Trigonometric Functions: Tangent

Flashcards covering key vocabulary and concepts related to the tangent trigonometric function, its properties, values, and transformations.

Tangent (tan
theta)

The third trigonometric function, defined as the ratio of the y-coordinate to the x-coordinate (y/x) on the unit circle.

Tangent as Sine/Cosine

Tangent of theta can also be expressed as sine theta divided by cosine theta (sin
theta / cos
theta).

Tangent and Slope

Geometrically, the tangent of an angle represents the slope of the line segment from the origin to the point on the unit circle corresponding to that angle.

Tangent of 5pi/4

Equals 1, because 5pi/4 is in the third quadrant where both sine and cosine are negative and equal in magnitude, resulting in a positive slope of 1.

Tangent at pi/2 (and 3pi/2)

Is undefined, as the x-coordinate (cosine) is zero at these angles, which leads to division by zero. The slope approaches positive and negative infinity from different sides.

Tangent of 11pi/6

Equals -1/sqrt(3) (or -sqrt(3)/3), as 11pi/6 is in the fourth quadrant where sine is negative and cosine is positive, resulting in a negative slope.

Tangent Zeros

Occur at all integer multiples of pi (n*pi), where the y-coordinate (sine) is zero.

Tangent equals 1

Occurs at angles such as pi/4 and 5pi/4, where the sine and cosine values are equal in magnitude and sign, yielding a slope of 1.

Period of Tangent Function

Pi (
), meaning the function's graph repeats its full cycle every
radians.

Amplitude of Tangent Function

Does not exist (DNE) because the function's range extends to positive and negative infinity, lacking defined maximum or minimum values.

Midline of Tangent Function

Officially does not exist due to the absence of a max and min, but it can be conceptually considered as y=0 for practical purposes.

Vertical Shift (Tangent)

A transformation that moves the entire graph of the tangent function up or down by a constant value (e.g., adding +10 shifts it up by 10 units).

Horizontal Compression (Tangent)

A transformation that squeezes the graph horizontally; a compression by a factor of 1/4 means multiplying the input variable by 4 inside the function (e.g., tan(4*theta)).

Vertical Stretch (Tangent)

A transformation that stretches the graph vertically, determined by how much the y-values are multiplied compared to the standard tangent function's output at key points (e.g., where tan(theta)=1).