The Range of arctan

Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 6 of 52.

Every number has an arctan. Between $\frac{-\pi}{2}$ and $\frac{\pi}{2}$ , the tangent takes every value exactly once. It goes from large negative numbers to large positive ones. So these angles are arctan's range, in radians.

$The tangent graph between minus pi and pi, with breaks at plus and minus pi over 2$

A negative input gives a negative angle: $\arctan\left(-1\right) = \frac{-\pi}{4}$ .

Challenge

Work out $x = \arctan\left(-\sqrt{3}\right)$ in radians, as a multiple of π. Write it as the next line.

Then press Check.

Try it yourself

x = \arctan\left(-\sqrt{3}\right)

Get $x$ alone

All lessons in Trig Equations and Inverses

1Your First Trig Equations

One Equation, Two Angles Reading Signs on the Circle Recap: First Equations

4Compositions

Draw the Triangle The Outer Function A Negative Inside When Undoing Does Not Undo Recap: Compositions

7General Solutions

Every Solution Tangent's Single Family Families in Radians Families, Rounded Recap: Every Solution

10Identities First

Write sin 2x Out Pick the Right cos 2x Swap cos²x for sin²x sec, csc and cot Recap: Identities First

2Inverses and Their Ranges

Why arcsin Gives One Angle The Range of arccosThe Range of arctanRecap: Inverse Ranges

5Equations on a Domain

A Negative Cosine Tangent Equations Equations in Radians Two Turns, Four Answers Recap: On a Domain

8Multiple Angles

Two Turns for 2x Solving sin 2x Half Angles Every Solution of sin 3x Recap: Multiple Angles

11Final Challenges

Final: The Bicycle Pedal Final: The Big Wheel Final: Every Solution Final: An Identity and More Final: An Inverse Inside

3Evaluating Inverse Trig

Rounded Angles in Radians An Angle From a Slope Recap: Inverse Values

6Off the Special Angles

Both Answers, Rounded A Negative Sine, Rounded Cosine, Rounded Tangent, Rounded Rounded, in Radians Recap: Off the Table

9Factoring Trig Equations

A Common Factor Never Divide by sin x A Quadratic in Sine A Case With No Angle Square Roots Give Two Signs Recap: Factoring