$Tan\left(\theta\right)$Tan(θ)
The tangent function can be defined as follows:
- The tangent of the angle can be geometrically defined to be $y$y-coordinate of point $Q$Q, where $Q$Q is the intersection of the extension of the line $OP$OP and the tangent of the circle at $\left(1,0\right)$(1,0)
- Using similar triangles we can also define this algebraically as the ratio: $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ
- This also represents the slope of the line that forms the angle $\theta$θ to the positive $x$x-axis
The graph of $\tan\theta$tanθ
Key features of the graph of $y=\tan\left(\theta\right)$y=tan(θ) are:
- Asymptotes: vertical asymptotes appear at $\theta=\frac{\pi}{2}+n\pi$θ=π2+nπ, for $n$n any integer
- Axis intercepts: vertical axis intercept: $\left(0,0\right)$(0,0), horizontal axis intercepts at: $\theta=n\pi$θ=nπ, for $n$n any integer - at each intercept there is a point of inflection
- Period: The period of this graph can be found as the distance between two successive asymptotes. Hence, the period is: $\pi$π
- Range: This graph is unbounded. Hence, the range is: $\left(-\infty,\infty\right)$(−∞,∞)
- Domain: The function is undefined at each vertical asymptote. Hence, the domain is: $\lbrace\theta:\theta\in\Re,\theta\ne n\pi\rbrace${θ:θ∈ℜ,θ≠nπ}, where $n$n is any integer
- Key points: useful for graphing, the function passes through $\left(\frac{\pi}{4},1\right)$(π4,1) and $\left(-\frac{\pi}{4},-1\right)$(−π4,−1)
- Symmetry: The tangent graph is an odd function, that is $\tan\left(-\theta\right)=-\tan\theta$tan(−θ)=−tanθ. This also means the graph has point symmetry about the origin (or any point of inflection) by $180^\circ$180°
The asymptotes of the function are where the angle $\theta$θ would cause the line $OP$OP to be vertical and hence the slope is undefined. We can also see this through the definition $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ. The function is undefined where $\cos\theta=0$cosθ=0 and the graph approaches the vertical lines at these values forming asymptotes.
The fact that the tangent function repeats at intervals of $\pi$π can be verified by considering the unit circle diagram. Either by considering the slope of the line $OP$OP and how the slope will be the same for a point at an angle of $\theta$θ or at at angle of $\theta+\pi$θ+π. Or we can consider this algebraically by looking at the ratio $\frac{\sin\theta}{\cos\theta}$sinθcosθ. If $\pi$π is added to an angle $\theta$θ, then the diagram below shows that $\sin(\theta+\pi)$sin(θ+π) has the same magnitude as $\sin\theta$sinθ but opposite sign. The same relation holds between $\cos(\theta+\pi)$cos(θ+π) and $\cos\theta$cosθ.
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We make use of the definition: $$
| $\tan(\theta+\pi)$tan(θ+π) | $=$= | $\frac{\sin(\theta+\pi)}{\cos(\theta+\pi)}$sin(θ+π)cos(θ+π) |
| $=$= | $\frac{-\sin\theta}{-\cos\theta}$−sinθ−cosθ | |
| $=$= | $\tan\theta$tanθ |
Practice question
question 1
Consider the graph of $y=\tan x$y=tanx for $-2\pi\le x\le2\pi$−2π≤x≤2π.
How would you describe the graph?
Periodic
ADecreasing
BEven
CLinear
DWhich of the following is not appropriate to refer to in regard to the graph of $y=\tan x$y=tanx?
Amplitude
ARange
BPeriod
CAsymptotes
DThe period of a periodic function is the length of $x$x-values that it takes to complete one full cycle.
Determine the period of $y=\tan x$y=tanx in radians.
State the range of $y=\tan x$y=tanx.
$-\infty
A$y>0$y>0
B$\frac{-\pi}{2}
C$-\pi
DAs $x$x increases, what would be the next asymptote of the graph after $x=\frac{7\pi}{2}$x=7π2?