Why does tan(x) have vertical asymptotes at odd multiples of π/2?
As θ approaches an odd multiple of π/2 the unit circle x-coordinate approaches zero while y is ±1, so tan(θ)=y/x becomes unbounded or undefined, producing vertical asymptotes.
Video Summary
Graphing the Tangent Function
Main takeaways
01
Tangent can be defined as the length of the tangent segment on the unit circle or as y/x for the point (x,y) on the unit circle.
02
Tangent values blow up where the circle's x-coordinate is zero, producing vertical asymptotes at odd multiples of π/2.
03
Typical values: tan(0)=0, tan(30°)≈0.577, tan(60°)≈1.732; tan(90°) is undefined.
04
Range of tan(x) is all real numbers; domain excludes x = (2n+1)π/2.
05
tan(x) is an odd function (symmetric about the origin) with period π and no amplitude.
Key moments
Questions answered
How can tangent be defined two different ways and why are they equal?
Tangent can be defined as the length of the tangent segment to the unit circle or as the ratio y/x for the point (x,y) on the unit circle. Similar triangles on the unit circle show these definitions produce the same value.
What is the domain and range of y = tan(x)?
The range is all real numbers. The domain is all real x except odd multiples of π/2, which can be written as x ≠ (2n+1)π/2 for integer n.
What is the period and symmetry of the tangent function?
tan(x) has period π and is an odd function (symmetric with respect to the origin), so the graph repeats every π and tan(-x) = -tan(x).
What sample values help sketch one cycle of tan(x)?
Key points include tan(0)=0, tan(30°)≈0.577, tan(60°)≈1.732, and undefined at 90°. Plotting these plus their negatives outlines one π-length cycle between vertical asymptotes.
Graphing the Tangent Function 00:05
"The goals of this video are to graph the tangent function on the coordinate plane using the unit circle."
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This video focuses on graphing the tangent function, its domain, and range.
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The tangent function is derived from the tangent line that intersects the unit circle at a single point, perpendicular to the radius extending from the origin.
Definition of Tangent Function 00:41
"The tangent function is defined as the length of the tangent segment."
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The tangent function can be defined through two perspectives: the length of the tangent segment drawn perpendicular to the x-axis at angle θ, or through the ratio of the coordinates (y/x) of the point where the terminal side of angle θ intersects the unit circle.
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The relationship between these two definitions can be demonstrated using similar triangles formed within the unit circle, confirming that tangent θ is the ratio of the lengths of sides in these triangles.
Behavior of the Tangent Function 02:38
"The graph of this function is a little more complicated than the graph of sine theta and cosine theta."
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The tangent function's graph differs from sine and cosine because it is represented as the ratio of y to x coordinates.
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As the angle θ approaches π/2 radians (90 degrees), the x-coordinate trends toward zero, causing the tangent value to become undefined, resulting in vertical asymptotes at those points.
Graphing Tangent Function by Hand 04:25
"I've already taken the time to complete this table for various angles and the value of tangent data."
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The presenter graphs the tangent function by first filling out a table of tangent values for selected angles, highlighting the function's behavior.
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Key angles such as 0 degrees (tangent is 0), 30 degrees (approximately 0.58), and 60 degrees (approximately 1.7) are noted, with special emphasis on the undefined value at 90 degrees due to the presence of a vertical asymptote.
Domain and Range of Tangent Function 06:48
"The range is all reals, but the domain is all x's such that x does not equal some odd multiple of π/2."
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The range of the tangent function encompasses all real numbers, indicating that the function can produce infinitely large values in both positive and negative directions.
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The domain excludes x-values that are odd multiples of π/2 (like ±π/2), which correspond to the vertical asymptotes present in the graph.
Characteristics of the Tangent Graph 07:54
"The graph is symmetrical with respect to the origin, and therefore is an odd function."
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The tangent function does not have an amplitude due to the absence of maximum or minimum values, signifying that its graph continues infinitely.
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The period of the tangent function is π radians, indicating that the graph repeats itself at intervals of π.