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Inverse Cosine Calculator

Mathematics - Inverse cosine function graph and unit circle

Inverse Cosine Calculator — Calculate arccos(x) in Degrees or Radians

Compute the inverse cosine (arccos) of any value between −1 and 1 with step-by-step explanation. Learn the domain, range, graph, and real-world applications of arccos with comprehensive FAQs. For related trigonometry tools, see our Pythagorean theorem calculator or the triangle area calculator.

Table of Contents

  1. 1What Is Inverse Cosine?
  2. 2Domain and Range
  3. 3Common arccos Values
  4. 4Graph of arccos(x)
  5. 5arccos in Right Triangles
  6. 6Properties of arccos
  7. 7Degrees vs. Radians
  8. 8Step-by-Step Examples
  9. 9Relationship with arcsin
  10. 10Common Mistakes
  11. 11Real-World Applications
  12. 12Practice Problems
  13. 13Tips for Mastery
  14. 14Final Thoughts
  15. 15FAQ: Common Questions

Calculate Inverse Cosine (arccos)

Enter a value between −1 and 1. The calculator will compute arccos(x) in your chosen unit and show the steps.

arccos(x) will appear here.
Note: arccos(x) = cos⁻¹(x) returns the angle whose cosine is x. The result is always between 0° and 180° (0 and π).
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Master Inverse Cosine (arccos)

90+ Interactive Trigonometry Exercises with Instant Feedback

Master Inverse Cosine (arccos) is a guided practice resource for students who want to build confidence with the inverse cosine function. It includes 90+ interactive exercises with instant feedback, covering everything from basic arccos values to real-world applications — without needing a tutor to check every answer.

  • ✓ 90+ self-checking exercises across 9 progressive sections
  • ✓ Instant feedback with hints and step-by-step explanations
  • ✓ Auto-saving progress so students can continue where they left off
  • ✓ Covers degrees, radians, negative inputs, and real-world applications
  • ✓ Works on any device — no sign-up required
$9.99 $4.99 50% OFF
Start Your Guided Practice

Inverse Cosine Function – Complete Explanation

The inverse cosine function, written as arccos(x) or cos⁻¹(x), answers the question: "What angle has a cosine of x?" It's the inverse operation of the cosine function, restricted to a specific domain and range to make it a proper function.

Inverse Cosine Definition

arccos(x) = θ  where  cos(θ) = x

Domain: −1 ≤ x ≤ 1
Range: 0 ≤ θ ≤ π (0° ≤ θ ≤ 180°)

1. What Is Inverse Cosine?

Inverse cosine (arccos) finds the angle when you know the cosine value. Since cosine is periodic and many angles have the same cosine value, the inverse cosine function is restricted to return only one value — the principal value between 0° and 180°.

Key characteristics:

  • Domain: −1 ≤ x ≤ 1
  • Range: 0 ≤ arccos(x) ≤ π (0° to 180°)
  • Notation: arccos(x) or cos⁻¹(x)
  • Principal value: Always returns the angle between 0° and 180°.

2. Domain and Range

Domain: −1 ≤ x ≤ 1

Range: 0 ≤ arccos(x) ≤ π

In degrees: 0° ≤ arccos(x) ≤ 180°

Values outside the domain (like 2 or −1.5) are undefined for arccos.

3. Common arccos Values

arccos(1) = 0°

arccos(√3/2) = 30° (π/6)

arccos(√2/2) = 45° (π/4)

arccos(1/2) = 60° (π/3)

arccos(0) = 90° (π/2)

arccos(−1/2) = 120° (2π/3)

arccos(−1) = 180° (π)

4. Graph of arccos(x)

The arccos graph is a smooth decreasing curve that starts at (−1, π), passes through (0, π/2), and ends at (1, 0).

Key features: Domain = [−1, 1], Range = [0, π], Continuous, Decreasing, Not symmetric about origin or y-axis.

5. arccos in Right Triangles

In a right triangle, arccos finds the angle when you know the adjacent side and hypotenuse.

