Graph of Tangent
Tangent repeats every π and has vertical asymptotes.
About this topic
The tangent graph, y = tan x, has period π and vertical asymptotes wherever cos x = 0, at x = π/2 + nπ. Between asymptotes it rises from −∞ to +∞ through the origin, so its range is all real numbers.
It is odd and has no amplitude. Tangent is the slope of the terminal ray of an angle, or sin x divided by cos x, so it blows up whenever the denominator is zero and equals zero whenever sine is zero. Its zeros are at multiples of π, and it equals 1 at π/4 and −1 at −π/4.
Example: at 89° the tangent is about 57, and at 89.9° it is about 573, showing the rapid growth toward the asymptote at 90°. A hill that rises 1 metre per metre horizontally has slope 1, an angle of 45°; as the slope grows without limit the angle approaches 90°.
Tangent graphs appear in modelling slopes, camera angles and rotating beacons shining on a wall. Students often draw the curve crossing the asymptotes; it never does. Also remember that tan has period π, not 2π, so the pattern repeats twice as fast as sine and cosine.
Key ideas
- tan x = sin x / cos x
- Asymptotes at π/2 + kπ
- Range is all real numbers
Quick check
Use cases: where Graph of Tangent helps
- Homework: review the key idea behind Graph of Tangent before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Graph of Tangent without notes.
- Visual learning: play with the interactive lab to see how Graph of Tangent changes when you adjust its inputs.
- Teaching a friend: explain Graph of Tangent 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 Graph of Tangent?
Tangent repeats every π and has vertical asymptotes.
What will I learn in Graph of Tangent?
tan x = sin x / cos x. Asymptotes at π/2 + kπ. Range is all real numbers.
Is Graph of Tangent 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 Graph of Tangent 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 Graph of Tangent?
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.