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§ Trigonometry·Grade 6

Inverse Trigonometry Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable inverse trigonometry worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from evaluate arcsin/arccos/arctan of a standard positive value, degrees at the easy level through to compositions like sin(arccos(v)) and arccos(sin(v)) at the advanced level.

CCSS.HSF.TF.B.6LK20.R1.invers_trigonometriLK20.R2.invers_trigonometri

What is inverse trigonometry?

Inverse trigonometric functions reverse the action of sine, cosine, and tangent by finding the angle that produces a given ratio. The three primary inverse functions are arcsin, arccos, and arctan, each with specific output ranges called principal values. These functions appear in CCSS.HSF.TF.B.6 as tools for solving trigonometric equations and analyzing periodic phenomena.

Why it matters

Inverse trigonometry enables engineers to calculate launch angles for projectiles, architects to determine roof slopes from height requirements, and GPS systems to triangulate positions from satellite distances. In physics, arctan helps find the direction of vector forces when components are known — for example, calculating that a force with components (3, 4) points at arctan(43) ≈ 53.1° above horizontal. Navigation systems use arcsin to determine elevation angles from altitude and distance measurements. These functions also appear in calculus integration formulas, particularly when evaluating integrals involving √(1-x²) or 1/(1+x²). Advanced mathematics relies on inverse trig functions for Fourier analysis, signal processing, and solving differential equations in engineering applications.

Common mistakes to watch for

  • ✗Confusing degree and radian outputs, such as writing arcsin(1/2) = 30 instead of π/6 radians
  • ✗Ignoring principal value ranges, like claiming arccos(-1/2) = 4π/3 instead of the correct 2π/3
  • ✗Mixing up function relationships, such as writing arctan(1) = π/4 but then incorrectly stating arctan(-1) = 3π/4 instead of -π/4

Questions teachers ask

What are the principal value ranges for inverse trig functions?+
arcsin outputs angles in [-π/2, π/2], arccos in [0, π], and arctan in (-π/2, π/2). These restricted ranges ensure each input produces exactly one output, making the functions well-defined mathematical operations.
How do you evaluate compositions like sin(arccos(1/3))?+
Let θ = arccos(1/3), so cos θ = 1/3. Use the Pythagorean identity sin²θ + cos²θ = 1 to find sin θ = √(1 - 1/9) = √(8/9) = 2√2/3.
Why can't arcsin accept values greater than 1?+
The sine function only produces outputs between -1 and 1, so arcsin can only accept inputs in this range. Values like arcsin(2) are undefined because no angle has a sine of 2.
What's the difference between arctan and tan⁻¹?+
These notations represent the same function — the inverse tangent. The notation tan⁻¹ uses the standard inverse function symbol, while arctan uses the arc prefix meaning 'the arc (angle) whose tangent is.'
How do you find exact values for non-standard inputs?+
Use reference triangles or the unit circle. For example, to find arcsin(√3/2), recognize that sin(π/3) = √3/2, so arcsin(√3/2) = π/3. Memorizing common angle values is essential for exact evaluation.
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Beginner

Generate →
Concepts
Evaluate arcsin/arccos/arctan of a standard positive value, degrees
Range
positive standard inputs
Steps
1–2 steps
Example
arcsin(1/2) = ?

Easy

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Concepts
Evaluate at any standard value, radians
Range
positive or negative standard inputs
Steps
1–2 steps
Example
arccos(−√2/2) = ?

Medium

Generate →
Concepts
Reinforce principal-value ranges
Range
tricky inputs near range boundaries
Steps
2–3 steps
Example
Explain why arcsin(−1) = −π/2 and not 3π/2.

Hard

Generate →
Concepts
Compositions like sin(arccos(v)) and arccos(sin(v))
Range
standard values
Steps
3 steps
Example
Evaluate sin(arccos(1/2)).

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