Inverse Trig Functions
arcsin, arccos and arctan return angles from ratios.
About this topic
Inverse trig functions find an angle from a ratio: sin⁻¹, cos⁻¹ and tan⁻¹ (also arcsin, arccos, arctan). Because the original functions repeat, their domains are restricted to make them one-to-one, so arcsin gives angles from −π/2 to π/2 and arccos from 0 to π. Arctan gives angles strictly between −π/2 and π/2.
Each answer is a single principal value, so sin⁻¹(1/2) = 30° even though 150° also has sine 1/2. The inputs for arcsin and arccos must lie between −1 and 1, while arctan accepts any real number. The notation sin⁻¹ means inverse function, not reciprocal; the reciprocal is 1/sin x or csc x.
Example: a ramp rises 1 m over a horizontal run of 12 m, so its angle is tan⁻¹(1/12), about 4.8°. Inverse functions are used by calculators to get angles, by pilots to find headings and by programmers to find the direction of a vector.
Tips: check that the answer lies in the correct range, and remember composite results such as sin(sin⁻¹ x) = x only when x is in the allowed domain. Sketch a right triangle to evaluate expressions like cos(sin⁻¹(3/5)), which equals 4/5.
Key ideas
- Domains are restricted to be functions
- arcsin(1/2) = 30°
- Used to solve triangles
Quick check
Use cases: where Inverse Trig Functions helps
- Homework: review the key idea behind Inverse Trig Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Inverse Trig Functions without notes.
- Visual learning: play with the interactive lab to see how Inverse Trig Functions changes when you adjust its inputs.
- Teaching a friend: explain Inverse Trig Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Inverse Trig Functions?
arcsin, arccos and arctan return angles from ratios.
What will I learn in Inverse Trig Functions?
Domains are restricted to be functions. arcsin(1/2) = 30°. Used to solve triangles.
Is Inverse Trig Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Inverse Trig Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Inverse Trig Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.