Model real-world periodic phenomena with trigonometric functions. Grade 12 Math 30-1.
Lesson 4.8 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.
The tangent function is defined using sine and cosine:
θ = ( θ)/( θ)
This means tangent represents the ratio of sine to cosine.
Since we're dividing by cosine, we cannot have θ = 0. Remember: division by zero is not allowed in mathematics - you cannot divide any number by 0. Therefore, θ cannot be (π)/(2), (3π)/(2), (5π)/(2), ... or any value where cosine equals zero.
Using the unit circle:
Tangent values become very large near (π)/(2) and -(π)/(2), which creates vertical asymptotes.
The graph of y = x has several important features:
It passes through the origin It increases continuously between asymptotes It repeats its pattern indefinitely
Unlike sine and cosine, the tangent graph is not bounded.
An asymptote is a line that a graph approaches but never touches. The function values become infinitely large (positive or negative) as they approach the asymptote.
Tangent is undefined whenever:
x = 0
This occurs at:
x = (π)/(2) + π n
where n Z
These values create vertical asymptotes.
In this graph, the red dotted lines represent the vertical asymptotes. Notice how the tangent curve approaches these lines but never crosses them, and the function values grow infinitely large near the asymptotes.
Domain. The domain of y = x is all real numbers except where asymptotes occur:
x ≠ (π)/(2) + π n
Range. The range of the tangent function is:
(-∞, ∞)
Tangent can produce any real value.
The graph of y = x repeats every:
π
So the period is π, which is different from sine and cosine.
A transformed tangent function can be written as:
y = a(b(x - c)) + d
Where:.
Period = (π)/(|b|)
Example:. y = (2x)
Period = (π)/(2)
The graph repeats twice as fast.
Determine the exact value of k if the point P((21π)/(4), k) lies on the graph of y = x.
Step 1: Reduce the angle.. (21π)/(4) = 5π + (π)/(4)
Since tangent has period π, this is equivalent to:
((π)/(4))
Step 2: Evaluate.. ((π)/(4)) = 1
So k = 1
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