Inverse Tan Formula Domain Range and Solved Examples
The concept of inverse tangent (also called arctan or tan–1x) plays a key role in mathematics, especially in trigonometry, and is widely useful in exam situations, science, engineering, and logical reasoning.
What Is Inverse Tangent (Arctan, Tan–1x)?
The inverse tangent is a special trigonometric function that helps you find the angle whose tangent value is a given number. In other words, if tan θ = x, then θ = tan–1x or arctan(x). This function is important in maths, science, engineering, and is also used in calculating angles in navigation, computer graphics, and even mobile apps. The terms arctan, tan inverse, and tan–1x all mean the same thing.
Key Formula for Inverse Tangent
Here’s the standard formula for inverse tangent:
If tan θ = x,
then θ = tan–1x = arctan(x)
The domain is all real x (–∞, ∞), and the range is (–π/2, π/2) or (–90°, 90°).
| x | tan–1x (Degrees) | tan–1x (Radians) |
|---|---|---|
| 0 | 0° | 0 |
| 1 | 45° | π/4 |
| –1 | –45° | –π/4 |
| ∞ | 90° | π/2 |
Cross-Disciplinary Usage
Inverse tangent is not only essential in mathematics, but it is also used in physics calculations (like resolving vectors and gradients), computer science (rotation in graphics), geography (navigation and bearings), and engineering. JEE, NEET, and various board exams ask direct and application-based questions using tan inverse.
Step-by-Step Illustration
- Suppose tan θ = 1. Find θ.
tan θ = 1
θ = tan–11
θ = 45° (or π/4 radians)
- Suppose tan θ = 0
θ = tan–10
θ = 0°
- Suppose tan θ = –1
θ = tan–1(–1)
θ = –45° (or –π/4 radians)
Speed Trick or Vedic Shortcut
To remember tan–1x for specific exam angles, use this trick: tan–11 = 45°, tan–11/√3 = 30°, tan–1√3 = 60°. Most calculators have a SHIFT or 2nd function key to access tan–1. Try it now for quick MCQ answers, just like in Vedantu classes for entrance exams!
Try These Yourself
- Find tan–1(1/√3) in degrees.
- If tan θ = 4, what is θ to two decimal places?
- Is tan–1(0) the same as 0?
- Find the domain and range of y = arctan(x).
Frequent Errors and Misunderstandings
- Mistaking tan–1x for 1/tan(x). (They are completely different!)
- Forgetting that the principal range for tan inverse is only (–90°, 90°).
- Mixing up degrees and radians on the calculator, leading to wrong answers.
- Using the ‘arctan’ function incorrectly on calculators (always check mode and range!).
Relation to Other Concepts
The idea of inverse tangent connects closely with inverse trigonometric functions in general, including arcsin and arccos. It’s also essential when solving trigonometric equations and using the trigonometry table for quick value lookups. Mastering this helps you with more advanced topics such as calculus and coordinate geometry.
Classroom Tip
A quick way to remember inverse tangent values: use the triangle sides. For tan–1x, draw a right triangle with the opposite side as x, adjacent as 1, and the hypotenuse as √(1+x²). It helps visualize and solve many trigonometry problems confidently. Vedantu teachers use such tricks in their live math sessions!
We explored inverse tangent (tan–1x or arctan) — from its meaning and formula to fast calculation and mistake avoidance. With regular practice and the right shortcuts, you can solve all types of inverse tangent questions for school, boards, and competitions. Keep learning with Vedantu for more such easy explanations and confidence in maths!
Related Internal Links
FAQs on Inverse Tan Function Explained Clearly
1. What is inverse tan?
The inverse tan or arctan function gives the angle whose tangent is a given number. It is written as tan⁻¹(x) or arctan(x).
- If tan θ = x, then θ = tan⁻¹(x).
- Its principal value lies between -π/2 and π/2 (or -90° and 90°).
- It is the inverse of the tangent function when the domain of tan is restricted.
2. What is the formula for inverse tan?
The basic formula for inverse tan is y = tan⁻¹(x), which means tan(y) = x for -π/2 < y < π/2.
- Domain of tan⁻¹(x): all real numbers.
- Range: (-π/2, π/2).
- In degrees: -90° < y < 90°.
3. What is the derivative of inverse tan?
The derivative of tan⁻¹(x) is 1 / (1 + x²). In calculus, this is written as:
- d/dx [tan⁻¹(x)] = 1 / (1 + x²)
4. What is the integral of inverse tan?
The integral of tan⁻¹(x) is x tan⁻¹(x) − (1/2) ln(1 + x²) + C. Using integration by parts:
- ∫ tan⁻¹(x) dx = x tan⁻¹(x) − (1/2) ln(1 + x²) + C
5. What is the domain and range of inverse tan?
The domain of tan⁻¹(x) is all real numbers, and its range is (-π/2, π/2).
- Domain: (−∞, ∞)
- Range: −π/2 < y < π/2
- In degrees: −90° < y < 90°
6. How do you evaluate tan⁻¹(1)?
The value of tan⁻¹(1) is π/4 or 45°. This is because:
- tan(π/4) = 1
- π/4 lies within the principal range (-π/2, π/2)
7. What is the difference between tan⁻¹(x) and 1/tan(x)?
The expression tan⁻¹(x) means inverse tangent, while 1/tan(x) means cotangent.
- tan⁻¹(x) gives an angle whose tangent is x.
- 1/tan(x) = cot(x), which is a trigonometric ratio.
8. How do you solve equations involving inverse tan?
To solve equations involving tan⁻¹(x), first isolate the inverse tan term and then apply tangent to both sides. Example:
- If tan⁻¹(x) = π/6
- Apply tan to both sides: x = tan(π/6)
- Since tan(π/6) = 1/√3, the solution is x = 1/√3
9. What is the graph of inverse tan?
The graph of tan⁻¹(x) is an increasing S-shaped curve with horizontal asymptotes at y = π/2 and y = -π/2.
- It passes through the point (0, 0).
- Domain: all real numbers.
- Range: (-π/2, π/2).
10. What are the properties of inverse tan?
The main properties of inverse tan include its domain, range, and odd function behavior.
- tan⁻¹(-x) = -tan⁻¹(x) (odd function).
- Domain: all real numbers.
- Range: (-π/2, π/2).
- Derivative: 1 / (1 + x²).
