How to Calculate the Inverse Tangent of a Number (tan⁻¹x)
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!
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FAQs on Inverse Tangent (Arctan, tan⁻¹x) – Meaning, Formula & Uses
1. What is the inverse tangent of x?
The inverse tangent of x, written as tan-1x or arctan(x), represents the angle whose tangent is x. It's the inverse function of the tangent function, providing the angle when you know the ratio of the opposite side to the adjacent side in a right-angled triangle.
2. How do you find tan-1x on a calculator?
Most calculators have a dedicated tan-1 or arctan button. You usually need to press a 2nd or shift function key first before selecting this button. Then, input the value of 'x' and press the equals button. The result will be the angle in either degrees or radians, depending on your calculator's mode setting.
3. What is the tan inverse of 1?
tan-1(1) = 45° or π/4 radians. This is because the tangent of 45° (or π/4 radians) is equal to 1.
4. Is tan-1x the same as 1/tan(x)?
No, they are different. tan-1x (or arctan x) represents the inverse tangent function—finding the angle. 1/tan(x) is the reciprocal of the tangent function, which is equivalent to cot(x) (cotangent).
5. What are the domain and range of arctan?
The domain of arctan(x) is all real numbers (-∞, ∞). The range is typically restricted to (-π/2, π/2) or (-90°, 90°) to ensure a unique output for each input.
6. What is the derivative of tan-1x?
The derivative of tan-1x with respect to x is 1/(1 + x²).
7. How is tan-1x derived from right triangles?
In a right-angled triangle, if the ratio of the opposite side to the adjacent side is 'x', then the angle θ is given by θ = tan-1(x).
8. Why can't tan-1x give angles beyond 90°?
The arctan function is defined to have a principal value range of (-90°, 90°) to provide a unique solution for every input. Angles outside this range can be found by adding or subtracting multiples of 180°.
9. Why do calculators sometimes give decimal answers instead of degrees?
Calculators often default to radians. Make sure your calculator is set to degree mode (DEG) to obtain answers in degrees.
10. Where is the inverse tangent function used in real life?
Inverse tangent finds applications in various fields, including: Calculating angles in engineering and physics problems (e.g., determining the angle of inclination or gradient), finding coordinate angles in navigation and surveying, and solving trigonometric equations.
11. What are some common values of arctan(x)?
Here are some common values to remember: arctan(0) = 0°, arctan(1) = 45°, arctan(-1) = -45°. You can use a calculator for other values.
12. What is the difference between tan x and arctan x?
tan x calculates the tangent of an angle x (ratio of opposite/adjacent). arctan x (or tan-1x) finds the angle whose tangent is x. They are inverse functions of each other.
