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Where Tangent Is Undefined

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 43 of 57.

Tangent divides by the across. At 90° and 270° the point is on the y-axis, the across is 0, and there is nothing to divide by: tan 90° and tan 270° are undefined. A vertical radius has no slope.

$Opposite points P at 45° and Q at 225° on one line through the centre$

P and Q lie on one line through the centre: one slope.

Tangent also repeats faster than sine and cosine. The points at θ and θ + 180° are opposite ends of one diameter, and a line has one slope: tan 45° = tan 225° = 1. So the angles with a given tangent come in pairs half a turn apart, not a full turn.

Challenge

Tap every angle in one turn with tan(θ) = √(3)/3, then press Check.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Angles and the Unit Circle

1A Point on a Circle

Height on the Circle Across: the Cosine Same Height, Two Angles Recap: Height and Across

4Radians

What a Radian Is Degrees to Radians Radians to Degrees Radians Without π Turns in Radians Recap: Radians

7Reference Angles and Signs

Reference Angles Reference in III and IV Signs by Quadrant Sign and Size Signs at Any Angle Recap: Reference Angles

10Six Functions, Any Circle

Any Circle, Any Radius From a Point to the Angle Secant, Cosecant, Cotangent Reciprocals in Radians Recap: Six Functions

2Standard Position

Angles in Standard Position Negative Angles Quadrants and Axes Recap: Placing Angles

5Arc Length and Sector Area

Arc Length Arcs From Degrees Finding the Angle Sector Area Recap: A Slice of Pizza

8All Four Quadrants

Points in Quadrant II Quadrants III and IV Every Angle for a Value Points in Radians Exact Values in Radians Recap: Four Quadrants

11Final Challenges

The Minute Hand The Ferris Wheel Seat Four Angles, Two Turns The Garden Sprinkler A Negative Radian Point Which Angle?

3Coterminal Angles

Coterminal Angles Back Into One Turn Values Repeat Every Turn Every Turn at Once Recap: Coterminal Angles

6The Unit Circle

Why (cos θ, sin θ)The 30° and 60° Points The 45° Point Points Off the Table Recap: Quadrant I Points

9Tangent on the Circle

Tangent Is y over xWhere Tangent Is UndefinedTangent in Radians Every Angle for a Tangent Recap: Tangent