tan-1
High School
Definition
Tangent's inverse function. It can help you calculate angles when certain information is given to you. When you know the sides of a right triangle, but not the angle, inverse tangent comes into use. The inverse tangent performs the opposite of the tangent function.
Worked examples
\(\tan^{-1}\left(\frac{5}{12}\right) \approx 22.6^\circ\)
Given opposite = 5 and adjacent = 12, inverse tangent returns the angle in the right triangle.
\(\tan(30^\circ) = \frac{1}{\sqrt{3}} \)→\( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = 30^\circ\)
Inverse tangent undoes tangent: it takes the ratio and gives back the angle.
Common mistakes
- \(\tan^{-1}(x) = \frac{1}{\tan(x)}\) → \(\tan^{-1}(x)\) is the inverse function, not \(\frac{1}{\tan(x)}\) (which is \(\cot(x)\)) The -1 exponent means inverse function, not reciprocal.
- \(\tan^{-1}\left(\frac{\)adjacent\(}{\)opposite\(}\right)\) → \(\tan^{-1}\left(\frac{\)opposite\(}{\)adjacent\(}\right)\) Tangent is opposite over adjacent, so inverse tangent takes the same ratio.
- \(\tan^{-1}(1.5)\) has no solution because tangent only works for angles less than 90° → \(\tan^{-1}(1.5) \approx 56.3^\circ\) Inverse tangent accepts any real number and returns an angle, typically between -90° and 90°.
Where you'll use it next
You'll use inverse tangent to solve trigonometric equations, find angles in vector problems, and work with polar coordinates in precalculus and calculus. It also appears in physics when resolving forces and analyzing projectile motion.
Found in 1 StudyPug lesson
Mastering Inverse Trigonometric Functions: Essential Techniques
11th Grade11thEssential Maths
Unlock the power of inverse trigonometric functions with our comprehensive guide. Learn evaluation techniques, domain restrictions, and real-world applications to excel in advanced mathematics.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026