If cos(θ) = adjacent/hypotenuse

Then θ = arccos(adjacent/hypotenuse)

Example: adjacent = 3, hypotenuse = 5 → θ = arccos(3/5) = arccos(0.6) ≈ 53.13°

6. Properties of arccos

  • Not odd or even: arccos(−x) = π − arccos(x)
  • Inverse relationship: cos(arccos(x)) = x for all x in domain
  • Complementary identity: arccos(x) + arcsin(x) = π/2 (90°)
  • Derivative: d/dx arccos(x) = −1/√(1−x²)
  • Integral: ∫ arccos(x) dx = x·arccos(x) − √(1−x²) + C

7. Degrees vs. Radians

The result of arccos can be expressed in either degrees or radians.

Radians: natural unit for mathematics (arccos(0) = π/2)

Degrees: more intuitive for real-world measurements (arccos(0) = 90°)

Convert: degrees = radians × 180/π

8. Step-by-Step Examples

Example 1: arccos(√3/2)

Step 1: Recognize that cos(30°) = √3/2.
Step 2: Since √3/2 is positive and within [−1, 1],
        arccos(√3/2) = 30°.

Answer: 30° or π/6 radians

Example 2: arccos(−1/2)

Step 1: Recognize that cos(120°) = −1/2.
Step 2: Since −1/2 is negative, the angle is between 90° and 180°.
        arccos(−1/2) = 120°.

Answer: 120° or 2π/3 radians

9. Relationship with arcsin

arccos and arcsin are complementary functions:

arccos(x) + arcsin(x) = π/2  (90°)

For example: arccos(0.5) = 60°, arcsin(0.5) = 30°, and 60° + 30° = 90°.

10. Common Mistakes

  • Input outside domain: arccos(2) is undefined. Only values between −1 and 1 are valid.
  • Confusing arccos with 1/cos: arccos(x) ≠ 1/cos(x). The notation cos⁻¹(x) means inverse function, not reciprocal.
  • Forgetting the range restriction: arccos only returns values between 0° and 180°.
  • Using wrong unit: Always check whether your calculator is in degree or radian mode.
  • Ignoring negative values: arccos(−x) = 180° − arccos(x).

11. Real-World Applications

  • Physics: Finding angles from known cosine ratios (dot products, work calculations).
  • Engineering: Calculating angles in trusses, ramps, and mechanical linkages.
  • Navigation: Determining bearings from distance measurements.
  • Computer graphics: Calculating angles between vectors.
  • Surveying: Finding angles from measured distances.
  • Robotics: Inverse kinematics for joint angle calculations.

12. Practice Problems

Problem 1: Find arccos(1). → 0° (0)

Problem 2: Find arccos(0). → 90° (π/2)

Problem 3: In a right triangle, the adjacent side is 4 and the hypotenuse is 5. Find the angle. → arccos(4/5) ≈ 36.87°

Problem 4: Find arccos(−√2/2). → 135° (3π/4)

13. Tips for Mastery

  • Memorize the common arccos values (0, ±1/2, ±√2/2, ±√3/2, ±1).
  • Remember the domain: −1 to 1. Anything else is invalid.
  • Remember the range: 0° to 180°.
  • Use the relationship arccos(x) + arcsin(x) = 90° to check your work.
  • Remember that arccos(−x) = 180° − arccos(x).

14. Final Thoughts

The inverse cosine function is essential in trigonometry and geometry. Understanding its domain, range, and relationship to the cosine function is crucial for solving triangles and real-world problems. Use the calculator above to check your work and practice until arccos becomes second nature.

15. Frequently Asked Questions

The domain is −1 ≤ x ≤ 1. Values outside this range are undefined.

The range is 0 to π (0° to 180°).

No! arccos(x) is the inverse function. 1/cos(x) is the reciprocal (secant).

arccos(0.5) = 60° or π/3 radians.

Because the arccos function is symmetric about the point (0, π/2).

arccos(x) + arcsin(x) = π/2 (90°). They are complementary.

No. The principal value is unique and always between 0° and 180°.

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Conclusion

The inverse cosine function is a cornerstone of trigonometry. Master its domain, range, and applications, and you'll be well-equipped for advanced mathematical studies.

Readers who wish to continue practicing can explore the companion learning resource for guided exercises.

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