kπ.
Use open endpoints for < or >, and closed endpoints for <= or >=.
Trigonometric Inequality Calculator
Solve sin, cos, and tan inequalities with a visual unit circle — see exactly which arc satisfies your inequality, plus the full periodic general solution.
trigonometric inequality calculatorsolve sin cos tan inequalitygeneral solution trig inequality
Unit-circle method
This page treats the answer as an arc first, then translates that visible arc into radians, degrees, and a repeating general solution.
Unit circle calculator
Solve sin, cos, and tan inequalities visually
Choose a trig function, comparison, and range. The unit circle marks critical angles, highlights satisfying arcs, and shows the full periodic solution.
Geometry / Periodic Inequalities
Function type
sin(x)
Solving range
Result
sin(x) > 1/2 is solved on [0, 2π) as \left(\frac{\pi}{6}, \frac{5\pi}{6}\right).
Blue arc
Satisfying angles
Endpoint
Filled means included
Red X
tan undefined
Principal interval
(30°, 150°)
Full periodic solution
Three-period preview
k = -1
k = 0
k = 1
The highlighted unit-circle arc repeats in adjacent periods. This preview limits the repetition to k = -1, 0, 1 so the general solution stays readable.
1
Find the critical angle(s)
Solve sin(x) = 1/2 first. These angles are where the inequality can switch from true to false.
2
Mark the unit circle
Plot the critical angle(s) on the unit circle, then compare the matching coordinate with the target value.
3
Keep the satisfying arc(s)
Shade the arc where sin(x) > 1/2. Strict inequalities use open endpoints; inclusive inequalities use closed endpoints.
4
Add periodic repetition
sin(x) repeats every 2π, so the general solution repeats the same arc for every integer k.
Unit Circle Visualizer
The answer is an arc before it is an interval
Critical angles
Solve equality first. These angles become the possible endpoints of the answer.
Highlighted arcs
The unit circle shows where sin, cos, or tan is above or below the target value.
Periodic copy
Once one cycle is understood, the same arc repeats with +2kπ for sin/cos or +kπ for tan.
Step-by-Step Breakdown
From trig inequality to repeating angle set
sin and cos
Sin reads the y-coordinate and cos reads the x-coordinate. Both repeat every 2π.
tan
Tan repeats every π and must exclude undefined angles, so answers often become unions.
Function Guide
Sin, cos, and tan use different visual cues
Cosine inequalities
Read the x-coordinate on the unit circle.
Repeat final arcs with +2kπ.
Left/right arcs often feel reversed compared with sine because the x-coordinate is being compared.
Tangent inequalities
Read slope y/x and split at undefined angles.
Repeat final arcs with +kπ.
Never include π/2 + kπ because tan is undefined there.
Common Mistakes
Period and endpoint checks before the final answer
Reporting only one principal interval
Add the periodic term so the solution covers every repeated arc on the real line.
Including tangent asymptotes
Angles where tan is undefined are always excluded, even when the inequality symbol is inclusive.
Using bracket endpoints for strict trig inequalities
Critical angles are open for < and > because equality is not part of the solution.
Examples
Common sin, cos, and tan inequality examples
sin inequality
sin(x) > 0.5
The critical angles are π/6 and 5π/6. On the unit circle, sin(x) is the y-coordinate, so the satisfying arc is between those two points.
Expected result
x ∈ (π/6, 5π/6)
Calculator output
cos inequality
cos(x) <= -0.5
The critical angles are 2π/3 and 4π/3. The inclusive symbol keeps both endpoints, so the arc is closed.
Expected result
x ∈ [2π/3, 4π/3]
Calculator output
tan domain break
tan(x) > 1
The critical angles are π/4 and 5π/4, but tan(x) is undefined at π/2 and 3π/2, so the solution breaks into a union of intervals.
Expected result
x ∈ (π/4, π/2) ∪ (5π/4, 3π/2)
Calculator output
Notice the solution breaks into two pieces because tan(x) is undefined at π/2. This is a union of intervals. Review union notation.
FAQ
Frequently Asked Questions
How do you solve a trigonometric inequality?
Find the critical angles where equality holds, mark them on the unit circle, test which arcs satisfy the inequality, then write the matching interval and repeat it by the function period.
What is the general solution of a trig inequality?
The general solution repeats the principal arc in every period. Sin and cos repeat with +2kπ, while tan repeats with +kπ, where k is any integer.
How does the unit circle help solve sin/cos inequalities?
On the unit circle, sin(x) is the y-coordinate and cos(x) is the x-coordinate. The solution is the arc where that coordinate is above, below, or equal to the target value.
Why does tan(x) inequality solutions have a gap or break?
tan(x) is undefined at π/2 + kπ. Those angles must be excluded, so tan inequality answers often split into multiple open intervals.
What's the difference between principal solution and general solution?
A principal solution reports the answer on one chosen interval such as [0, 2π). A general solution adds the periodic term so every repeated arc on the real line is included.
How do you convert radians to degrees in the final answer?
Multiply radians by 180/π. For example, π/6 is 30°, π/4 is 45°, π/2 is 90°, and 5π/6 is 150°.